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Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
MCS 22E47
Invariant nite-dimensional spaces of functions
on two-dimensional algebras 1
c
D. S. Tugaryov
S.-Petersburg Acad. University, S.-Petersburg, Russia
A description of nite dimensional spaces of functions on algebras of generalized complex
numbers invariant with respect to a motion group is presented
Keywords:
two-dimensional algebras, groups of motions, Laplace operator
Let A be an algebra of generalized complex numbers z = x + iy , where x, y ? R,
with a relation i2 = ? + 2?i. It is isomorphic to one of three algebras C, D, ?:
the algebra of complex (i2 = ?1), double (i2 = 1) and dual (i2 = 0) numbers
respectively. For a number z = x + iy , the number z = x ? iy is called conjugated
to z . As coordinates on A one can also take z and z .
The exponential function ez is dened as the sum of a series:
?
X
zn
.
e =
n!
n=0
z
Let G be the group of "motions" of the algebra A: it is generated by parallel
translations z 7? z + a, a ? A, and "rotations" z 7? eit z , t ? R.
The Laplace operator
?2
?z?z
is invariant with respect to the group G. In coordinates x, y on algebras C, D, ? the
Laplace operator is relatively
?=
?2
?2
?2
?2
?2
+
,
?
,
.
?x2 ?y 2 ?x2 ?y 2 ?x2
For the algebra ?, an invariant operator is also ?1 = ?/?x.
1 Supported
by the Russian Foundation for Basic Research (RFBR): grant 09-01-00325-а, Sci.
Progr. "Development of Scientic Potential of Higher School": project 1.1.2/9191, Fed. Object
Progr. 14.740.11.0349 and Templan 1.5.07
1780
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
Let us describe nite dimensional spaces V of functions f ? C ? (A), invariant
with respect to G.
A space V , invariant with respect to G, is called indecomposable, if it cannot be
represented as a direct sum of subspaces V1 и V2 , invariant with respect to G.
Let us call an indecomposable space V , invariant with respect to G, weakly
indecomposable, if there are no subspaces V1 and V2 of V , invariant with respect to
G, such that V /V0 = V1 /V0 + V2 /V0 where V0 = V1 ? V2 .
Theorem 1.1
Let
A
be
C or D. Any nite dimensional weakly indecomposable
G space V consists of polynomials f (z, z) of degree 6 k
6 m in z . Its dimension is equal to (k + 1)(m + 1).
invariant with respect to
in
z
and of degree
Thus, the space V is the space of solutions of a system of equations:
?
?z
k+1
f = 0,
?
?z
m+1
f = 0.
Let us take a basis in V consisting of monomials z r z s , r 6 k , s 6 m. In this
basis, the Laplace operator ? has a Jordan normal form, the number of Jordan
boxes is equal to k + m + 1. A basis for a Jordan box is formed by monomials z r z s
with xed dierence r ? s. For all boxes, their eigenvalues are equal to zero.
Theorem 1.2
A = ?. Any nite dimensional weakly indecomposable invariant
with respect to G space V consists of polynomials f (x, y) of degree 6 m in y and of
degree 6 k + m in x, y together. Its dimension is equal to (k + 1 + m/2)(m + 1).
Let
Thus, the space V is the space of solutions of a system of equations:
?
?y
m+1
f = 0,
?
?
+
?x ?y
k+m+1
f = 0.
Let us take a basis in V consisting of monomials xr y s , r + s 6 k + m, s 6 m.
In this basis, the operator ?1 has a Jordan normal form. A basis for a Jordan box
is formed by monomials xr y s with s xed, the number of Jordan boxes is equal to
m + 1. For all boxes, their eigenvalues are equal to zero.
1781
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