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Hopf algebras and the duality to logarithmic CFT - How to make

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Hopf algebras and the duality to logarithmic CFT
How to make complicated things simple
AM Semikhatov
Theory Department
Lebedev Physics Institute
Supersymmetries & Quantum Symmetries
Dubna 2009
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Hopf algebras and the duality to logarithmic CFT
How to make some complicated things somewhat simpler
AM Semikhatov
Theory Department
Lebedev Physics Institute
Supersymmetries & Quantum Symmetries
Dubna 2009
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
“Nonunitary evolution” etH
Г№Г± applications to
models with disorder
systems with transient and recurrent states (sand-pile model:
Bak, Dhar, Ruelle, Priezzhev, . . . ),
percolation (R Langlands, P Pouliot, and Y Saint-Aubin), . . .
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
“Nonunitary evolution” etH
Г№Г± applications to
models with disorder
systems with transient and recurrent states (sand-pile model:
Bak, Dhar, Ruelle, Priezzhev, . . . ),
percolation (R Langlands, P Pouliot, and Y Saint-Aubin), . . .
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
“Nonunitary evolution” etH
Г№Г± applications to
models with disorder
systems with transient and recurrent states (sand-pile model:
Bak, Dhar, Ruelle, Priezzhev, . . . ),
percolation (R Langlands, P Pouliot, and Y Saint-Aubin), . . .
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
“Nonunitary evolution” etH
Г№Г± applications to
models with disorder
systems with transient and recurrent states (sand-pile model:
Bak, Dhar, Ruelle, Priezzhev, . . . ),
percolation (R Langlands, P Pouliot, and Y Saint-Aubin), . . .
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
“Nonunitary evolution” etH
Г№Г± applications to
models with disorder
systems with transient and recurrent states (sand-pile model:
Bak, Dhar, Ruelle, Priezzhev, . . . ),
percolation (R Langlands, P Pouliot, and Y Saint-Aubin), . . .
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
log: whence?
Let L0 вњ’ z
вќ‡
вќ‡z
have a size-2 Jordan cell:
zhвњ¶ в™Јz q вњЏ О”hв™Јz qВ g в™Јz q,
zg вњ¶ в™Јz q вњЏ О”g в™Јz q.
Solution:
вњ‚
О” 1
0 О”
вњЎ
g в™Јx q вњЏ B x О” ,
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
log: whence?
Let L0 вњ’ z
вќ‡
вќ‡z
have a size-2 Jordan cell:
zhвњ¶ в™Јz q вњЏ О”hв™Јz qВ g в™Јz q,
zg вњ¶ в™Јz q вњЏ О”g в™Јz q.
Solution:
вњ‚
О” 1
0 О”
вњЎ
g в™Јx q вњЏ B x О” ,
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
log: whence?
Let L0 вњ’ z
вќ‡
вќ‡z
have a size-2 Jordan cell:
zhвњ¶ в™Јz q вњЏ О”hв™Јz qВ g в™Јz q,
zg вњ¶ в™Јz q вњЏ О”g в™Јz q.
Solution:
вњ‚
О” 1
0 О”
вњЎ
g в™Јx q вњЏ B x О” ,
hв™Јx q вњЏ A x О” В B x О” logв™Јx q.
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Logarithmic вњ‘ nonsemisimple
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of a
Hamiltonian
log: whence?
Let L0 вњ’ z
вќ‡
вќ‡z
have a size-2 Jordan cell:
zhвњ¶ в™Јz q вњЏ О”hв™Јz qВ g в™Јz q,
zg вњ¶ в™Јz q вњЏ О”g в™Јz q.
Solution:
вњ‚
О” 1
0 О”
вњЎ
g в™Јx q вњЏ B x О” ,
hв™Јx q вњЏ A x О” В B x О” logв™Јx q.
“Logarithmic” = “Nonsemisimple”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Feigin, Gainutdinov, Semikhatov, Tipunin
1 Modular group representations and fusion in logarithmic conformal field theories
and in the quantum group center, Commun. Math. Phys. 265 (2006) 47–93
2 Kazhdan–Lusztig correspondence for the representation category of the triplet
W -algebra in logarithmic CFT, Theor. Math. Phys. 148 (2006) 1210–1235
3 Logarithmic extensions of minimal models: characters and modular
transformations, Nucl. Phys. B 757 (2006) 303–343
4 Kazhdan–Lusztig-dual quantum group for logarithmic extensions of Virasoro
minimal models, J. Math. Phys. 48 (2007) 032303
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Quantum Groups and the Duality
Quantum groups at roots of unity capture much of the LCFT
structure
even roots of unity: q 2p вњЏ 1.
ribbon
factorizable
hence with a modular group action
— correspondence between the representation categories: up to
an equivalence
— isomorphic fusion
— identical modular group representations
Other essential work:
J Fjelstad, J Fuchs, S Hwang, AM Semikhatov, and IYu Tipunin, Logarithmic
conformal field theories via logarithmic deformations, Nucl. Phys. B633 (2002)
379–413
J Fuchs, S Hwang, AM Semikhatov, and IYu Tipunin, Nonsemisimple fusion
algebras and the Verlinde formula, Commun. Math. Phys. 247 (2004) 713–742
AM Gainutdinov, A generalization of the Verlinde formula in logarithmic CFT,
Teor. Mat. Fiz. 159 (2009) 193–205.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Some previous works
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)
Pierce and Rasmussen (dense polymers)
M Jeng, G Piroux, P Ruelle (sand-pile model)
F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)
vertex-operator algebras with nonsemisimple representation categories:
YZ Huang, J Lepowsky, and L Zhang;
J Fuchs; M Miyamoto; A Milas;
M Flohr, N Carqueville
D AdamoviВґc and A Milas
YZ Huang
K Nagatomo and A Tsuchiya
V Schomerus and H Saleur (supergeometry Г№Г± logs)
M Gaberdiel and I Runkel (boundary logarithmic theories)
M Flohr and M Gaberdiel (torus amplitudes)
H Eberle and M Flohr (fusion)
N Read and H Saleur (statistical-mechanics construction)
P Mathieu and D Ridout (percolation)
PA Pearce, J Rasmussen, and P Ruelle; Rasmussen++
MR Gaberdiel, I Runkel, and S Wood (в™Ј2, 3q model)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Some previous works
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)
Pierce and Rasmussen (dense polymers)
M Jeng, G Piroux, P Ruelle (sand-pile model)
F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)
vertex-operator algebras with nonsemisimple representation categories:
YZ Huang, J Lepowsky, and L Zhang;
J Fuchs; M Miyamoto; A Milas;
M Flohr, N Carqueville
D AdamoviВґc and A Milas
YZ Huang
K Nagatomo and A Tsuchiya
V Schomerus and H Saleur (supergeometry Г№Г± logs)
M Gaberdiel and I Runkel (boundary logarithmic theories)
M Flohr and M Gaberdiel (torus amplitudes)
H Eberle and M Flohr (fusion)
N Read and H Saleur (statistical-mechanics construction)
P Mathieu and D Ridout (percolation)
PA Pearce, J Rasmussen, and P Ruelle; Rasmussen++
MR Gaberdiel, I Runkel, and S Wood (в™Ј2, 3q model)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Some previous works
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)
Pierce and Rasmussen (dense polymers)
M Jeng, G Piroux, P Ruelle (sand-pile model)
F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)
vertex-operator algebras with nonsemisimple representation categories:
YZ Huang, J Lepowsky, and L Zhang;
J Fuchs; M Miyamoto; A Milas;
M Flohr, N Carqueville
D AdamoviВґc and A Milas
YZ Huang
K Nagatomo and A Tsuchiya
V Schomerus and H Saleur (supergeometry Г№Г± logs)
M Gaberdiel and I Runkel (boundary logarithmic theories)
M Flohr and M Gaberdiel (torus amplitudes)
H Eberle and M Flohr (fusion)
N Read and H Saleur (statistical-mechanics construction)
P Mathieu and D Ridout (percolation)
PA Pearce, J Rasmussen, and P Ruelle; Rasmussen++
MR Gaberdiel, I Runkel, and S Wood (в™Ј2, 3q model)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Some previous works
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)
Pierce and Rasmussen (dense polymers)
M Jeng, G Piroux, P Ruelle (sand-pile model)
F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)
vertex-operator algebras with nonsemisimple representation categories:
YZ Huang, J Lepowsky, and L Zhang;
J Fuchs; M Miyamoto; A Milas;
M Flohr, N Carqueville
D AdamoviВґc and A Milas
YZ Huang
K Nagatomo and A Tsuchiya
V Schomerus and H Saleur (supergeometry Г№Г± logs)
M Gaberdiel and I Runkel (boundary logarithmic theories)
M Flohr and M Gaberdiel (torus amplitudes)
H Eberle and M Flohr (fusion)
N Read and H Saleur (statistical-mechanics construction)
P Mathieu and D Ridout (percolation)
PA Pearce, J Rasmussen, and P Ruelle; Rasmussen++
MR Gaberdiel, I Runkel, and S Wood (в™Ј2, 3q model)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Some previous works
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)
Pierce and Rasmussen (dense polymers)
M Jeng, G Piroux, P Ruelle (sand-pile model)
F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)
vertex-operator algebras with nonsemisimple representation categories:
YZ Huang, J Lepowsky, and L Zhang;
J Fuchs; M Miyamoto; A Milas;
M Flohr, N Carqueville
D AdamoviВґc and A Milas
YZ Huang
K Nagatomo and A Tsuchiya
V Schomerus and H Saleur (supergeometry Г№Г± logs)
M Gaberdiel and I Runkel (boundary logarithmic theories)
M Flohr and M Gaberdiel (torus amplitudes)
H Eberle and M Flohr (fusion)
N Read and H Saleur (statistical-mechanics construction)
P Mathieu and D Ridout (percolation)
PA Pearce, J Rasmussen, and P Ruelle; Rasmussen++
MR Gaberdiel, I Runkel, and S Wood (в™Ј2, 3q model)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Some previous works
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)
Pierce and Rasmussen (dense polymers)
M Jeng, G Piroux, P Ruelle (sand-pile model)
F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)
vertex-operator algebras with nonsemisimple representation categories:
YZ Huang, J Lepowsky, and L Zhang;
J Fuchs; M Miyamoto; A Milas;
M Flohr, N Carqueville
D AdamoviВґc and A Milas
YZ Huang
K Nagatomo and A Tsuchiya
V Schomerus and H Saleur (supergeometry Г№Г± logs)
M Gaberdiel and I Runkel (boundary logarithmic theories)
M Flohr and M Gaberdiel (torus amplitudes)
H Eberle and M Flohr (fusion)
N Read and H Saleur (statistical-mechanics construction)
P Mathieu and D Ridout (percolation)
PA Pearce, J Rasmussen, and P Ruelle; Rasmussen++
MR Gaberdiel, I Runkel, and S Wood (в™Ј2, 3q model)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Some previous works
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)
Pierce and Rasmussen (dense polymers)
M Jeng, G Piroux, P Ruelle (sand-pile model)
F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)
vertex-operator algebras with nonsemisimple representation categories:
YZ Huang, J Lepowsky, and L Zhang;
J Fuchs; M Miyamoto; A Milas;
M Flohr, N Carqueville
D AdamoviВґc and A Milas
YZ Huang
K Nagatomo and A Tsuchiya
V Schomerus and H Saleur (supergeometry Г№Г± logs)
M Gaberdiel and I Runkel (boundary logarithmic theories)
M Flohr and M Gaberdiel (torus amplitudes)
H Eberle and M Flohr (fusion)
N Read and H Saleur (statistical-mechanics construction)
P Mathieu and D Ridout (percolation)
PA Pearce, J Rasmussen, and P Ruelle; Rasmussen++
MR Gaberdiel, I Runkel, and S Wood (в™Ј2, 3q model)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Last year:
D. AdamoviВґc and A. Milas, Lattice construction of logarithmic modules
for certain vertex algebras, arXiv:0902.3417
— Construction of projective modules
turned out to be as expected on the quantum group basis [FGST]
K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W в™Јpq
and the restricted quantum group at root of unity, arXiv:0902.4607
— Proof of the FGST equivalence conjecture by extending the
“log-deformation” construction of [FFHST]
Y.-Z. Huang, Generalized twisted modules associated to general
automorphisms of a vertex operator algebra arXiv:0905.0514
H. Kondo and Y. Saito, Indecomposable decomposition of tensor
products of modules over the restricted quantum universal enveloping
algebra associated to рќ–�рќ–‘2 , arXiv:0901.4221 [math.QA]
— a nonbraided category
M.R. Gaberdiel, I. Runkel, and S. Wood, Fusion rules and boundary
conditions in the c вњЏ 0 triplet model, arXiv:0905.0916
— Nonrigid category for the W2,3 algebra of FGST3.
— Making it rigid ùñ the FGST3 “QG” fusion
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Last year:
D. AdamoviВґc and A. Milas, Lattice construction of logarithmic modules
for certain vertex algebras, arXiv:0902.3417
— Construction of projective modules
turned out to be as expected on the quantum group basis [FGST]
K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W в™Јpq
and the restricted quantum group at root of unity, arXiv:0902.4607
— Proof of the FGST equivalence conjecture by extending the
“log-deformation” construction of [FFHST]
Y.-Z. Huang, Generalized twisted modules associated to general
automorphisms of a vertex operator algebra arXiv:0905.0514
H. Kondo and Y. Saito, Indecomposable decomposition of tensor
products of modules over the restricted quantum universal enveloping
Moral:
algebra associated to рќ–�рќ–‘2 , arXiv:0901.4221 [math.QA]
1 “Engineering”
equivalence
math. proof of the equivalence
— a nonbraidedthe
category
2 Further
predictions
for Logarithmic
CFT
M.R. Gaberdiel,
I. Runkel,
and S. Wood,
Fusion rules and boundary
(Bushlanov,
Tipunin
’09)
conditions inFeigin,
the c вњЏGainutdinov,
0 triplet model,
arXiv:0905.0916
—
Nonrigid
category
for
the
W
algebra
ofgroup
FGST3.
2,3 quantum
3 Motivation for some particular
structures
— Making it rigid ùñ the FGST3 “QG” fusion
Г№Г±
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Last year:
D. AdamoviВґc and A. Milas, Lattice construction of logarithmic modules
for certain vertex algebras, arXiv:0902.3417
— Construction of projective modules
turned out to be as expected on the quantum group basis [FGST]
K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W в™Јpq
and the restricted quantum group at root of unity, arXiv:0902.4607
— Proof of the FGST equivalence conjecture by extending the
“log-deformation” construction of [FFHST]
Y.-Z. Huang, Generalized twisted modules associated to general
automorphisms of a vertex operator algebra arXiv:0905.0514
H. Kondo and Y. Saito, Indecomposable decomposition of tensor
products of modules over the restricted quantum universal enveloping
Moral:
algebra associated to рќ–�рќ–‘2 , arXiv:0901.4221 [math.QA]
1 “Engineering”
equivalence
math. proof of the equivalence
— a nonbraidedthe
category
2 Further
predictions
for Logarithmic
CFT
M.R. Gaberdiel,
I. Runkel,
and S. Wood,
Fusion rules and boundary
(Bushlanov,
Tipunin
’09)
conditions inFeigin,
the c вњЏGainutdinov,
0 triplet model,
arXiv:0905.0916
—
Nonrigid
category
for
the
W
algebra
ofgroup
FGST3.
2,3 quantum
3 Motivation for some particular
structures
— Making it rigid ùñ the FGST3 “QG” fusion
Г№Г±
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Last year:
D. AdamoviВґc and A. Milas, Lattice construction of logarithmic modules
for certain vertex algebras, arXiv:0902.3417
— Construction of projective modules
turned out to be as expected on the quantum group basis [FGST]
K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W в™Јpq
and the restricted quantum group at root of unity, arXiv:0902.4607
— Proof of the FGST equivalence conjecture by extending the
“log-deformation” construction of [FFHST]
Y.-Z. Huang, Generalized twisted modules associated to general
automorphisms of a vertex operator algebra arXiv:0905.0514
H. Kondo and Y. Saito, Indecomposable decomposition of tensor
products of modules over the restricted quantum universal enveloping
Moral:
algebra associated to рќ–�рќ–‘2 , arXiv:0901.4221 [math.QA]
1 “Engineering”
equivalence
math. proof of the equivalence
— a nonbraidedthe
category
2 Further
predictions
for Logarithmic
CFT
M.R. Gaberdiel,
I. Runkel,
and S. Wood,
Fusion rules and boundary
(Bushlanov,
Tipunin
’09)
conditions inFeigin,
the c вњЏGainutdinov,
0 triplet model,
arXiv:0905.0916
—
Nonrigid
category
for
the
W
algebra
ofgroup
FGST3.
2,3 quantum
3 Motivation for some particular
structures
— Making it rigid ùñ the FGST3 “QG” fusion
Г№Г±
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Last year:
D. AdamoviВґc and A. Milas, Lattice construction of logarithmic modules
for certain vertex algebras, arXiv:0902.3417
— Construction of projective modules
turned out to be as expected on the quantum group basis [FGST]
K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W в™Јpq
and the restricted quantum group at root of unity, arXiv:0902.4607
— Proof of the FGST equivalence conjecture by extending the
“log-deformation” construction of [FFHST]
Y.-Z. Huang, Generalized twisted modules associated to general
automorphisms of a vertex operator algebra arXiv:0905.0514
H. Kondo and Y. Saito, Indecomposable decomposition of tensor
products of modules over the restricted quantum universal enveloping
Moral:
algebra associated to рќ–�рќ–‘2 , arXiv:0901.4221 [math.QA]
1 “Engineering”
equivalence
math. proof of the equivalence
— a nonbraidedthe
category
2 Further
predictions
for Logarithmic
CFT
M.R. Gaberdiel,
I. Runkel,
and S. Wood,
Fusion rules and boundary
(Bushlanov,
Tipunin
’09)
conditions inFeigin,
the c вњЏGainutdinov,
0 triplet model,
arXiv:0905.0916
—
Nonrigid
category
for
the
W
algebra
ofgroup
FGST3.
2,3 quantum
3 Motivation for some particular
structures
— Making it rigid ùñ the FGST3 “QG” fusion
Г№Г±
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Last year:
D. AdamoviВґc and A. Milas, Lattice construction of logarithmic modules
for certain vertex algebras, arXiv:0902.3417
— Construction of projective modules
turned out to be as expected on the quantum group basis [FGST]
K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W в™Јpq
and the restricted quantum group at root of unity, arXiv:0902.4607
— Proof of the FGST equivalence conjecture by extending the
“log-deformation” construction of [FFHST]
Y.-Z. Huang, Generalized twisted modules associated to general
automorphisms of a vertex operator algebra arXiv:0905.0514
H. Kondo and Y. Saito, Indecomposable decomposition of tensor
products of modules over the restricted quantum universal enveloping
Moral:
algebra associated to рќ–�рќ–‘2 , arXiv:0901.4221 [math.QA]
1 “Engineering”
equivalence
math. proof of the equivalence
— a nonbraidedthe
category
2 Further
predictions
for Logarithmic
CFT
M.R. Gaberdiel,
I. Runkel,
and S. Wood,
Fusion rules and boundary
(Bushlanov,
Tipunin
’09)
conditions inFeigin,
the c вњЏGainutdinov,
0 triplet model,
arXiv:0905.0916
—
Nonrigid
category
for
the
W
algebra
ofgroup
FGST3.
2,3 quantum
3 Motivation for some particular
structures
— Making it rigid ùñ the FGST3 “QG” fusion
Г№Г±
Semikhatov
Hopf algebras and the duality to logarithmic CFT
This talk:
AM Semikhatov, A Heisenberg double addition to the logarithmic
Kazhdan–Lusztig duality, arXiv:0905.2215
AM Semikhatov, A differential рќ’°-module algebra for рќ’° вњЏ рќ’°q sв„“в™Ј2q at an
even root of unity, arXiv:0809.0144
(Theor. Math. Phys. 159 (2009) 1–24)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
indecomposable
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
вњ©
...
вњ«
...............
pвњ¶ вњЃ1
вњЄ
...
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Rational models: basic inputs
c
Virasoro algebra rLm , Ln s вњЏ в™Јm вњЃ nqLmВ n В 12
в™Јm3 вњЃ mqОґmВ n,0
Highest-weight modules Ln➙1 ⑤Δ② ✏ 0, L0 ⑤Δ② ✏ Δ⑤Δ②
Verma
irreducible
вњ¶
Rational в™Јp, pвњ¶ q-models at c вњЏ 13 вњЃ ppвњ¶ вњЃ pp :
Kac table of “good” modules:
1
вњ¶
2 в™Јp вњЃ 1q вњ‚ в™Јp вњЃ 1q nonisomorphic
Virasoro irreps
pвњ¶ 1
вњЃ
вњ©
вњ«
...
...............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
These irreps have no extensions among themselves
semisimple (diagonalizable)
Г№Г±
Г№Г± chiral space of states вњЏ вћљв™Јirrepsq
Г№Г± numerous deep properties of RCFT. . .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
pвњ¶ 1
вњЃ
adding 1 row and 1 column:
A new possibility: в™Јp, 1q models
with the extended Kac table
вњ©
вњ«
...
..............
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
N ONLOGARITHMIC CONTENT: void
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences:
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences:
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences:
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
P ROJECTIVE MODULES: home for logarithmic partners
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences:
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences:
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™Јprojective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences:
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™ЈW -algebra projective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
LCFT: “minimal” extension of rational models
...
...
. . . . . . . . . . . . . .. . .
pвњ¶ вњЃ1
...
вњЄ
...
❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥
вњ©
вњ«
adding 1 row and 1 column:
“only” p  p✶ ✁ 1 new boxes
A new possibility: в™Јp, 1q models
with the extended Kac table
вњЃ
p 1
...
❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥
p
Drastic consequences:
Representations admit indecomposable extensions
Г№Г± chiral space of states вњЏ вћљв™ЈW -algebra projective modulesq
The symmetry extends from Virasoro to a larger W -algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3]
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3]
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3]
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
Gaberdiel–Kausch++, FHST (kernel of screening),
Carqueville–Flohr (C2 cofiniteness),
Adamovi´c–Milas++, Nagatomo–Tsuchiya
a triplet algebra Wp.pвњ¶ [FGST3]
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
Gaberdiel–Kausch++, FHST (kernel of screening),
Carqueville–Flohr (C2 cofiniteness),
Adamovi´c–Milas++, Nagatomo–Tsuchiya
a triplet algebra Wp.pвњ¶ [FGST3]
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
Gaberdiel–Kausch++, FHST (kernel of screening),
Carqueville–Flohr (C2 cofiniteness),
Adamovi´c–Milas++, Nagatomo–Tsuchiya
a triplet algebra Wp.pвњ¶ [FGST3]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
Gaberdiel–Kausch++, FHST (kernel of screening),
Carqueville–Flohr (C2 cofiniteness),
Adamovi´c–Milas++, Nagatomo–Tsuchiya
a triplet algebra Wp.pвњ¶ [FGST3], used in [GRW]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Г№Г± dimension 15 in the lowest case в™Јp вњЏ 2, p вњЏ 3q
вњ¶
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
Gaberdiel–Kausch++, FHST (kernel of screening),
Carqueville–Flohr (C2 cofiniteness),
Adamovi´c–Milas++, Nagatomo–Tsuchiya
a triplet algebra Wp.pвњ¶ [FGST3], used in [GRW]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Г№Г± dimension 15 in the lowest case в™Јp вњЏ 2, p вњЏ 3q
вњ¶
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3], used in [GRW]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Г№Г± dimension 15 in the lowest case в™Јp вњЏ 2, p вњЏ 3q
вњ¶
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
♣p, p✶ q: a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3], used in [GRW]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Г№Г± dimension 15 in the lowest case в™Јp вњЏ 2, p вњЏ 3q
вњ¶
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
♣p, p✶ q: a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3], used in [GRW]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3], used in [GRW]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Two series of LCFT models:
LCFTs generally have extended symmetry algebras.
в™Јp, 1q:
в™Јp, pвњ¶ q:
“the” triplet algebra W ♣pq:
currents W В , W 0 , W вњЃ , of dimension 2p вњЃ 1
a triplet algebra Wp.pвњ¶ [FGST3], used in [GRW]
currents W В , W 0 , W вњЃ of dimension в™Ј2p вњЃ 1qв™Ј2pвњ¶ вњЃ 1q
Dual quantum groups:
в™Јp, 1q:
рќ’°рќ”® sв„“в™Ј2q at рќ”® вњЏ eiПЂ в‘Јp , dim вњЏ 2p3
first appeared in Alekseev–Gluschenkov–Lyakhovskaya (1994);
rediscovered in FGST, FGST2 (2005);
Arike (2006); Furlan–Hadjiivanov–Todorov (2007);
Kondo–Saito (2009); Suter (1994); Xiao (1997)
в™Јp, pвњ¶ q:
a quantum group 𝔤p,p✶ , dim ✏ 2p3 p✶ 3
FGST3, FGST4 (2006);
Arike (2009)
QG vs LCFTs:
Equivalence or something very near it
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
Semikhatov
вњЏ в‘ ОІ , abв‘Ў .
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
в™Јp, 1q case:
kE вњЏ рќ”®Ek , E p вњЏ 0, k 4p вњЏ 1; рќ”® вњЏ eiПЂ в‘Јp
comultiplication: О”в™ЈE q вњЏ 1 вќњ E В E вќњ k 2 , О”в™Јk q вњЏ k вќњ k.
write О”b вњЏ
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
Semikhatov
вћ¦
bвњ¶ вќњ bвњ· вњЏ bвњ¶ вќњ bвњ·
вњЏ в‘ ОІ , abв‘Ў .
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
в™Јp, 1q case:
kE вњЏ рќ”®Ek , E p вњЏ 0, k 4p вњЏ 1; рќ”® вњЏ eiПЂ в‘Јp
comultiplication: О”в™ЈE q вњЏ 1 вќњ E В E вќњ k 2 , О”в™Јk q вњЏ k вќњ k.
write О”b вњЏ
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
Semikhatov
вћ¦
bвњ¶ вќњ bвњ· вњЏ bвњ¶ вќњ bвњ·
вњЏ в‘ ОІ , abв‘Ў .
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
в™Јp, 1q case:
kE вњЏ рќ”®Ek , E p вњЏ 0, k 4p вњЏ 1; рќ”® вњЏ eiПЂ в‘Јp
comultiplication: О”в™ЈE q вњЏ 1 вќњ E В E вќњ k 2 , О”в™Јk q вњЏ k вќњ k.
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
вњЏ в‘ ОІ , abв‘Ў .
в™Јp, 1q case:
вњЃn
в‘ F , E m k n в‘Ў вњЏ Оґm,1 рќ”® вњЃрќ”® рќ”®вњЃ1 , в‘ П°, E m k n в‘Ў вњЏ Оґm,0 рќ”®вњЃnв‘Ј2 ,
then П°F вњЏ рќ”®F П°, F p вњЏ 0, П° 4p вњЏ 1; О”в™ЈF q вњЏ П° 2 вќњ F В F вќњ 1, О”в™ЈП° q вњЏ П° вќњ П°.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
вњЏ в‘ ОІ , abв‘Ў .
в™Јp, 1q case:
вњЃn
в‘ F , E m k n в‘Ў вњЏ Оґm,1 рќ”® вњЃрќ”® рќ”®вњЃ1 , в‘ П°, E m k n в‘Ў вњЏ Оґm,0 рќ”®вњЃnв‘Ј2 ,
then П°F вњЏ рќ”®F П°, F p вњЏ 0, П° 4p вњЏ 1; О”в™ЈF q вњЏ П° 2 вќњ F В F вќњ 1, О”в™ЈП° q вњЏ П° вќњ П°.
Some standard notation:
B acts on B вњќ : h ГЎ ОІ
B вњќ acts on B:
вњЏ в‘ ОІ вњ·, hв‘Ў ОІ вњ¶ (left regular action)
ОІ Г h вњЏ в‘ ОІ вњ¶ , hв‘Ў ОІ вњ· (right regular action)
ОІ ГЎ a вњЏ в‘ ОІ , a вњ· в‘Ў a вњ¶
a Г ОІ вњЏ в‘ ОІ , a вњ¶ в‘Ў a вњ·
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
вњЏ в‘ ОІ , abв‘Ў .
Some standard notation:
B acts on B вњќ : h ГЎ ОІ
вњЏ в‘ ОІ вњ·, hв‘Ў ОІ вњ¶ (left regular action)
ОІ Г h вњЏ в‘ ОІ вњ¶ , hв‘Ў ОІ вњ· (right regular action)
B вњќ acts on B: ОІ ГЎ a вњЏ в‘ ОІ , aвњ· в‘Ў aвњ¶
a Г ОІ вњЏ в‘ ОІ , a вњ¶ в‘Ў a вњ·
Then the Drinfeld double: рќ’џв™ЈB q вњЏ B вњќ вќњ B with the composition
в™ЈО± вќњ aqв™ЈОІ вќњ bq вњЏ в™Јq.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
вњЏ в‘ ОІ , abв‘Ў .
Some standard notation:
B acts on B вњќ : h ГЎ ОІ
вњЏ в‘ ОІ вњ·, hв‘Ў ОІ вњ¶ (left regular action)
ОІ Г h вњЏ в‘ ОІ вњ¶ , hв‘Ў ОІ вњ· (right regular action)
B вњќ acts on B: ОІ ГЎ a вњЏ в‘ ОІ , aвњ· в‘Ў aвњ¶
a Г ОІ вњЏ в‘ ОІ , a вњ¶ в‘Ў a вњ·
Then the Drinfeld double: рќ’џв™ЈB q вњЏ B вњќ вќњ B with the composition
в™ЈО± вќњ aqв™ЈОІ вќњ bq вњЏ О± в™Јqвќњ b.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
вњЏ в‘ ОІ , abв‘Ў .
Some standard notation:
B acts on B вњќ : h ГЎ ОІ
вњЏ в‘ ОІ вњ·, hв‘Ў ОІ вњ¶ (left regular action)
ОІ Г h вњЏ в‘ ОІ вњ¶ , hв‘Ў ОІ вњ· (right regular action)
B вњќ acts on B: ОІ ГЎ a вњЏ в‘ ОІ , aвњ· в‘Ў aвњ¶
a Г ОІ вњЏ в‘ ОІ , a вњ¶ в‘Ў a вњ·
Then the Drinfeld double: рќ’џв™ЈB q вњЏ B вњќ вќњ B with the composition
в™ЈО± вќњ aqв™ЈОІ вќњ bq вњЏ О± в™Јaвњ¶ ГЎ ОІ qвќњ aвњ·b.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
вњЏ в‘ ОІ , abв‘Ў .
Some standard notation:
B acts on B вњќ : h ГЎ ОІ
вњЏ в‘ ОІ вњ·, hв‘Ў ОІ вњ¶ (left regular action)
ОІ Г h вњЏ в‘ ОІ вњ¶ , hв‘Ў ОІ вњ· (right regular action)
B вњќ acts on B: ОІ ГЎ a вњЏ в‘ ОІ , aвњ· в‘Ў aвњ¶
a Г ОІ вњЏ в‘ ОІ , a вњ¶ в‘Ў a вњ·
Then the Drinfeld double: рќ’џв™ЈB q вњЏ B вњќ вќњ B with the composition
в™ЈО± вќњ aqв™ЈОІ вќњ bq вњЏ О± в™Јaвњ¶ ГЎ ОІ Г в™Јaвњёqqвќњ aвњ·b.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
вњЏ в‘ ОІ , abв‘Ў .
Some standard notation:
B acts on B вњќ : h ГЎ ОІ
вњЏ в‘ ОІ вњ·, hв‘Ў ОІ вњ¶ (left regular action)
ОІ Г h вњЏ в‘ ОІ вњ¶ , hв‘Ў ОІ вњ· (right regular action)
B вњќ acts on B: ОІ ГЎ a вњЏ в‘ ОІ , aвњ· в‘Ў aвњ¶
a Г ОІ вњЏ в‘ ОІ , a вњ¶ в‘Ў a вњ·
Then the Drinfeld double: рќ’џв™ЈB q вњЏ B вњќ вќњ B with the composition
в™ЈО± вќњ aqв™ЈОІ вќњ bq вњЏ О± в™Јaвњ¶ ГЎ ОІ Г SвњЃ1в™Јaвњёqqвќњ aвњ·b.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
How?
Serendipitous finding of FGST: Drinfeld double
Start with the algebra B of screening(s) and zero mode(s).
в™Јp, 1q case:
kE вњЏ рќ”®Ek , E p вњЏ 0, k 4p вњЏ 1; рќ”® вњЏ eiПЂ в‘Јp
comultiplication: О”в™ЈE q вњЏ 1 вќњ E В E вќњ k 2 , О”в™Јk q вњЏ k вќњ k.
Take the dual space B вњќ ,
then в‘ ОІ Оі, bв‘Ў вњЏ в‘ ОІ , bвњ¶ в‘Ўв‘ Оі, bвњ· в‘Ў and в‘ О”в™ЈОІ q, a вќњ bв‘Ў
Then the Drinfeld double: рќ’џв™ЈB q вњЏ B вњќ вќњ B
вњЏ в‘ ОІ , abв‘Ў .
в™Јp, 1q example: E вњЏ Оµ вќњ E, k вњЏ Оµ вќњ k, F вњЏ F вќњ 1, П° вњЏ П° вќњ 1
kE вњЏ рќ”®Ek , E p вњЏ 0, k 4p вњЏ 1,
П°F вњЏ рќ”®F П°, F p вњЏ 0, П° 4p вњЏ 1,
k 2 вњЃ П°2
k П° вњЏ П°k, kFk вњЃ1 вњЏ рќ”®вњЃ1 F , П°EП° вњЃ1 вњЏ рќ”®вњЃ1 E, rE, F s вњЏ
.
рќ”® вњЃ рќ”®вњЃ1
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
Drinfeld double рќ’џв™ЈB q
вњЏ рќ”®Ek ,
П°F вњЏ рќ”®F П°,
kFk вњЃ1 вњЏ рќ”®вњЃ1 F ,
kE
kП° вњЏ П°k,
E p вњЏ 0,
F p вњЏ 0,
k 4p вњЏ 1,
П° 4p вњЏ 1,
П°EП° вњЃ1 вњЏ рќ”®вњЃ1 E,
1
too big for us!
2
Semikhatov
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®П°вњЃ1 .
2
рќ’џв™Ј B q
вњЏ рќ’џв™ЈBqв‘Јв™ЈП° k вњЃ 1q
рќ’°рќ”® sв„“в™Ј2q в‘Ё рќ’џв™ЈB q : only k even
Hopf algebras and the duality to logarithmic CFT
2
Dual quantum group
Drinfeld double рќ’џв™ЈB q
вњЏ рќ”®Ek ,
П°F вњЏ рќ”®F П°,
kFk вњЃ1 вњЏ рќ”®вњЃ1 F ,
kE
kП° вњЏ П°k,
рќ’°рќ”®sв„“в™Ј2q at 2pth root of unity
k 2E
E p вњЏ 0,
F p вњЏ 0,
k 4p вњЏ 1,
П° 4p вњЏ 1,
П°EП° вњЃ1 вњЏ рќ”®вњЃ1 E,
Гі
1
2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®П°вњЃ1 .
2
рќ’џв™Ј B q
вњЏ рќ’џв™ЈBqв‘Јв™ЈП° k вњЃ 1q
рќ’°рќ”® sв„“в™Ј2q в‘Ё рќ’џв™ЈB q : only k even
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
2
Dual quantum group
Drinfeld double рќ’џв™ЈB q
вњЏ рќ”®Ek ,
П°F вњЏ рќ”®F П°,
kFk вњЃ1 вњЏ рќ”®вњЃ1 F ,
kE
kП° вњЏ П°k,
рќ’°рќ”®sв„“в™Ј2q at 2pth root of unity
k 2E
E p вњЏ 0,
F p вњЏ 0,
k 4p вњЏ 1,
П° 4p вњЏ 1,
П°EП° вњЃ1 вњЏ рќ”®вњЃ1 E,
Гі
1
2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®П°вњЃ1 .
2
рќ’џв™Ј B q
вњЏ рќ’џв™ЈBqв‘Јв™ЈП° k вњЃ 1q
рќ’°рќ”® sв„“в™Ј2q в‘Ё рќ’џв™ЈB q : only k even
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
2
Dual quantum group
Drinfeld double рќ’џв™ЈB q
вњЏ рќ”®Ek ,
П°F вњЏ рќ”®F П°,
kFk вњЃ1 вњЏ рќ”®вњЃ1 F ,
kE
kП° вњЏ П°k,
рќ’°рќ”®sв„“в™Ј2q at 2pth root of unity
k 2E
E p вњЏ 0,
F p вњЏ 0,
k 4p вњЏ 1,
П° 4p вњЏ 1,
П°EП° вњЃ1 вњЏ рќ”®вњЃ1 E,
Гі
1
2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®П°вњЃ1 .
2
рќ’џв™Ј B q
вњЏ рќ’џв™ЈBqв‘Јв™ЈП° k вњЃ 1q
рќ’°рќ”® sв„“в™Ј2q в‘Ё рќ’џв™ЈB q : only k even
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
2
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Properties:
The R-matrix of the double рќ’џв™ЈB q вњ• B вњќ вќњ B
RвњЏ
вћі
в™ЈОµ вќњ eJ qвќњв™ЈeJ вќњ 1q ВЂ рќ’џв™ЈBqвќњ рќ’џв™ЈBq,
J
(teJ вњ‰ is a basis of B and teJ вњ‰ its dual basis in B вњќ )
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Properties:
The R-matrix of the double рќ’џв™ЈB q вњ• B вњќ вќњ B
RвњЏ
pвњЃ1 4pвњЃ1
1 вћі вћі в™Јрќ”® вњЃ рќ”®вњЃ1 qm
4p
rms!
вњЏ
вњЏ
рќ”® 2 mв™ЈmвњЃ1qВ mв™Јi вњЃj qвњЃ 2 E m k i вќњ F m П° вњЃj ВЂ рќ’џв™ЈB qвќњ рќ’џв™ЈB q,
ij
1
m 0 i,j 0
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Properties:
The R-matrix of the double рќ’џв™ЈB q вњ• B вњќ вќњ B pushes forward to рќ’џв™ЈB q ,
RвњЏ
pвњЃ1 4pвњЃ1
1 вћі вћі в™Јрќ”® вњЃ рќ”®вњЃ1 qm
4p
rms!
вњЏ
вњЏ
рќ”® 2 mв™ЈmвњЃ1qВ mв™Јi вњЃj qвњЃ 2 E m k i вќњ F m k j ВЂ рќ’џв™ЈB q вќњ рќ’џв™ЈB q ,
ij
1
m 0 i,j 0
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Properties:
The R-matrix of the double рќ’џв™ЈB q вњ• B вњќ вќњ B pushes forward to рќ’џв™ЈB q ,
RвњЏ
pвњЃ1 4pвњЃ1
1 вћі вћі в™Јрќ”® вњЃ рќ”®вњЃ1 qm
4p
rms!
вњЏ
вњЏ
рќ”® 2 mв™ЈmвњЃ1qВ mв™Јi вњЃj qвњЃ 2 E m k i вќњ F m k j ВЂ рќ’џв™ЈB q вќњ рќ’џв™ЈB q ,
ij
1
m 0 i,j 0
but does not restrict to рќ’°рќ”® sв„“в™Ј2q.
рќ’°рќ”® sв„“в™Ј2q is not quasitriangular.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
But 𝒰𝔮 sℓ♣2q is factorizable: a nondegenerate “M-matrix”
M
вњЏ R21R12 ВЂ рќ’°рќ”®sв„“в™Ј2qвќњ рќ’°рќ”®sв„“в™Ј2q
the map
рќ’°рќ”® sв„“в™Ј2qвњќ Г‘ рќ’°рќ”® sв„“в™Ј2q, ОІ ГћГ‘ в™ЈОІ вќњ idqM
is a linear isomorphism;
вњ’
an algebra isomorphism qChar Г‘ рќ’µ by restriction.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
But 𝒰𝔮 sℓ♣2q is factorizable: a nondegenerate “M-matrix”
M
вњЏ
pвњЃ1 pвњЃ1 2pвњЃ1
1 вћі вћі вћі в™Јрќ”® вњЃ рќ”®вњЃ1 qmВ n
2p
rms!rns!
вњЏ вњЏ
вњЏ
m 0 n 0 i,j 0
рќ”®
в™Ј вњЃ q В nв™ЈnвњЃ1q вњЃmв™Јj В mqВ i в™ЈmвњЃj вњЃ2nqВ 2jn
2
m m 1
2
вњ‚ F m E n k 2j вќњ E m F n k 2i
the map
рќ’°рќ”® sв„“в™Ј2qвњќ Г‘ рќ’°рќ”® sв„“в™Ј2q, ОІ ГћГ‘ в™ЈОІ вќњ idqM
is a linear isomorphism;
вњ’
an algebra isomorphism qChar Г‘ рќ’µ by restriction.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
— “almost” factorizable: the equation for M
ВЂ рќ’°рќ”®sв„“в™Ј2qвќњ рќ’°рќ”®sв„“в™Ј2q,
в™ЈО” вќњ idqв™ЈM q вњЏ R32M13R23,
MО”в™Јx q вњЏ О”в™Јx qM вќ…x,
involves R.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
рќ’°рќ”® sв„“в™Ј2q is also ribbon:
вќ‰ v ВЂ рќ’µв™Јрќ’°рќ”®sв„“в™Ј2qq,
v 2 вњЏ uS в™Јu q, S в™Јv q вњЏ v , Оµ в™Јv q вњЏ 1,
Semikhatov
О”в™Јv q вњЏ M вњЃ1 в™Јv вќњ v q.
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
рќ’°рќ”® sв„“в™Ј2q is also ribbon:
вќ‰ v ВЂ рќ’µв™Јрќ’°рќ”®sв„“в™Ј2qq,
v 2 вњЏ uS в™Јu q, S в™Јv q вњЏ v , Оµ в™Јv q вњЏ 1,
v
вњЏ
pвњЃ1 2pвњЃ1
1 вњЃ i вћі вћі в™Јрќ”® вњЃ рќ”®вњЃ1 qm
вќ„
2 p
rms!
вњЏe
вњЏ вњЏ
О”в™Јv q вњЏ M вњЃ1 в™Јv вќњ v q.
рќ”®вњЃ 2 В mj В 2 в™Јj В pВ 1q F m E m k 2j
m
1
2
m 0 j 0
2iПЂL0
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Modular group action:
Lyubashenko–Majid:
ribbon В M-matrix В integral В cointegral Г№Г±
SL♣2, ℤq
Semikhatov
Г± рќ’µ,
dim рќ’µ вњЏ 3p вњЃ 1.
Hopf algebras and the duality to logarithmic CFT
Dual quantum group
рќ’°рќ”® sв„“в™Ј2q at 2pth root of unity
k 2E
вњЏ рќ”®Ek 2 , E p вњЏ 0, k 4p вњЏ 1,
k 2 F вњЏ рќ”®вњЃ2 Fk 2 , F p вњЏ 0,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1 .
Modular group action:
Lyubashenko–Majid:
ribbon В M-matrix В integral В cointegral Г№Г±
SL♣2, ℤq
Г± рќ’µ,
dim рќ’µ вњЏ 3p вњЃ 1.
The same SL♣2, ℤq representation is induced from the W ♣pq
characters.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ ?
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ ?
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ ?
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ ?
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ в™Јaвњ¶ вћЌ П† qв™Јaвњ· вћЌ П€ q
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ в™Јaвњ¶ вћЌ П† qв™Јaвњ· вћЌ П€ q
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ в™Јaвњ¶ вћЌ П† qв™Јaвњ· вћЌ П€ q
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Algebraic analog of fractional statistics
Can we have a manifestly QG-invariant description of LCFT?
—in terms of “quasiparticle” excitations with “fractional statistics”—
What would be an algebraic analogue of the algebra of fields in such
description?
Assuming that the QG acts on fields, it has to act on products of fields:
a вћЌ в™ЈП† П€ q вњЏ в™Јaвњ¶ вћЌ П† qв™Јaвњ· вћЌ П€ q
So we need a module algebra
— to begin with, a module algebra over the Drinfeld double 𝒟♣B q
Theorem
For a Hopf algebra B with invertible antipode, the Heisenberg double
в„‹в™ЈB вњќ q is a рќ’џв™ЈB q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Heisenberg double
в„‹в™ЈB вњќ q вњЏ B вњќ вќњ B as a vector space
(the same vector space as рќ’џв™ЈB q вњЏ B вњќ вќњ B )
with the composition law
♣α � aq♣β � bq ✏ α ♣a✶ á β q� a✷b,
О±, ОІ
ВЂ Bвњќ,
a, b ВЂ B.
Theorem (continued):
The рќ’џв™ЈB q action on в„‹в™ЈB вњќ q is given by
В where
вњџ
В вњџ
в™Ј Вµ вќњ m q вћЌ в™ЈО± в�… a q вњЏ в™Ј Вµ вќњ m q вњ¶ ГЎ О± в�… в™Ј Вµ вќњ m q вњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ·в™Јm ГЎ О± qSвњќвњЃ1в™ЈВµ вњ¶q,
в™ЈВµ вќњ mq вћЌ a вњЏ в™Јmвњ¶aSв™Јmвњ·qqГ SвњќвњЃ1в™ЈВµ q.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Heisenberg double
в„‹в™ЈB вњќ q вњЏ B вњќ в�… B as a vector space
(the same vector space as рќ’џв™ЈB q вњЏ B вњќ вќњ B )
with the composition law
♣α � aq♣β � bq ✏ α ♣a✶ á β q� a✷b,
О±, ОІ
ВЂ Bвњќ,
a, b ВЂ B.
Theorem (continued):
The рќ’џв™ЈB q action on в„‹в™ЈB вњќ q is given by
В where
вњџ
В вњџ
в™Ј Вµ вќњ m q вћЌ в™ЈО± в�… a q вњЏ в™Ј Вµ вќњ m q вњ¶ ГЎ О± в�… в™Ј Вµ вќњ m q вњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ·в™Јm ГЎ О± qSвњќвњЃ1в™ЈВµ вњ¶q,
в™ЈВµ вќњ mq вћЌ a вњЏ в™Јmвњ¶aSв™Јmвњ·qqГ SвњќвњЃ1в™ЈВµ q.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Heisenberg double
в„‹в™ЈB вњќ q вњЏ B вњќ в�… B as a vector space
(the same vector space as рќ’џв™ЈB q вњЏ B вњќ вќњ B )
with the composition law
♣α � aq♣β � bq ✏ α ♣a✶ á β q� a✷b,
О±, ОІ
ВЂ Bвњќ,
a, b ВЂ B.
Theorem (continued):
The рќ’џв™ЈB q action on в„‹в™ЈB вњќ q is given by
В where
вњџ
В вњџ
в™Ј Вµ вќњ m q вћЌ в™ЈО± в�… a q вњЏ в™Ј Вµ вќњ m q вњ¶ ГЎ О± в�… в™Ј Вµ вќњ m q вњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ·в™Јm ГЎ О± qSвњќвњЃ1в™ЈВµ вњ¶q,
в™ЈВµ вќњ mq вћЌ a вњЏ в™Јmвњ¶aSв™Јmвњ·qqГ SвњќвњЃ1в™ЈВµ q.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Heisenberg double
в„‹в™ЈB вњќ q вњЏ B вњќ в�… B as a vector space
(the same vector space as рќ’џв™ЈB q вњЏ B вњќ вќњ B )
with the composition law
♣α � aq♣β � bq ✏ α ♣a✶ á β q� a✷b, α, β € B✝,
whereas in рќ’џв™ЈB q,
в™ЈО± вќњ aqв™ЈОІ вќњ bq вњЏ О± в™Јaвњ¶ ГЎ ОІ Г SвњЃ1в™Јaвњёqqвќњ aвњ·b
a, b ВЂ B.
Theorem (continued):
The рќ’џв™ЈB q action on в„‹в™ЈB вњќ q is given by
В where
вњџ
В вњџ
в™Ј Вµ вќњ m q вћЌ в™ЈО± в�… a q вњЏ в™Ј Вµ вќњ m q вњ¶ ГЎ О± в�… в™Ј Вµ вќњ m q вњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ·в™Јm ГЎ О± qSвњќвњЃ1в™ЈВµ вњ¶q,
в™ЈВµ вќњ mq вћЌ a вњЏ в™Јmвњ¶aSв™Јmвњ·qqГ SвњќвњЃ1в™ЈВµ q.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Heisenberg double
в„‹ в™ЈB вњќ q вњЏ B вњќ в�… B
as a vector space
(the same vector space as рќ’џв™ЈB q вњЏ B вњќ вќњ B
with the composition law
♣α � aq♣β � bq ✏ α ♣a✶ á β q� a✷b,
)
О±, ОІ
ВЂ Bвњќ,
a, b ВЂ B.
Theorem (continued):
The рќ’џв™ЈB q action on в„‹в™ЈB вњќ q is given by
В where
вњџ
В вњџ
в™Ј Вµ вќњ m q вћЌ в™ЈО± в�… a q вњЏ в™Ј Вµ вќњ m q вњ¶ ГЎ О± в�… в™Ј Вµ вќњ m q вњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ·в™Јm ГЎ О± qSвњќвњЃ1в™ЈВµ вњ¶q,
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Heisenberg double
в„‹в™ЈB вњќ q вњЏ B вњќ в�… B as a vector space
(the same vector space as рќ’џв™ЈB q вњЏ B вњќ вќњ B )
with the composition law
♣α � aq♣β � bq ✏ α ♣a✶ á β q� a✷b,
О±, ОІ
ВЂ Bвњќ,
a, b ВЂ B.
invented in: A Alekseev, L Faddeev, Commun. Math. Phys. 141 (1991) 413–422;
N Reshetikhin, M Semenov-Tian-Shansky, Lett. Math. Phys. 19 (1990) 133–142;
M Semenov-Tyan-Shanskii, Theor. Math. Phys. 93 (1992) 1292–1307.
Interpretation:
View a, b ВЂ B as operators and О±, ОІ
Theorem (continued):
The рќ’џв™ЈB q action on в„‹в™ЈB вњќ q is given by
в™ЈВµ вќњ mq вћЌ в™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™Ј B q , О± в�… a ВЂ в„‹ в™Ј B вњќ q ,
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q,
вњЃ1 q. to logarithmic CFT
algebras вњќ
в™ЈВµ вќњ mq вћЌSemikhatov
a вњЏ в™Јmвњ¶ aS в™ЈHopf
mвњ· qqГ S and theв™ЈВµduality
В where
€ B✝ as functions: “Leibnitz rule”
вњџ
В вњџ
Heisenberg double
в„‹в™ЈB вњќ q вњЏ B вњќ в�… B as a vector space
(the same vector space as рќ’џв™ЈB q вњЏ B вњќ вќњ B )
with the composition law
♣α � aq♣β � bq ✏ α ♣a✶ á β q� a✷b,
О±, ОІ
ВЂ Bвњќ,
a, b ВЂ B.
Theorem (continued):
The рќ’џв™ЈB q action on в„‹в™ЈB вњќ q is given by
В where
вњџ
В вњџ
в™Ј Вµ вќњ m q вћЌ в™ЈО± в�… a q вњЏ в™Ј Вµ вќњ m q вњ¶ ГЎ О± в�… в™Ј Вµ вќњ m q вњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ·в™Јm ГЎ О± qSвњќвњЃ1в™ЈВµ вњ¶q,
в™ЈВµ вќњ mq вћЌ a вњЏ в™Јmвњ¶aSв™Јmвњ·qqГ SвњќвњЃ1в™ЈВµ q.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
рќ’џв™ЈB q action on B вњќ
Take the left regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ вњ• B вќњ B вњќ ,
в™ЈВµ вќњ mqГЎв™Јa вќњ О± q вњЏ в™ЈВµ вњ· ГЎ aqвќњ Вµ вњё в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q.
Restrict it to 1 вќњ B вњќ :
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, О± ВЂ Bвњќ ,
— B ✝ becomes a quantum commutative 𝒟♣B q-module algebra.
рќ’џв™ЈB q action on B:
Take the right regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ ,
restrict it to B вќњ Оµ
convert into a left action using the antipode:
в™ЈВµ вќњ mqвћЌ a вњЏ в™Јmвњ¶ aSв™Јmвњ· qqГ Sвњќ вњЃ1 в™ЈВµ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, a ВЂ B.
— B becomes a quantum commutative 𝒟♣B q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
рќ’џв™ЈB q action on B вњќ
Take the left regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ вњ• B вќњ B вњќ ,
в™ЈВµ вќњ mqГЎв™Јa вќњ О± q вњЏ в™ЈВµ вњ· ГЎ aqвќњ Вµ вњё в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q.
Restrict it to 1 вќњ B вњќ :
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, О± ВЂ Bвњќ ,
— B ✝ becomes a quantum commutative 𝒟♣B q-module algebra.
рќ’џв™ЈB q action on B:
Take the right regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ ,
restrict it to B вќњ Оµ
convert into a left action using the antipode:
в™ЈВµ вќњ mqвћЌ a вњЏ в™Јmвњ¶ aSв™Јmвњ· qqГ Sвњќ вњЃ1 в™ЈВµ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, a ВЂ B.
— B becomes a quantum commutative 𝒟♣B q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
рќ’џв™ЈB q action on B вњќ
Take the left regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ вњ• B вќњ B вњќ ,
в™ЈВµ вќњ mqГЎв™Јa вќњ О± q вњЏ в™ЈВµ вњ· ГЎ aqвќњ Вµ вњё в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q.
Restrict it to 1 вќњ B вњќ :
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, О± ВЂ Bвњќ ,
— B ✝ becomes a quantum commutative 𝒟♣B q-module algebra.
рќ’џв™ЈB q action on B:
Take the right regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ ,
restrict it to B вќњ Оµ
convert into a left action using the antipode:
в™ЈВµ вќњ mqвћЌ a вњЏ в™Јmвњ¶ aSв™Јmвњ· qqГ Sвњќ вњЃ1 в™ЈВµ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, a ВЂ B.
— B becomes a quantum commutative 𝒟♣B q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
рќ’џв™ЈB q action on B вњќ
Take the left regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ вњ• B вќњ B вњќ ,
в™ЈВµ вќњ mqГЎв™Јa вќњ О± q вњЏ в™ЈВµ вњ· ГЎ aqвќњ Вµ вњё в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q.
Restrict it to 1 вќњ B вњќ :
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, О± ВЂ Bвњќ ,
— B ✝ becomes a quantum commutative 𝒟♣B q-module algebra.
рќ’џв™ЈB q action on B:
Take the right regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ ,
restrict it to B вќњ Оµ
convert into a left action using the antipode:
в™ЈВµ вќњ mqвћЌ a вњЏ в™Јmвњ¶ aSв™Јmвњ· qqГ Sвњќ вњЃ1 в™ЈВµ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, a ВЂ B.
— B becomes a quantum commutative 𝒟♣B q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
рќ’џв™ЈB q action on B вњќ
Take the left regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ вњ• B вќњ B вњќ ,
в™ЈВµ вќњ mqГЎв™Јa вќњ О± q вњЏ в™ЈВµ вњ· ГЎ aqвќњ Вµ вњё в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q.
Restrict it to 1 вќњ B вњќ :
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, О± ВЂ Bвњќ ,
— B ✝ becomes a quantum commutative 𝒟♣B q-module algebra.
рќ’џв™ЈB q action on B:
Take the right regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ ,
restrict it to B вќњ Оµ
convert into a left action using the antipode:
в™ЈВµ вќњ mqвћЌ a вњЏ в™Јmвњ¶ aSв™Јmвњ· qqГ Sвњќ вњЃ1 в™ЈВµ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, a ВЂ B.
— B becomes a quantum commutative 𝒟♣B q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
рќ’џв™ЈB q action on B вњќ
Take the left regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ вњ• B вќњ B вњќ ,
в™ЈВµ вќњ mqГЎв™Јa вќњ О± q вњЏ в™ЈВµ вњ· ГЎ aqвќњ Вµ вњё в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q.
Restrict it to 1 вќњ B вњќ :
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, О± ВЂ Bвњќ ,
— B ✝ becomes a quantum commutative 𝒟♣B q-module algebra.
рќ’џв™ЈB q action on B:
Take the right regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ ,
restrict it to B вќњ Оµ
convert into a left action using the antipode:
в™ЈВµ вќњ mqвћЌ a вњЏ в™Јmвњ¶ aSв™Јmвњ· qqГ Sвњќ вњЃ1 в™ЈВµ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, a ВЂ B.
— B becomes a quantum commutative 𝒟♣B q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Explanation:
В вњџ
В вњџ
в™ЈВµ вќњ mqвћЌв™ЈО± в�… aq вњЏ в™ЈВµ вќњ mqвњ¶ ГЎ О± в�… в™ЈВµ вќњ mqвњ· вћЌ a ,
Вµ вќњ m ВЂ рќ’џв™ЈB q, О± в�… a ВЂ в„‹в™ЈB вњќ q,
рќ’џв™ЈB q action on B вњќ
Take the left regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ вњ• B вќњ B вњќ ,
в™ЈВµ вќњ mqГЎв™Јa вќњ О± q вњЏ в™ЈВµ вњ· ГЎ aqвќњ Вµ вњё в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q.
Restrict it to 1 вќњ B вњќ :
в™ЈВµ вќњ mqГЎ О± вњЏ Вµ вњ· в™Јm ГЎ О± qSвњќ вњЃ1 в™ЈВµ вњ¶ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, О± ВЂ Bвњќ ,
— B ✝ becomes a quantum commutative 𝒟♣B q-module algebra.
рќ’џв™ЈB q action on B:
Take the right regular action of рќ’џв™ЈB q on рќ’џв™ЈB qвњќ ,
restrict it to B вќњ Оµ
convert into a left action using the antipode:
в™ЈВµ вќњ mqвћЌ a вњЏ в™Јmвњ¶ aSв™Јmвњ· qqГ Sвњќ вњЃ1 в™ЈВµ q, Вµ вќњ m ВЂ рќ’џв™ЈBq, a ВЂ B.
— B becomes a quantum commutative 𝒟♣B q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
“Truncation” to a smaller algebra:
Recall that we had
Dually, we have
рќ’џв™ЈB q
в„‹ в™ЈB q
рќ’°рќ”® sв„“в™Ј2q
в„‹рќ”® sв„“в™Ј2q
a quotient, then a subspace
a subspace, then a quotient
dim рќ’°рќ”® sв„“в™Ј2q вњЏ 2p3 ,
dim в„‹рќ”® sв„“в™Ј2q вњЏ 2p3
Гі
Гі
Basis: E, F , k 2 ,
E p вњЏ 0,
k 2E
k 4p вњЏ 1,
Fp вњЏ 0
вњЏ рќ”®Ek 2, k 2F вњЏ рќ”®вњЃ2Fk 2,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1
Semikhatov
Basis: z, вќ‡ , О»
z p вњЏ 0,
О» 2p вњЏ 1,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вњЃ
1
вќ‡ z вњЏ в™Ј рќ”® вњЃ рќ”® q1 В рќ”® вњЃ2 z вќ‡
Hopf algebras and the duality to logarithmic CFT
“Truncation” to a smaller algebra:
Recall that we had
Dually, we have
рќ’џв™ЈB q
в„‹ в™ЈB q
рќ’°рќ”® sв„“в™Ј2q
в„‹рќ”® sв„“в™Ј2q
a quotient, then a subspace
a subspace, then a quotient
dim рќ’°рќ”® sв„“в™Ј2q вњЏ 2p3 ,
dim в„‹рќ”® sв„“в™Ј2q вњЏ 2p3
Гі
Гі
Basis: E, F , k 2 ,
E p вњЏ 0,
k 2E
k 4p вњЏ 1,
Fp вњЏ 0
вњЏ рќ”®Ek 2, k 2F вњЏ рќ”®вњЃ2Fk 2,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1
Semikhatov
Basis: z, вќ‡ , О»
z p вњЏ 0,
О» 2p вњЏ 1,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вњЃ
1
вќ‡ z вњЏ в™Ј рќ”® вњЃ рќ”® q1 В рќ”® вњЃ2 z вќ‡
Hopf algebras and the duality to logarithmic CFT
“Truncation” to a smaller algebra:
Recall that we had
Dually, we have
рќ’џв™ЈB q
в„‹ в™ЈB q
рќ’°рќ”® sв„“в™Ј2q
в„‹рќ”® sв„“в™Ј2q
a quotient, then a subspace
a subspace, then a quotient
dim рќ’°рќ”® sв„“в™Ј2q вњЏ 2p3 ,
dim в„‹рќ”® sв„“в™Ј2q вњЏ 2p3
Гі
Гі
Basis: E, F , k 2 ,
E p вњЏ 0,
k 2E
k 4p вњЏ 1,
Fp вњЏ 0
вњЏ рќ”®Ek 2, k 2F вњЏ рќ”®вњЃ2Fk 2,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1
Semikhatov
Basis: z, вќ‡ , О»
z p вњЏ 0,
О» 2p вњЏ 1,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вњЃ
1
вќ‡ z вњЏ в™Ј рќ”® вњЃ рќ”® q1 В рќ”® вњЃ2 z вќ‡
Hopf algebras and the duality to logarithmic CFT
“Truncation” to a smaller algebra:
Recall that we had
Dually, we have
рќ’џв™ЈB q
в„‹ в™ЈB q
рќ’°рќ”® sв„“в™Ј2q
в„‹рќ”® sв„“в™Ј2q
a quotient, then a subspace
a subspace, then a quotient
dim рќ’°рќ”® sв„“в™Ј2q вњЏ 2p3 ,
dim в„‹рќ”® sв„“в™Ј2q вњЏ 2p3
Гі
Гі
Basis: E, F , k 2 ,
E p вњЏ 0,
k 2E
k 4p вњЏ 1,
Fp вњЏ 0
вњЏ рќ”®Ek 2, k 2F вњЏ рќ”®вњЃ2Fk 2,
2
вњЃ2
rE, F s вњЏ kрќ”® вњЃвњЃрќ”®kвњЃ1
Semikhatov
Basis: z, вќ‡ , О»
z p вњЏ 0,
О» 2p вњЏ 1,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вњЃ
1
вќ‡ z вњЏ в™Ј рќ”® вњЃ рќ”® q1 В рќ”® вњЃ2 z вќ‡
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
The reduced Heisenberg double:
в„‹рќ”® sв„“в™Ј2q вњЏ в™Јв„‚rО» sв‘Јв™ЈО» 2p вњЃ 1qqвќњ в„‚рќ”® rz, вќ‡s,
О» 2p вњЏ 1,
z p вњЏ 0,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
with the рќ’°рќ”® sв„“в™Ј2q action
вњЏ рќ”® В 1 1 О» z,
k 2 вћЌ О» вњЏ рќ”®вњЃ1 О» ,
E вћЌ z m вњЏ вњЃ рќ”®m rmsz mВ 1 , k 2 вћЌ z m вњЏ рќ”®2m z m ,
E вћЌвќ‡ n вњЏ рќ”®1вњЃn rnsвќ‡ nвњЃ1 , k 2 вћЌвќ‡ n вњЏ рќ”®вњЃ2n вќ‡ n ,
E вћЌО»
в„‹рќ”® sв„“в™Ј2q вњЏ
p
Г вњЏ
n 1
Semikhatov
n рќ’«В n вќµ
p
Г вњЏ
F вћЌО»
вњЏ вњЃ рќ”® В рќ”® 1 вќ‡ О» ,
F вћЌ z m вњЏ rmsрќ”®1вњЃm z mвњЃ1 ,
F вћЌвќ‡ n вњЏ вњЃ рќ”®n rnsвќ‡ nВ 1
n рќ’«вњЃ
n
n 1
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
The reduced Heisenberg double:
в„‹рќ”® sв„“в™Ј2q вњЏ в™Јв„‚rО» sв‘Јв™ЈО» 2p вњЃ 1qqвќњ в„‚рќ”® rz, вќ‡s,
О» 2p вњЏ 1,
z p вњЏ 0,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
with the рќ’°рќ”® sв„“в™Ј2q action
вњЏ рќ”® В 1 1 О» z,
k 2 вћЌ О» вњЏ рќ”®вњЃ1 О» ,
E вћЌ z m вњЏ вњЃ рќ”®m rmsz mВ 1 , k 2 вћЌ z m вњЏ рќ”®2m z m ,
E вћЌвќ‡ n вњЏ рќ”®1вњЃn rnsвќ‡ nвњЃ1 , k 2 вћЌвќ‡ n вњЏ рќ”®вњЃ2n вќ‡ n ,
E вћЌО»
Semikhatov
F вћЌО»
вњЏ вњЃ рќ”® В рќ”® 1 вќ‡ О» ,
F вћЌ z m вњЏ rmsрќ”®1вњЃm z mвњЃ1 ,
F вћЌвќ‡ n вњЏ вњЃ рќ”®n rnsвќ‡ nВ 1
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
The reduced Heisenberg double:
в„‹рќ”® sв„“в™Ј2q вњЏ в™Јв„‚rО» sв‘Јв™ЈО» 2p вњЃ 1qqвќњ в„‚рќ”® rz, вќ‡s,
О» 2p вњЏ 1,
z p вњЏ 0,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
with the рќ’°рќ”® sв„“в™Ј2q action
вњЏ рќ”® В 1 1 О» z,
k 2 вћЌ О» вњЏ рќ”®вњЃ1 О» ,
E вћЌ z m вњЏ вњЃ рќ”®m rmsz mВ 1 , k 2 вћЌ z m вњЏ рќ”®2m z m ,
E вћЌвќ‡ n вњЏ рќ”®1вњЃn rnsвќ‡ nвњЃ1 , k 2 вћЌвќ‡ n вњЏ рќ”®вњЃ2n вќ‡ n ,
E вћЌО»
F вћЌО»
вњЏ вњЃ рќ”® В рќ”® 1 вќ‡ О» ,
F вћЌ z m вњЏ rmsрќ”®1вњЃm z mвњЃ1 ,
F вћЌвќ‡ n вњЏ вњЃ рќ”®n rnsвќ‡ nВ 1
dim в„‚рќ”® rz, вќ‡s вњЏ p2 ,
Semikhatov
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
The reduced Heisenberg double:
в„‹рќ”® sв„“в™Ј2q вњЏ в™Јв„‚rО» sв‘Јв™ЈО» 2p вњЃ 1qqвќњ в„‚рќ”® rz, вќ‡s,
О» 2p вњЏ 1,
z p вњЏ 0,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
with the рќ’°рќ”® sв„“в™Ј2q action
вњЏ рќ”® В 1 1 О» z,
k 2 вћЌ О» вњЏ рќ”®вњЃ1 О» ,
E вћЌ z m вњЏ вњЃ рќ”®m rmsz mВ 1 , k 2 вћЌ z m вњЏ рќ”®2m z m ,
E вћЌвќ‡ n вњЏ рќ”®1вњЃn rnsвќ‡ nвњЃ1 , k 2 вћЌвќ‡ n вњЏ рќ”®вњЃ2n вќ‡ n ,
dim в„‚рќ”® rz, вќ‡s вњЏ p2 ,
E вћЌО»
F вћЌО»
вњЏ вњЃ рќ”® В рќ”® 1 вќ‡ О» ,
F вћЌ z m вњЏ rmsрќ”®1вњЃm z mвњЃ1 ,
F вћЌвќ‡ n вњЏ вњЃ рќ”®n rnsвќ‡ nВ 1
algebra of “q-differential operators on a line”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
The reduced Heisenberg double:
в„‹рќ”® sв„“в™Ј2q вњЏ в™Јв„‚rО» sв‘Јв™ЈО» 2p вњЃ 1qqвќњ в„‚рќ”® rz, вќ‡s,
О» 2p вњЏ 1,
z p вњЏ 0,
вќ‡p вњЏ 0,
О» z вњЏ zО» ,
О» вќ‡ вњЏ вќ‡О» ,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
with the рќ’°рќ”® sв„“в™Ј2q action
вњЏ рќ”® В 1 1 О» z,
k 2 вћЌ О» вњЏ рќ”®вњЃ1 О» ,
E вћЌ z m вњЏ вњЃ рќ”®m rmsz mВ 1 , k 2 вћЌ z m вњЏ рќ”®2m z m ,
E вћЌвќ‡ n вњЏ рќ”®1вњЃn rnsвќ‡ nвњЃ1 , k 2 вћЌвќ‡ n вњЏ рќ”®вњЃ2n вќ‡ n ,
dim в„‚рќ”® rz, вќ‡s вњЏ p2 ,
E вћЌО»
F вћЌО»
вњЏ вњЃ рќ”® В рќ”® 1 вќ‡ О» ,
F вћЌ z m вњЏ rmsрќ”®1вњЃm z mвњЃ1 ,
F вћЌвќ‡ n вњЏ вњЃ рќ”®n rnsвќ‡ nВ 1
algebra of “q-differential operators on a line”
a рќ’°рќ”® sв„“в™Ј2q-module algebra
Semikhatov
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
z p вњЏ 0,
вќ‡p вњЏ 0,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
with the рќ’°рќ”® sв„“в™Ј2q action
E вћЌ z m вњЏ вњЃ рќ”®m rmsz mВ 1 , k 2 вћЌ z m вњЏ рќ”®2m z m ,
E вћЌвќ‡ n вњЏ рќ”®1вњЃn rnsвќ‡ nвњЃ1 ,
k 2 вћЌвќ‡ n вњЏ рќ”®вњЃ2n вќ‡ n ,
a рќ’°рќ”® sв„“в™Ј2q-module algebra isomorphism
рќ’°рќ”® sв„“в™Ј2q action on matrices:
i, j вњЃ 1
в™Ј вњЃj вњЃ1q
вњЃ рќ”®рќ”® вњЃ рќ”®вњЃ
2 i
F:
1
i, j o
F вћЌвќ‡ n вњЏ
в™Јрќ”®вњЃрќ”®вњЃ1 qвњЃ1
вњЃ рќ”®n rnsвќ‡nВ 1
в„‚рќ”® rz, вќ‡s вњ• Matp в™Јв„‚q
i вњЃ 1, j
E:
F вћЌ z m вњЏ rmsрќ”®1вњЃm z mвњЃ1 ,
i, j В 1
Semikhatov
вњЃрќ”®j вњЃ2i rj вњЃ1s G
i, j
y
рќ”®1вњЃi i
rs
i В 1, j
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
z p вњЏ 0,
вќ‡p вњЏ 0,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
a рќ’°рќ”® sв„“в™Ј2q-module algebra isomorphism
в„‚рќ”® rz, вќ‡s вњ• Matp в™Јв„‚q
As a рќ’°рќ”® sв„“в™Ј2q-module,
В В в„‚рќ”® rz, вќ‡s вњЏ рќ’«В 1 вќµ рќ’«3 вќµв�Ћв�Ћв�Ћвќµ рќ’«ОЅ .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
в„‹рќ”® sв„“в™Ј2q
z p вњЏ 0,
вќ‡p вњЏ 0,
вќ‡z вњЏ в™Јрќ”® вњЃ рќ”®вњЃ1 q1 В рќ”®вњЃ2 z вќ‡
a рќ’°рќ”® sв„“в™Ј2q-module algebra isomorphism
As a рќ’°рќ”® sв„“в™Ј2q-module,
в„‚рќ”® rz, вќ‡s вњ• Matp в™Јв„‚q
В В в„‚рќ”® rz, вќ‡s вњЏ рќ’«В 1 вќµ рќ’«3 вќµв�Ћв�Ћв�Ћвќµ рќ’«ОЅ .
Projective module structrure, рќ’«В 1:
вњЃ
pвћ¦1
1 i i
z вќ‡
r
is
i вњЏ1
z p вњЃ1
Г•
z pвњЃ2
Г• ... Г•
~
~~
~~~
z II
II
II
F I6
E
Semikhatov
1
@@
@@F
@2
вќ‡
uu
uu
u
u
zu E
Г• ... Г• вќ‡ вњЃ Г•
p 2
вќ‡pвњЃ1
Hopf algebras and the duality to logarithmic CFT
Notes
p
Г в„‹рќ”® sв„“в™Ј2q вњЏ
рќ’°рќ”® sв„“в™Ј2q вњЏ
вњЏ
n рќ’«В n вќµ
p
Г вњЏ
n рќ’«вњЃ
n
n 1
n 1
p
Г p
Г вњЏ
n 1
n рќ’«В n вќµ
вњЏ
n рќ’«вњЃ
n (regular representation)
n 1
— two algebraic structures on this sum, one a Hopf algebra, the other its
module algebra.
в„‹в™ЈB вњќ q is a Hopf algebroid over B вњќ [JH Lu,. . . ]
—“quantization” of a Poisson groupoids — not used here.
в„‹рќ”® sв„“в™Ј2q extends to a differential рќ’°рќ”® sв„“в™Ј2q-module algebra.
“Pentagon”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
p
Г в„‹рќ”® sв„“в™Ј2q вњЏ
рќ’°рќ”® sв„“в™Ј2q вњЏ
вњЏ
n рќ’«В n вќµ
p
Г вњЏ
n рќ’«вњЃ
n
n 1
n 1
p
Г p
Г вњЏ
n 1
n рќ’«В n вќµ
вњЏ
n рќ’«вњЃ
n (regular representation)
n 1
— two algebraic structures on this sum, one a Hopf algebra, the other its
module algebra.
в„‹в™ЈB вњќ q is a Hopf algebroid over B вњќ [JH Lu,. . . ]
—“quantization” of a Poisson groupoids — not used here.
в„‹рќ”® sв„“в™Ј2q extends to a differential рќ’°рќ”® sв„“в™Ј2q-module algebra.
“Pentagon”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
p
Г в„‹рќ”® sв„“в™Ј2q вњЏ
рќ’°рќ”® sв„“в™Ј2q вњЏ
вњЏ
n рќ’«В n вќµ
p
Г вњЏ
n рќ’«вњЃ
n
n 1
n 1
p
Г p
Г вњЏ
n 1
n рќ’«В n вќµ
вњЏ
n рќ’«вњЃ
n (regular representation)
n 1
— two algebraic structures on this sum, one a Hopf algebra, the other its
module algebra.
в„‹в™ЈB вњќ q is a Hopf algebroid over B вњќ [JH Lu,. . . ]
—“quantization” of a Poisson groupoids — not used here.
в„‹рќ”® sв„“в™Ј2q extends to a differential рќ’°рќ”® sв„“в™Ј2q-module algebra.
“Pentagon”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
p
Г в„‹рќ”® sв„“в™Ј2q вњЏ
рќ’°рќ”® sв„“в™Ј2q вњЏ
вњЏ
n рќ’«В n вќµ
p
Г вњЏ
n рќ’«вњЃ
n
n 1
n 1
p
Г p
Г вњЏ
n 1
n рќ’«В n вќµ
вњЏ
n рќ’«вњЃ
n (regular representation)
n 1
— two algebraic structures on this sum, one a Hopf algebra, the other its
module algebra.
в„‹в™ЈB вњќ q is a Hopf algebroid over B вњќ [JH Lu,. . . ]
—“quantization” of a Poisson groupoids — not used here.
в„‹рќ”® sв„“в™Ј2q extends to a differential рќ’°рќ”® sв„“в™Ј2q-module algebra.
“Pentagon”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
p
Г в„‹рќ”® sв„“в™Ј2q вњЏ
рќ’°рќ”® sв„“в™Ј2q вњЏ
вњЏ
n рќ’«В n вќµ
p
Г вњЏ
n рќ’«вњЃ
n
n 1
n 1
p
Г p
Г вњЏ
n 1
n рќ’«В n вќµ
вњЏ
n рќ’«вњЃ
n (regular representation)
n 1
— two algebraic structures on this sum, one a Hopf algebra, the other its
module algebra.
в„‹в™ЈB вњќ q is a Hopf algebroid over B вњќ [JH Lu,. . . ]
—“quantization” of a Poisson groupoids — not used here.
в„‹рќ”® sв„“в™Ј2q extends to a differential рќ’°рќ”® sв„“в™Ј2q-module algebra.
“Pentagon”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
p
Г в„‹рќ”® sв„“в™Ј2q вњЏ
рќ’°рќ”® sв„“в™Ј2q вњЏ
вњЏ
n рќ’«В n вќµ
p
Г вњЏ
n рќ’«вњЃ
n
n 1
n 1
p
Г p
Г вњЏ
n 1
n рќ’«В n вќµ
вњЏ
n рќ’«вњЃ
n (regular representation)
n 1
— two algebraic structures on this sum, one a Hopf algebra, the other its
module algebra.
в„‹в™ЈB вњќ q is a Hopf algebroid over B вњќ [JH Lu,. . . ]
—“quantization” of a Poisson groupoids — not used here.
в„‹рќ”® sв„“в™Ј2q extends to a differential рќ’°рќ”® sв„“в™Ј2q-module algebra.
“Pentagon”
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Differential algebra
О©в„‹рќ”® sв„“в™Ј2q, de Rham complex of в„‹рќ”® sв„“в™Ј2q:
dz 2 вњЏ 0, d вќ‡ 2 вњЏ 0, d вќ‡ dz вњЏ вњЃрќ”®вњЃ2 dz d вќ‡ ,
dz z вњЏ рќ”®вњЃ2 z dz,
dz вќ‡ вњЏ рќ”®2 вќ‡ dz,
E вћЌ dz вњЏ вњЃr2sz dz,
E вћЌ d вќ‡ вњЏ 0,
k 2 вћЌ dz вњЏ рќ”®2 dz,
k вћЌ d вќ‡ вњЏ рќ”®вњЃ2 d вќ‡ ,
2
Semikhatov
d вќ‡вќ‡ вњЏ рќ”®2 вќ‡ d вќ‡ ,
d вќ‡ z вњЏ рќ”®вњЃ2 z d вќ‡ .
F вћЌ dz вњЏ 0,
F вћЌ d вќ‡ вњЏ вњЃрќ”®2 r2sвќ‡ d вќ‡
Hopf algebras and the duality to logarithmic CFT
Differential algebra
О©в„‹рќ”® sв„“в™Ј2q, de Rham complex of в„‹рќ”® sв„“в™Ј2q:
dz 2 вњЏ 0, d вќ‡ 2 вњЏ 0, d вќ‡ dz вњЏ вњЃрќ”®вњЃ2 dz d вќ‡ ,
dz z вњЏ рќ”®вњЃ2 z dz,
dz вќ‡ вњЏ рќ”®2 вќ‡ dz,
E вћЌ dz вњЏ вњЃr2sz dz,
E вћЌ d вќ‡ вњЏ 0,
k 2 вћЌ dz вњЏ рќ”®2 dz,
k вћЌ d вќ‡ вњЏ рќ”®вњЃ2 d вќ‡ ,
2
d вќ‡вќ‡ вњЏ рќ”®2 вќ‡ d вќ‡ ,
d вќ‡ z вњЏ рќ”®вњЃ2 z d вќ‡ .
F вћЌ dz вњЏ 0,
F вћЌ d вќ‡ вњЏ вњЃрќ”®2 r2sвќ‡ d вќ‡
dО» commutes with z and вќ‡ and anticommutes with dz and d вќ‡ ,
dО» О»
вњЏ рќ”®вњЃ1 О» dО»
(whence, in particular, d в™ЈО» 2n q вњЏ 0),
О» commutes with dz and d вќ‡ .
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Differential algebra
О©в„‹рќ”® sв„“в™Ј2q, de Rham complex of в„‹рќ”® sв„“в™Ј2q:
dz 2 вњЏ 0, d вќ‡ 2 вњЏ 0, d вќ‡ dz вњЏ вњЃрќ”®вњЃ2 dz d вќ‡ ,
dz z вњЏ рќ”®вњЃ2 z dz,
dz вќ‡ вњЏ рќ”®2 вќ‡ dz,
E вћЌ dz вњЏ вњЃr2sz dz,
E вћЌ d вќ‡ вњЏ 0,
k 2 вћЌ dz вњЏ рќ”®2 dz,
k вћЌ d вќ‡ вњЏ рќ”®вњЃ2 d вќ‡ ,
2
d вќ‡вќ‡ вњЏ рќ”®2 вќ‡ d вќ‡ ,
d вќ‡ z вњЏ рќ”®вњЃ2 z d вќ‡ .
F вћЌ dz вњЏ 0,
F вћЌ d вќ‡ вњЏ вњЃрќ”®2 r2sвќ‡ d вќ‡
dО» commutes with z and вќ‡ and anticommutes with dz and d вќ‡ ,
dО» О»
вњЏ рќ”®вњЃ1 О» dО»
(whence, in particular, d в™ЈО» 2n q вњЏ 0),
О» commutes with dz and d вќ‡ .
Then the рќ’°рќ”® sв„“в™Ј2q action
E вћЌ dО» вњЏ
в™Ј
1
z dО»
рќ”®В 1
В О» dz q,
k 2 вћЌ dО» вњЏ рќ”®вњЃ1 dО» ,
F вћЌ dО» вњЏ вњЃ
рќ”®
рќ”®В 1
в™Јвќ‡ dО» В О» d вќ‡q
endows О©в„‹рќ”® sв„“в™Ј2q with the structure of a differential рќ’°рќ”® sв„“в™Ј2q-module algebra.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
YB and pentagon:
As vector spaces,
рќ’џв™ЈB q вњ• B вњќ вќњ B вњ• в„‹в™ЈB вњќ q.
The canonical element
рќ’џв™ЈB qвќњ рќ’џв™ЈB q в—— R вњЏ
вћі
в™ЈОµ вќњ eJ qвќњв™ЈeJ вќњ 1q вњЏ W ВЂ в„‹в™ЈBвњќqвќњ в„‹в™ЈBвњќq
J
Yang–Baxter equation:
R12 R13 R23 вњЏ R23 R13 R12
Semikhatov
Pentagon equation:
W12 W13 W23 вњЏ W23 W12
Hopf algebras and the duality to logarithmic CFT
YB and pentagon:
As vector spaces,
рќ’џв™ЈB q вњ• B вњќ вќњ B вњ• в„‹в™ЈB вњќ q.
The canonical element
рќ’џв™ЈB qвќњ рќ’џв™ЈB q в—— R вњЏ
вћі
в™ЈОµ вќњ eJ qвќњв™ЈeJ вќњ 1q вњЏ W ВЂ в„‹в™ЈBвњќqвќњ в„‹в™ЈBвњќq
J
Yang–Baxter equation:
R12 R13 R23 вњЏ R23 R13 R12
Semikhatov
Pentagon equation:
W12 W13 W23 вњЏ W23 W12
Hopf algebras and the duality to logarithmic CFT
YB and pentagon:
As vector spaces,
рќ’џв™ЈB q вњ• B вњќ вќњ B вњ• в„‹в™ЈB вњќ q.
The canonical element
рќ’џв™ЈB qвќњ рќ’џв™ЈB q в—— R вњЏ
вћі
в™ЈОµ вќњ eJ qвќњв™ЈeJ вќњ 1q вњЏ W ВЂ в„‹в™ЈBвњќqвќњ в„‹в™ЈBвњќq
J
Yang–Baxter equation:
R12 R13 R23 вњЏ R23 R13 R12
Semikhatov
Pentagon equation:
W12 W13 W23 вњЏ W23 W12
Hopf algebras and the duality to logarithmic CFT
YB and pentagon:
As vector spaces,
рќ’џв™ЈB q вњ• B вњќ вќњ B вњ• в„‹в™ЈB вњќ q.
The canonical element
рќ’џв™ЈB qвќњ рќ’џв™ЈB q в—— R вњЏ
вћі
в™ЈОµ вќњ eJ qвќњв™ЈeJ вќњ 1q вњЏ W ВЂ в„‹в™ЈBвњќqвќњ в„‹в™ЈBвњќq
J
Yang–Baxter equation:
R12 R13 R23 вњЏ R23 R13 R12
Semikhatov
Pentagon equation:
W12 W13 W23 вњЏ W23 W12
Hopf algebras and the duality to logarithmic CFT
YB and pentagon:
As vector spaces,
рќ’џв™ЈB q вњ• B вњќ вќњ B вњ• в„‹в™ЈB вњќ q.
The canonical element
рќ’џв™ЈB qвќњ рќ’џв™ЈB q в—— R вњЏ
вћі
в™ЈОµ вќњ eJ qвќњв™ЈeJ вќњ 1q вњЏ W ВЂ в„‹в™ЈBвњќqвќњ в„‹в™ЈBвњќq
J
Yang–Baxter equation:
R12 R13 R23 вњЏ R23 R13 R12
Semikhatov
Pentagon equation:
W12 W13 W23 вњЏ W23 W12
Hopf algebras and the duality to logarithmic CFT
From fermions to parafermions
p вњЏ 2, the simplest case, рќ”® вњЏ eiПЂ в‘Ј2 :
z 2 вњЏ 0,
E вћЌ z вњЏ 0,
E вћЌвќ‡ вњЏ 1,
вќ‡2 вњЏ 0,
вќ‡ z В z вќ‡ вњЏ 2i
k 2 вћЌ z вњЏ вњЃ z, F вћЌ z вњЏ 1,
k 2 вћЌвќ‡ вњЏ вњЃвќ‡ , F вћЌвќ‡ вњЏ 0
Free fermions:
Semikhatov
Hopf algebras and the duality to logarithmic CFT
From fermions to parafermions
p вњЏ 2, the simplest case, рќ”® вњЏ eiПЂ в‘Ј2 :
z 2 вњЏ 0,
вќ‡2 вњЏ 0,
вќ‡ z В z вќ‡ вњЏ 2i
finite-dimensional counterpart of free fermions, which are known to describe
the в™Јp вњЏ 2, 1q logarithmic conformal field model, c вњЏ вњЃ2.
(general в™Јp, 1q models: c вњЏ 13 вњЃ 6p вњЃ p )
6
Free fermions:
Semikhatov
Hopf algebras and the duality to logarithmic CFT
From fermions to parafermions
p вњЏ 2, the simplest case, рќ”® вњЏ eiПЂ в‘Ј2 :
z 2 вњЏ 0,
вќ‡2 вњЏ 0,
вќ‡ z В z вќ‡ вњЏ 2i
finite-dimensional counterpart of free fermions, which are known to describe
the в™Јp вњЏ 2, 1q logarithmic conformal field model, c вњЏ вњЃ2.
(general в™Јp, 1q models: c вњЏ 13 вњЃ 6p вњЃ p )
6
Free fermions:
Оѕ в™Јu q О· в™Јv q вњЏ u вњЃ v ,
u, v
Virasoro with central charge c вњЏ вњЃ2:
T в™Јu q вњЏ вњЃО· в™Јu qвќ‡ Оѕ в™Јu q.
Screening:
вћЅ
1
E
вњЏ
Semikhatov
О·
ВЂ в„‚.
вњЏ О·0 .
Hopf algebras and the duality to logarithmic CFT
From fermions to parafermions
Free fermions:
Оѕ в™Јu q О· в™Јv q вњЏ u вњЃ v ,
u, v
Virasoro with central charge c вњЏ вњЃ2:
T в™Јu q вњЏ вњЃО· в™Јu qвќ‡ Оѕ в™Јu q.
Screening:
вћЅ
1
E
вњЏ
О·
ВЂ в„‚.
вњЏ О·0 .
The complex of (Feigin–Fuchs) Virasoro modules:
o
вњЊy o
1
вњ†
О·
вњЊy o
вњ†
вњЊy o
вќ‡Оѕ Оѕ
вќ‡О·О·
вњЊy o
вњ†
вњЊy o
вњ†
вњЊy o
вњ†
вњЊ o
вњ†
вњЊ o
вњ†
О·Оѕ
Semikhatov
Оѕ
вњ†
вќ‡2 Оѕ вќ‡ Оѕ
вќ‡2 Оѕ вќ‡ Оѕ Оѕ
вњ†
вњЊ o
Hopf algebras and the duality to logarithmic CFT
вњ†
From fermions to parafermions
Free fermions:
The complex of (Feigin–Fuchs) Virasoro modules:
вќ‡О·О·
вњЊy o
вњЊy o
1
вњ†
О·
вњЊy o
вњ†
вњЊy o
вќ‡Оѕ Оѕ
вњ†
вњЊy o
вњ†
вњЊy o
Оѕ
О·Оѕ
вњЊ o
вњ†
вњЊ o
вњ†
Extension: Оґ в™Јu q вњЏ вќ‡ вњЃ1 О· в™Јu q,
Оѕ в™Јu qОґ в™Јv q вњЏ logв™Јu вњЃ v q
вќ‡вњЃ1 О· в™Јuq
o
вњ†
вњ†
вќ‡2 Оѕ вќ‡ Оѕ
вќ‡2 Оѕ вќ‡ Оѕ Оѕ
вњ†
вњЊ o
LвњЃ 1
О· в™Јu q
Semikhatov
Hopf algebras and the duality to logarithmic CFT
вњ†
From fermions to parafermions
Free fermions:
The complex of (Feigin–Fuchs) Virasoro modules:
o
Then
вњЊy o
1
вњ†
О·
вњЊy o
вњ†
вњЊy o
вќ‡Оѕ Оѕ
вќ‡О·О·
вњЊy o
вњ†
вњЊy o
вњ†
вњЊy o
вњ†
вњЊ o
вњ†
вњЊ o
вњ†
О·Оѕ
Оѕ
вњ†
вќ‡2 Оѕ вќ‡ Оѕ
вќ‡2 Оѕ вќ‡ Оѕ Оѕ
вњ†
вњЊ o
bbbbbbb Оґ в™Јu q Оѕ в™Јu q \\\\\\\\\\\\\\F
]]bb
aa Оѕ в™Јu q
]]bb
]]]]]]]]]]]]F paaaaaaaaaaaaaaa
Оґ в™Јu q pb]bb
1
Semikhatov
Hopf algebras and the duality to logarithmic CFT
вњ†
p вњЏ 2, the simplest case:
Free fermions:
Bosonization:
Оѕ в™Јu q вњЏ eП• в™Јu q ,
then
О· в™Јu q вњЏ eвњЃП• в™Јu q ,
Оґ в™Јu qвќ‡ Оѕ в™Јu q
NNN
NNN
9
вќ‡П• в™Јuq вњЏ вњЃО· в™ЈuqОѕ в™Јuq,
e2П• в™Јu q
t
t
y tt
t
вќ‡ Оѕ в™Јu q
Alternative bosonization: introduce the scalar field as
вќ‡П† в™Јuq вњЏ Оґ в™Јuqвќ‡Оѕ в™Јuq,
e2П† в™Јu qI
III
I6
Semikhatov
О· в™Јu q
О· в™Јu qОѕ в™Јu q
qq
q
q
q
x
Hopf algebras and the duality to logarithmic CFT
p вњЏ 2, the simplest case:
Free fermions:
Bosonization:
Оѕ в™Јu q вњЏ eП• в™Јu q ,
then
О· в™Јu q вњЏ eвњЃП• в™Јu q ,
Оґ в™Јu qвќ‡ Оѕ в™Јu q
NNN
NNN
9
вќ‡П• в™Јuq вњЏ вњЃО· в™ЈuqОѕ в™Јuq,
e2П• в™Јu q
t
t
y tt
t
вќ‡ Оѕ в™Јu q
Alternative bosonization: introduce the scalar field as
вќ‡П† в™Јuq вњЏ Оґ в™Јuqвќ‡Оѕ в™Јuq,
e2П† в™Јu qI
III
I6
Semikhatov
О· в™Јu q
О· в™Јu qОѕ в™Јu q
qq
q
q
q
x
Hopf algebras and the duality to logarithmic CFT
From p вњЏ 2 to general p:
From “fermionic statistics”
вќ‡ z В z вќ‡ вњЏ 2i
to “anyonic” (“parafermionic”) statistics
вќ‡
m
z
n
вњЏ
вћі
вћ™
j в™Јj вњЃ1q
рќ”®в™Јj вњЃ2mqnВ jmвњЃ 2
j 0
Г•
m
j
В n
j ! рќ”®
j
r s вњЃ рќ”®вњЃ1
вњџj nвњЃj mвњЃj
z
вќ‡
вњЃ
pвћ¦1
1 i i
z вќ‡
r
is
i вњЏ1
0-form diagram:
z pвњЃ1
вњ’ вњљвњ’ вњљ
z pвњЃ2
1-form diagrams:
Г• ... Г•
{
{{
}{{
z II
II
II
F I6
E
1
Оґ в™Јu qвќ‡ Оѕ в™Јu q
II
I6
CC
CCF
C3
uвќ‡
uu
u
u
u
zu E
Г• ... Г• вќ‡ вњЃ Г•
p 2
вќ‡pвњЃ1
e2П• в™Јu q
}
}
~}
вќ‡ Оѕ в™Јu q
e2П† в™Јu>q
>>
0
Semikhatov
О· в™Јu qОѕ в™Јu q
x algebras and the duality to logarithmic CFT
|xxHopf
From p вњЏ 2 to general p:
From “fermionic statistics”
вќ‡ z В z вќ‡ вњЏ 2i
to “anyonic” (“parafermionic”) statistics
вќ‡m z n вњЏ
вћі
вћ™
рќ”®в™Јj вњЃ2mqnВ jmвњЃ
в™Ј вњЃ1q вњ’mвњљвњ’nвњљ
j j
2
j
j 0
j
В rj s! рќ”® вњЃ рќ”®вњЃ1
вњџj nвњЃj mвњЃj
z
вќ‡
0-form diagram:
bbbbbbb Оґ в™Јu q Оѕ в™Јu q \\\\\\\\\\\\\\F
]]bb
aa Оѕ в™Јu q
]]bb
]]]]]]]]]]]]F paaaaaaaaaaaaaaa
Оґ в™Јu q pb]bb
1
1-form diagrams:
Оґ в™Јu qвќ‡ Оѕ в™Јu q
II
I6
e2П• в™Јu q
}
}
~}
вќ‡ Оѕ в™Јu q
О· в™Јu qОѕ в™Јu q
e2П† в™Јu>q
>>
0
|xxx
О· в™Јu q
Semikhatov
Hopf algebras and the duality to logarithmic CFT
From p вњЏ 2 to general p:
From “fermionic statistics”
вќ‡ z В z вќ‡ вњЏ 2i
to “anyonic” (“parafermionic”) statistics
вќ‡
m
z
n
вњЏ
вћі
вћ™
j в™Јj вњЃ1q
рќ”®в™Јj вњЃ2mqnВ jmвњЃ 2
j 0
Г•
m
j
В n
j ! рќ”®
j
r s вњЃ рќ”®вњЃ1
вњџj nвњЃj mвњЃj
z
вќ‡
вњЃ
pвћ¦1
1 i i
z вќ‡
r
is
i вњЏ1
0-form diagram:
z pвњЃ1
вњ’ вњљвњ’ вњљ
z pвњЃ2
1-form diagrams:
Г• ... Г•
{
{{
}{{
z II
II
II
F I6
E
1
Оґ в™Јu qвќ‡ Оѕ в™Јu q
II
I6
CC
CCF
C3
uвќ‡
uu
u
u
u
zu E
Г• ... Г• вќ‡ вњЃ Г•
p 2
вќ‡pвњЃ1
e2П• в™Јu q
}
}
~}
вќ‡ Оѕ в™Јu q
e2П† в™Јu>q
>>
0
Semikhatov
О· в™Јu qОѕ в™Јu q
x algebras and the duality to logarithmic CFT
|xxHopf
From p вњЏ 2 to general p:
“anyonic” (“parafermionic”) statistics
вќ‡m z n вњЏ
вћі
вћ™
рќ”®в™Јj вњЃ2mqnВ jmвњЃ
j 0
Г•
j
j
В rj s! рќ”® вњЃ рќ”®вњЃ1
вњџj nвњЃj mвњЃj
z
вќ‡
вњЃ
1 i i
z вќ‡
r
is
i вњЏ1
z pвњЃ2
Г• ... Г•
1-form diagrams:
Оґ в™Јu qвќ‡ Оѕ в™Јu q
II
I6
2
pвћ¦1
0-form diagram:
z pвњЃ1
в™Ј вњЃ1q вњ’mвњљвњ’nвњљ
j j
в™Јq
e2П• u
}
~}}
вќ‡ Оѕ в™Јu q
вњЃ
pвћі1
вњЏ
i 1
{
{{
}{{
z II
II
II
F I6
E
1
CC
CCF
C3
вќ‡
uu
uu
u
zuu E
Г• ... Г• вќ‡ вњЃ Г•
p 2
1 i
i
ri s z d в™Јвќ‡ q
EE F
EE
E4
e2П† в™Јu>q
>>
Semikhatov
0
вќ‡pвњЃ1
вќ‡ p вњЃ1 d вќ‡
zz
zz
z
|z
вќ‡pвњЃ2 d вќ‡
E
dвќ‡
Г• вќ‡ dвќ‡ Г• ... Г•
О· в™Јu qОѕ в™Јu q
x algebras and the duality to logarithmic CFT
|xxHopf
From p вњЏ 2 to general p:
вњЃ
pвћ¦1
1 i i
z вќ‡
r
is
i вњЏ1
0-form diagram:
Г•
z pвњЃ1
z pвњЃ2
Г• ... Г•
1-form diagrams:
Оґ в™Јu qвќ‡ Оѕ в™Јu q
II
I6
в™Јq
e2П• u
}
}
~}
вќ‡Оѕ в™Јuq
вњЃ
вњЃ
pвћі1
вњЏ
i 1
{
{{
}{{
z II
II
II
F I6
E
1
CC
CCF
C3
вќ‡
u
uu
uuE
u
u
z
p 2
1 i
i
ri s z d в™Јвќ‡ q
EE F
EE
E4
zz
zz
z
|z
вќ‡pвњЃ2 d вќ‡
E
dвќ‡
Г• вќ‡ dвќ‡ Г• ... Г•
вњЃ
pвћі1
вњЏ
i 1
E
Г• . . . Г• z dz Г• dz
Semikhatov
вќ‡pвњЃ1
вќ‡ p вњЃ1 d вќ‡
z p 1 dz
DD
DDF
DD
4
z pвњЃ2 dz
Г• ... Г• вќ‡ вњЃ Г•
y
yy
|yy
ri s d в™Јz qвќ‡
1
i
i
О· в™Јu qОѕ в™Јu q
e2П† в™Јu>q
>>
0
|xxx
О· в™Јu q
Hopf algebras and the duality to logarithmic CFT
From p вњЏ 2 to general p:
вњЃ
pвћ¦1
1 i i
z вќ‡
r
is
i вњЏ1
0-form diagram:
Г•
z pвњЃ1
z pвњЃ2
Г• ... Г•
1-form diagrams:
Оґ в™Јu qвќ‡ Оѕ в™Јu q
II
I6
в™Јq
e2П• u
}
}
~}
вќ‡Оѕ в™Јuq
вњЃ
вњЃ
pвћі1
вњЏ
i 1
{
{{
}{{
z II
II
II
F I6
E
1
CC
CCF
C3
вќ‡
u
uu
uuE
u
u
z
p 2
1 i
i
ri s z d в™Јвќ‡ q
EE F
EE
E4
zz
zz
z
|z
вќ‡pвњЃ2 d вќ‡
E
dвќ‡
Г• вќ‡ dвќ‡ Г• ... Г•
вњЃ
pвћі1
вњЏ
i 1
E
Г• . . . Г• z dz Г• dz
Semikhatov
вќ‡pвњЃ1
вќ‡ p вњЃ1 d вќ‡
z p 1 dz
DD
DDF
DD
4
z pвњЃ2 dz
Г• ... Г• вќ‡ вњЃ Г•
y
yy
|yy
ri s d в™Јz qвќ‡
1
i
i
О· в™Јu qОѕ в™Јu q
e2П† в™Јu>q
>>
0
|xxx
О· в™Јu q
Hopf algebras and the duality to logarithmic CFT
Notes
OPE:
вќ‡m в™Јuq z n в™Јv q вњЏ rmsОґ m,n logв™Јu вњЃ v q.
Logarithmic partner of the identity operator:
вњЃ
i
i
ri s z в™Јu qвќ‡ в™Јu q
i вњЏ1
I
pвћ¦1 1
z p вњЃ1 в™Ј u q
Г• z вњЃ в™Јu q Г• . . . Г•
p 2
uu
zuuu
z в™Јu q N
NNN
NNN
F
N9
E
IIIF
I6
1
pp
ppEp
p
p
wp
вќ‡в™Јuq Г• . . . Г• вќ‡pвњЃ2 в™Јuq Г• вќ‡pвњЃ1 в™Јuq
One of the two currents:
рќ’Ґв™Ј u q вњ‘
вњЃ
pвћі1
вњЏ
рќ”®nвњЃ1 z n в™Јu qd вќ‡ n в™Јu q
n 1
рќ’Ґв™Јu q рќ’Ґв™Јv q вњЏ
1
в™Јu вњЃ v q2 .
TTTT
TTTFT
TA
e
qq
xqqq
вќ„2p П• в™Јuq
E
d вќ‡ 1 в™Јu q
Г• . . . Г• d вќ‡ вњЃ в™Јu q,
p 1
I need the understanding of what property — kind of
“braided commutativity” — replaces quantum commutativity. OR, what is an
abstract “OPE-sttructure” on 𝒰𝔮 sℓ♣2q-module algebras.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
OPE:
вќ‡m в™Јuq z n в™Јv q вњЏ rmsОґ m,n logв™Јu вњЃ v q.
Logarithmic partner of the identity operator:
вњЃ
i
i
ri s z в™Јu qвќ‡ в™Јu q
i вњЏ1
I
pвћ¦1 1
z p вњЃ1 в™Ј u q
Г• z вњЃ в™Јu q Г• . . . Г•
p 2
uu
zuuu
z в™Јu q N
NNN
NNN
F
N9
E
IIIF
I6
1
pp
ppEp
p
p
wp
вќ‡в™Јuq Г• . . . Г• вќ‡pвњЃ2 в™Јuq Г• вќ‡pвњЃ1 в™Јuq
One of the two currents:
рќ’Ґв™Ј u q вњ‘
вњЃ
pвћі1
вњЏ
рќ”®nвњЃ1 z n в™Јu qd вќ‡ n в™Јu q
n 1
рќ’Ґв™Јu q рќ’Ґв™Јv q вњЏ
1
в™Јu вњЃ v q2 .
TTTT
TTTFT
TA
e
qq
xqqq
вќ„2p П• в™Јuq
E
d вќ‡ 1 в™Јu q
Г• . . . Г• d вќ‡ вњЃ в™Јu q,
p 1
I need the understanding of what property — kind of
“braided commutativity” — replaces quantum commutativity. OR, what is an
abstract “OPE-sttructure” on 𝒰𝔮 sℓ♣2q-module algebras.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
OPE:
вќ‡m в™Јuq z n в™Јv q вњЏ rmsОґ m,n logв™Јu вњЃ v q.
Logarithmic partner of the identity operator:
вњЃ
i
i
ri s z в™Јu qвќ‡ в™Јu q
i вњЏ1
I
pвћ¦1 1
z p вњЃ1 в™Ј u q
Г• z вњЃ в™Јu q Г• . . . Г•
p 2
uu
zuuu
z в™Јu q N
NNN
NNN
F
N9
E
IIIF
I6
1
pp
ppEp
p
p
wp
вќ‡в™Јuq Г• . . . Г• вќ‡pвњЃ2 в™Јuq Г• вќ‡pвњЃ1 в™Јuq
One of the two currents:
рќ’Ґв™Ј u q вњ‘
вњЃ
pвћі1
вњЏ
рќ”®nвњЃ1 z n в™Јu qd вќ‡ n в™Јu q
n 1
рќ’Ґв™Јu q рќ’Ґв™Јv q вњЏ
1
в™Јu вњЃ v q2 .
TTTT
TTTFT
TA
e
qq
xqqq
вќ„2p П• в™Јuq
E
d вќ‡ 1 в™Јu q
Г• . . . Г• d вќ‡ вњЃ в™Јu q,
p 1
I need the understanding of what property — kind of
“braided commutativity” — replaces quantum commutativity. OR, what is an
abstract “OPE-sttructure” on 𝒰𝔮 sℓ♣2q-module algebras.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Notes
OPE:
вќ‡m в™Јuq z n в™Јv q вњЏ rmsОґ m,n logв™Јu вњЃ v q.
Logarithmic partner of the identity operator:
вњЃ
i
i
ri s z в™Јu qвќ‡ в™Јu q
i вњЏ1
I
pвћ¦1 1
z p вњЃ1 в™Ј u q
Г• z вњЃ в™Јu q Г• . . . Г•
p 2
uu
zuuu
z в™Јu q N
NNN
NNN
F
N9
E
IIIF
I6
1
pp
ppEp
p
p
wp
вќ‡в™Јuq Г• . . . Г• вќ‡pвњЃ2 в™Јuq Г• вќ‡pвњЃ1 в™Јuq
One of the two currents:
рќ’Ґв™Ј u q вњ‘
вњЃ
pвћі1
вњЏ
рќ”®nвњЃ1 z n в™Јu qd вќ‡ n в™Јu q
n 1
рќ’Ґв™Јu q рќ’Ґв™Јv q вњЏ
1
в™Јu вњЃ v q2 .
TTTT
TTTFT
TA
e
qq
xqqq
вќ„2p П• в™Јuq
E
d вќ‡ 1 в™Јu q
Г• . . . Г• d вќ‡ вњЃ в™Јu q,
p 1
I need the understanding of what property — kind of
“braided commutativity” — replaces quantum commutativity. OR, what is an
abstract “OPE-sttructure” on 𝒰𝔮 sℓ♣2q-module algebras.
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Conclusion
Duality to LCFT is based on the pair
в™Јрќ’џв™ЈBq, в„‹в™ЈBвњќqq
в™ЈHopf algebra, its module algebraq
рќ’џв™ЈB q: well known by now (FGST(2005))
в„‹в™ЈB вњќ q: new
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Conclusion
Duality to LCFT is based on the pair
в™Јрќ’џв™ЈBq, в„‹в™ЈBвњќqq
в™ЈHopf algebra, its module algebraq
рќ’џв™ЈB q: well known by now (FGST(2005))
в„‹в™ЈB вњќ q: new
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Conclusion
Duality to LCFT is based on the pair
в™Јрќ’џв™ЈBq, в„‹в™ЈBвњќqq
в™ЈHopf algebra, its module algebraq
рќ’џв™ЈB q: well known by now (FGST(2005))
в„‹в™ЈB вњќ q: new
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Conclusion
Duality to LCFT is based on the pair
в™Јрќ’џв™ЈBq, в„‹в™ЈBвњќqq
в™ЈHopf algebra, its module algebraq
рќ’џв™ЈB q: well known by now (FGST(2005))
в„‹в™ЈB вњќ q: new
“Use:”
Helps study Logarithmic CFTs
Motivates studies of some classes of quantum groups and related
structures
Semikhatov
Hopf algebras and the duality to logarithmic CFT
My Feeling:
Semikhatov
Hopf algebras and the duality to logarithmic CFT
My Feeling:
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Engineers’ approach:
W в™ЈpВ вњЏ 2, pвњЃ вњЏ 3q:
WВ вњЏ
вњЃ
35 В 4 вњџ2 56 5
28 6
8 7
280 В 3 вњџ2
П•
П• 3П•
П• 2П•
П• П•
П•
27
27
27
27
9 3
280 4
56 5
28 6 В 70 4 В 2 вњџ2
П•
П•
П• 3П• П•
П• 2П• П•
П•
3 3
9 3
3 3
9 3
В вњџ
В В В вњџ
35 2 4 280 3
280 3 2
140 4
2
2
П• 2П•
П•
П• 2П•
П•
П•
П•
3
3
9
3
В вњџ2
56 5 В вњџ3 140 В 2 вњџ3 В вњџ2
560 3
70 4 В 2
вќ‡
вњЃ вќ„ вќ‡
В вќ‡ вќ‡ В вќ‡ вќ‡ В вќ‡ вќ‡ вњЃ вќ„ вќ‡
вќ‡
вњЃ вќ„ вќ‡ вќ‡ вќ‡ вњЃ вќ„ вќ‡ вќ‡ вќ‡ вњЃ вќ„ вќ‡
вќ‡2 П•
вќ‡П•
вњџ2
вњџ
В вќ‡ q В вќ‡ вќ‡ вќ‡ В вќ‡ q вќ‡ В вќ‡ вќ‡ в™Јвќ‡П• 2
вњџ
В 9 вќ‡ П• вќ‡П• вњЃ вќ„3 вќ‡ П• вќ‡П• вњЃ 3вќ„3 вќ‡ П• вќ‡ П• вќ‡П• вњЃ 3вќ„3 вќ‡ П• вќ‡П• 4
В вњџ В вњџ
В В вњџ
В В 70 вќ‡2 П• 2 вќ‡П• 4 В 563 вќ‡3 П• вќ‡П• q5 вњЃ вќ„283 вќ‡2 П• вќ‡П• 6 В вќ‡П• q8 вњЃ 271вќ„3 вќ‡8 П•
Semikhatov
вњ вќ„3П•
e2
Hopf algebras and the duality to logarithmic CFT
,
Engineers’ approach:
W в™ЈpВ вњЏ 2, pвњЃ вњЏ 3q:
WвњЃ
вњЏ
вњЃ
В 5 вњџ2
П•
2653 6
23 7
11
1
вњЃ 3456
вќ‡ П• вќ‡4 П• вњЃ 384
вќ‡ П• вќ‡3 П• вњЃ 1152
вќ‡8 П• вќ‡2 П• вњЃ 768
вќ‡9 П• вќ‡П•
В вњџ
В вњџ
13475
2555
вќ„3 вќ‡4 П• вќ‡3 П• 2 вњЃ 576
вќ„3 вќ‡4 П• 2 вќ‡2 П• В 642695
вќ„3 вќ‡5 П• вќ‡3 П• вќ‡2 П• В 192
вќ„3 вќ‡ 5 П• вќ‡ 4 П• вќ‡ П•
вњЃ 641225
В вњџ
В вњџ
1351
2891
вќ„3 вќ‡6 П• вќ‡2 П• 2 вњЃ 192
вќ„3 вќ‡6 П• вќ‡3 П• вќ‡П• вњЃ 192103вќ„3 вќ‡7 П• вќ‡2 П• вќ‡П• вњЃ 38413вќ„3 вќ‡8 П• вќ‡П• 2
вњЃ 576
В 3 вњџ2 В 2 вњџ2 735 В 3 вњџ3
В 2 вњџ3 245 4
4
3
2
В 3535
вќ‡ П• вњЃ 16 вќ‡ П• вќ‡П• вњЃ 3395
32 вќ‡ П•
54 вќ‡ П• вќ‡ П• В 24 вќ‡ П• вќ‡ П• вќ‡ П• вќ‡ П•
В вњџ
В вњџ
В вњџ
В вњџ
2
2
2
2
4
5
2
5
3
В 12635
вќ‡П• В 245
вќ‡П• В 105
576 вќ‡ П•
12 вќ‡ П• вќ‡ П•
32 вќ‡ П• вќ‡ П• вќ‡ П•
В вњџ2 19 7 В вњџ3
В 2 вњџ5
В 2
13405
2443 6
вќ„3 вќ‡3 П• вќ‡2 П• q3 вќ‡П•
вњЃ 288 вќ‡ П• вќ‡ П• вќ‡П• вњЃ 96 вќ‡ П• вќ‡П• вњЃ 144вќ„3 вќ‡ П• В 248225
вќ„
В вњџ
вќ„3 вќ‡ 4 П• вќ‡ 3 П• вќ‡ П• 3
вњЃ 1054 3 В вќ‡3 П• вњџ2 вќ‡2 П• В вќ‡П• вњџ2 В 24665вќ„3 вќ‡4 П• В вќ‡2 П• вњџ2 В вќ‡П• вњџ2 В 2245
В вњџ
В вњџ
В 2 вњџ4 В вњџ2 385 3 В 2 вњџ2 В вњџ3
вќ„3 вќ‡5 П• вќ‡2 П• вќ‡П• 3 вњЃ 2491вќ„3 вќ‡6 П• вќ‡П• 4 В 16205
вќ‡П• В 4 вќ‡ П• вќ‡ П• вќ‡П•
вњЃ 8245
144 вќ‡ П•
В вњџ В вњџ
В вњџ
В вњџ
В вњџ
В 2
4
4
3
4
2
5
вќ„3 вќ‡ 2 П• 3 вќ‡ П• 4
В 525
вќ‡П• В 353 вќ‡ П• вќ‡ П• вќ‡П• вњЃ 7 вќ‡ П• вќ‡П• вњџ5 В 3665
8 вќ‡ П•
вќ„
В 2 вњџ2 В вњџ6
В 1052 3 вќ‡3 П• вќ‡2 П• В вќ‡П• вњџ5 вњЃ 335вќ„3 вќ‡4 П• В вќ‡П• вњџ6 В 455
вќ‡П• В 5 вќ‡3 П• В вќ‡П• вњџ7
6 вќ‡ П•
вњ вќ„
В вќ„253 вќ‡2 П• В вќ‡П• вњџ8 В В вќ‡П• вњџ10 вњЃ 138241 вќ„3 вќ‡10 П• eвњЃ2 3П• ,
217
192
вќ‡
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Engineers’ approach:
W в™ЈpВ вњЏ 2, pвњЃ вњЏ 3q: with the OPE
W В в™Јz q W вњЃ в™Јw q вњЏ 27 в�Ћ 3 в�Ћ 53 в�Ћ 72 в�Ћ 11 в�Ћ 17
T в™Јz q вњЏ
T в™Јw q
в™Јz вњЃ w q28 В . . . ,
вќ‡ в™Ј qвќ‡П• в™Јz qвњЃ 2вќ„1 3 вќ‡2 П• в™Јz q.
1
П• z
2
Semikhatov
Hopf algebras and the duality to logarithmic CFT
Engineers’ approach:
Lattice construction:
Return
[r,s;0]
Return
b Г—0
~~ 00
~~
0
~
00
~
[p+ в€’r,s;1]
00
66
00
66
0#
66
[r,pв€’ в€’s;1]
66
T•
n
Wв€’
66 nnnn r y В‡
n6
W+
n
Wв€’
n
n 66
n
n
[r,s;2]
[r,s;2]
66
в—¦
в—¦y 6n
66
66
q **
66
66 Wв€’
**
W+
**
W+
66 в€’ в€’s;2] '
[p+ в€’r,p
[p+ в€’r,pв€’ в€’s;2]
6
*
n
b 0
66 nnn U
**
~~ 00
n
n
6
~
n
*
~
00
** [p в€’r,s;3] nnnnn 666
~~
[p+ в€’r,s;3] 00
66
** +
66
55
66
00
66
**
66
55
6
00
66
6
*
66[r,p в€’s;3] & 55
66
# [r,pв€’ в€’s;3]
66 − nT •
55
66
U•
T•
66 nnnn y
55nnnnn y
66 nnnnn y
n
n
n
n
6
n
5
6
nn 66
nn 66
nnn 55
[r,s;4]
[r,s;4] nn
[r,s;4] nnn
[r,s;4] nnn
55
66
66
в—¦
в—¦y 5
в—¦y 6
в—¦y 5
Wв€’
55
66
66
55
55
66
q ***
55
66
66
55
55
66
**
'
&
&
5
5
6
[p+ в€’r,p
[p+ в€’r,p
[p+ в€’r,p
**
55 в€’ в€’s;4]U
55 в€’ в€’s;4]U
66 в€’ в€’s;4]U
b 0
55 oooo
55 nnnn
66 nnnn
**
~~ 00
n6n6n
** W+
oo5o55
~~
nn5n55
n
00
n
o
n
n
o
66
~~
n
**
oo
n
55
5
nnn
00
55
6
55
**
66
55
55
66
6[p6 + в€’r,s;5] 66
00
5
55
55
66
66
55
55
66
66
**
00
5& 5& 66
55
55
66
66[r,p в€’s;5]6& Г™
#
в€’
55
5
6
6
6
T •y
T •y
T•
U •y
U •y
6
5
6
6
n
n
n
n
n
n
n
n
n
n
55 nn
55 nn
66 nnn
66 nnn y
66 nnn
n
n
n
n
n
n5
nn5
nnn 666
nnn 666
nnn 666
nnn 55
nnnn 55
[r,s;6] nn
[r,s;6] nnn
[r,s;6] nnn
[r,s;6] nnn
66
55
55
66
66
в—¦
в—¦
в—¦
в—¦
в—¦
66
55
55
66
66
66
55
55
66
66
&
&
&
&
'
Semikhatov
algebras and the
duality
to logarithmic CFT
[p+ в€’r,pв€’ в€’s;6]
[p+ в€’r,pв€’
в€’s;6]
[p+Hopf
в€’r,pв€’ в€’s;6]
[p+ в€’r,p
в€’ в€’s;6]
Semikhatov
Hopf algebras and the duality to logarithmic CFT
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