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Communications in Algebra
ISSN: 0092-7872 (Print) 1532-4125 (Online) Journal homepage: http://www.tandfonline.com/loi/lagb20
Isoclinism between pairs of n-Lie algebras and
their properties
Azam K. Mousavi & Mohammad Reza R. Moghaddam
To cite this article: Azam K. Mousavi & Mohammad Reza R. Moghaddam (2017): Isoclinism
between pairs of n-Lie algebras and their properties, Communications in Algebra, DOI:
10.1080/00927872.2017.1384000
To link to this article: http://dx.doi.org/10.1080/00927872.2017.1384000
Accepted author version posted online: 27
Sep 2017.
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Date: 28 October 2017, At: 00:13
Isoclinism between pairs of n-Lie algebras and their properties
1 International
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Azam K. Mousavi1 and Mohammad Reza R. Moghaddam2,3
Campus, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad,
Mashhad, Iran
of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures,
Ferdowsi University of Mashhad, Mashhad, Iran
3 Department
of Mathematics, Khayyam University, Mashhad, Iran
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Abstract
In this paper, we investigate the notion of isoclinism on a pair of n-Lie algebras, which forms an
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2 Department
equivalence relation. In addition, we prove that each equivalence class contains a stem pair of nLie algebras, which has minimal dimension amongst the finite dimensional pairs of n-Lie algebras.
Finally, some more results are obtained when two isoclinic pairs of n-Lie algebras are given.
KEYWORDS: n-Lie algebra; isoclinism; nilpotent n-Lie algebra
Received 22 April 2017; Revised 24 August 2017
Mohammad Reza R. Moghaddam rezam@ferdowsi.um.ac.ir; m.r.moghaddam@khayyam.ac.ir
Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran; Department of Mathematics,
Khayyam University, Mashhad, Iran.
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2010 Mathematics Subject Classification:
Primary 17B05; Secondary 17B30
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1. INTRODUCTION
In 1987, Filippov [3] introduced the notion of n-Lie algebras and classified all n-Lie algebras of
dimension n + 1 over an algebraically closed field. Also, Kasymov [4] defined the property of
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nilpotency in n-Lie algebras. The structure of n-Lie algebras may be far from that of Lie algebras,
for instance the sum of some nilpotent ideals in an n-Lie algebra need not be nilpotent (see [10] for
An n-Lie algebra is a vector space L over a field F together with the following n-linear map
[?, . . . , ?] : L О и и и О L ?? L
M
given by
(x1 , . . . , xn ) 7 ?? [x1 , . . . , xn ],
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more detail).
for all xi ? L such that the following conditions hold:
(i) [x1 , . . . , xi , . . . , xj , . . . , xn ] = 0, when xi = xj ; and
(ii) [[x1 , . . . , xn ], y2 , . . . , yn ] =
Pn
i=1 [x1 , . . . , [xi , y2 , . . . , yn ], . . . , xn ],
for all xi , yj ? L, 1 ? i ? n and 2 ? j ? n.
3
A subspace S of n-Lie algebra L, which is closed under the n-Lie product is said to be n-Lie
subalgebra of L. Also, n-Lie subalgebra I of L is called n-Lie ideal of L if [I, |L, .{z
. . , L} ] ? I.
(n?1)?times
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Finally, the centre of the n-Lie algebra L is defined to be
Z(L) = {x ? L : [x, |L, .{z
. . , L} ] = 0}.
(n?1)?times
Moghaddam and Saeedi [2] generalized the notion of isoclinism to n-Lie algebras. They proved
that the notion of isoclinism and isomorphism are equivalent for any two n-Lie algebras of the same
dimensions.
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The n-Lie algebras L and M are said to be isoclinic, denoted by L ? M, if there exist algebra
isomorphisms ? : L/Z(L) ?? M/Z(M) and ? : L2 ?? M 2 such that the following diagram is
commutative.
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The notion of isoclinism for Lie algebras is introduced, by Moneyhun [6] in 1994. In 2016, Eshrati,
L
Z(L)
О
L
Z(L)
О иии О
?
/ [L, L, . . . , L]
? n?1
?
M
Z(M)
L
Z(L)
О
M
Z(M)
О иии О
M
Z(M)
/ [M, M, . . . , M].
Also in 2009, Moghaddam and Parvaneh [5] extended the notion of isoclinism to the pairs of Lie
algebras. A pair of n-Lie algebras is defined as follows:
4
Definition. Let M be an ideal of an n-Lie algebra L. Then (M, L) is considered to be a pair of n-Lie
algebra and one may define the commutator and the centre of the pair (M, L) as follows:
and
In this paper, we introduce the notion of isoclinism for the pairs of n-Lie algebras. Also, by using
the notion of stem pair of n-Lie algebras, we shall prove that; any two stem pairs of n-Lie algebras
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(M, L) and (J, K) are isomorphic if and only if they are isoclinic (see Theorem 3.6, below).
Definition. Two pairs of n-Lie algebras (M1 , L1 ) and (M2 , L2 ) are called isoclinic, denoted by
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Z(M, L) = {m ? M : [m, l, . . . , l] = 0, ?l ? L}.
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[M, L, . . . , L] = h[m, l, . . . , l] : m ? M, l ? Li
(M1 , L1 ) ? (M2 , L2 ), if there exist a pair of isomorphisms (?, ?), in which ? : L1 /Z(M1 , L1 ) ??
L2 /Z(M2 , L2 ) with ?(M 1 ) = M 2 , M j =
Mj
Z(Mj ,Lj ) ,
j = 1, 2, and ? : [M1 , L1 , . . . , L1 ] ??
[M2 , L2 , . . . , L2 ] satisfies ?([m1 , l2 , . . . , ln ]) = [m2 , l2? , . . . , ln? ], for all m1 ? M1 , li ? L1 , m2 ?
?(m1 ), li? ? ?(li ) and 2 ? i ? n, and that the following diagram is commutative:
M1
Z(M1 ,L1 )
О
L1
Z(M1 ,L1 )
О иии О
?
/ [M1 , L1 , . . . , L1 ]
? n?1
M2
Z(M2 ,L2 )
L1
Z(M1 ,L1 )
?
О
L2
Z(M2 ,L2 )
О иии О
L2
Z(M2 ,L2 )
/ [M2 , L2 , . . . , L2 ],
5
in which the horizontal maps are defined naturally.
(M2 , L2 ). Then
(ii) ?([x, l2 , . . . , ln ]) = [?(x), l2? , . . . , ln? ] with li? ? ?(li ),
Proof.
L1
Z(M1 ,L1 ) , 2
M
for all x ? [M1 , L1 , . . . , L1 ], li ? L1 and li ? L1 =
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(i) ?(x + Z(M1 , L1 )) = ?(x) + Z(M2 , L2 );
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Lemma 1.1. Assume (?, ?) is an isoclinism between the pairs of n-Lie algebras (M1 , L1 ) and
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? i ? n.
(i) If x ? [M1 , L1 , . . . , L1 ], then x =
n. Then
?(x + Z(M1 , L1 )) = ?
k
X
Pk
i=1 [mi , li2 , . . . , lin ],
for some mi ? M1 and lij ? L1 , 2 ? j ?
[mi , li2 , . . . , lin ] + Z(M1 , L1 )
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i=1
!
k
X
[?(mi + Z(M1 , L1 )), ?(li2 + Z(M1 , L1 )), . . . , ?(lin + Z(M1 , L1 ))]
=
k
X
?
?
=
[m?i + Z(M2 , L2 ), li2
+ Z(M2 , L2 ), . . . , lin
+ Z(M2 , L2 )]
i=1
=
k
X
?
?
] + Z(M2 , L2 )
, . . . , lin
[m?i , li2
i=1
=
k
X
?[mi , li2 , . . . , lin ] + Z(M2 , L2 )
M
i=1
!
k
X
[mi , li2 , . . . , lin ] + Z(M2 , L2 )
= ?
i=1
= ?(x) + Z(M2 , L2 ),
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i=1
where m?i ? ?(mi + Z(M1 , L1 )), and lij? ? ?(lij + Z(M1 , L1 )).
(ii) Let lij ? L1 , lij? ? ?(lij +Z(M1 , L1 )) and x ? [M1 , L1 , . . . , L1 ]. Utilizing (i) we observe that ?(x+
Z(M1 , L1 )) = ?(x) + Z(M2 , L2 ) and the result follows from the commutativity of the diagram.
Lemma 1.2. Let (M1 , L1 ) and (M2 , L2 ) be pairs of n-Lie algebras. Then (M1 , L1 ) ? (M2 , L2 ) if
and only if there exist ideals N1 ? Z(M1 , L1 ) and N2 ? Z(M2 , L2 ) of L1 and L2 , respectively, such
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that ? : L1 /N1 ?? L2 /N2 and ? : [M1 , L1 , . . . , L1 ] ?? [M2 , L2 , . . . , L2 ] satisfying
?
?
],
?(M1 /N1 ) = M2 /N2 ; ?([m1 , li2 , . . . , lin ]) = [m2 , li2
, . . . , lin
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where lij? ? ?(lij + Z(M1 , L1 )), m2 ? ?(m1 ) and 2 ? j ? n.
Z(M2 , L2 ). To prove the converse, assume m1 ? Z(M1 , L1 ). Then [m1 , l2 , . . . , ln ] = 0.
0 = ?(0) = ?([m1 , l2 , . . . , ln ]) = [m2 , l2? , . . . , ln? ],
where li? ? ?(li ). On the other hand, ?(m1 + N1 ) = m2 + N2 for some m2 ? Z(M2 , L2 ), implies that
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?(Z(M1 , L1 )/N1 ) ? Z(M2 , L2 )/N2 . The reverse inclusion follows by the same argument applied to
? ?1 .
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Proof. If (M1 , L1 ) ? (M2 , L2 ), then the result follows by choosing N1 = Z(M1 , L1 ) and N2 =
2. SOME PROPERTIES OF PAIRS OF N-LIE ALGEBRAS
This section is devoted to showing some properties of a pair of n-Lie algebras. Particularly, a stem
pair of n-Lie algebras will be introduced. It is also shown that any family of isoclinism pairs of
n-Lie algebras contains a stem pair of n-Lie algebras.
Definition. The pair (N, K) of n-Lie algebras is called abelian if Z(N, K) = N or equivalently
[N, K, . . . , K] = 0.
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Remark. It is known that for any n-Lie algebra L, there exists an isoclinism L ? L ? A, for any
abelian n-Lie algebra A. However, this result does not hold for the pairs of n-Lie algebras. If (M, L)
to (M ? N, L ? K), because
so that
L?K
L
L?K
?
6=
.
=
Z(M ? N, L ? K)
Z(M, L) ? N
Z(M, L)
To see this, we note that in the case of finite dimensions,
L?K
L
= (dim L ? dim Z(M, L)) + (dim K ? dim N) 6 = dim
.
Z(M ? N, L ? K)
Z(M, L)
M
dim
Theorem 2.1. Let (?, ?) be the isoclinism pair for the pairs of n-Lie algebras (M, L) and (N, K).
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Z(M ? N, L ? K) = Z(M, L) ? N
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is a pair of n-Lie algebras and (N, K) is an abelian pair of n-Lie algebras, then (M, L) is not isoclinic
(i) If L1 is a subalgebra of L containing Z(M, L) such that ?(L1 /Z(M, L)) = K1 /Z(N, K), then
(L1 ? M, L1 ) ? (K1 ? N, K1 ).
(ii) If M1 ? M is contained in [M, L, . . . , L], then (M/M1 , L/M1 ) ? (N/?(M1 ), K/?(M1 )).
Proof.
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(i) Assume M1 = Z(L1 ? M, L) ? Z(L1 ? M, L1 ) and N1 = Z(K1 ? N, K) ? Z(K1 ? N, K1 ). We
define the maps ? : L1 /M1 ?? K1 /N1 and ? : [L1 ? M, L1 , . . . , L1 ] ?? [K1 ? N, K1 , . . . , K1 ]
as ?(l1 + M1 ) = k1 + N1 and ?(x) = ?(x), for all l1 ? L1 , k1 ? ?(l1 + Z(M, L)) and x ?
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[L1 ? M, L1 , . . . , L1 ]. It is easy to see that ? and ? are isomorphisms yielding the corresponding
(ii) Since ? is an isomorphism, we have that ?(M1 ) ? K. Put N = N/?(M1 ), K = K/?(M1 ),
M = M/M1 and L = L/M1 . Then M 1 = (Z(M, L) + M1 )/M1 ? Z(M, L) and N 1 = (Z(N, K) +
?(M1 ))/?(M1 )?Z(N, K). Now we define the maps ? : L/M 1 ?? K/N 1 and ? : [M, L, . . . , L] ??
[N, K, . . . , K] given by ?(l + M 1 ) = k + N 1 and ?(x) = ?(x), for all l ? L, k ? ?(l + Z(M, L))
corresponding isoclinism arises.
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and x ? [M, L, . . . , L]. One can easily verify that ? and ? are isomorphisms, from which the
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isoclinism.
Theorem 2.2. Let (M, L) be a pair of n-Lie algebras, K a subalgebra of L and N ? M be an ideal
of L. Then
(i) (K ? M, K) ? (K ? M + Z(M, L), K + Z(M, L)). In particular, if L = K + Z(M, L), then
(K ? M, K) ? (M, L). Conversely, if K/Z(K ? M, K) satisfies the descending chain condition
on ideals and (K ? M, K) ? (M, L), then L = K + Z(M, L).
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(ii) (M/N, L/N) ? (M/N ?[N, L, . . . , L], L/N ?[N, L, . . . , L]). In particular, if N ?[N, L, . . . , L] =
0, then (M/N, L/N) ? (M, L). Conversely, if [M, L, . . . , L] satisfies the increasing chain
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condition on ideals and that (M/N, L/N) ? (M, L), then N ? [N, L, . . . , L] = 0.
(i) Assume M1 = Z(K ? M, L) ? N1 = Z(K ? M + Z(M, L), L). We define the maps ? and ? as
follows
? : MK1 ?? K+Z(M,L)
N1
k + M1 7 ?? k + N1
M
and
=[K?M,K,...,K]
z
}|
{
? : [K ? M, K, . . . , K] ?? [K ? M + Z(M, L), K + Z(M, L), . . . , K + Z(M, L)]
x 7 ?? x.
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Proof.
One can easily see that ? and ? are the corresponding isomorphisms. Clearly, if L = K + Z(M, L),
then M = K ? M + Z(M, L).
Conversely, assume that K1 = K + Z(M, L). We show that L = K1 . It is obvious that (K +
Z(M, L))/Z(M, L) ? L/Z(M, L) and
(M, L) ? (K ? M, K) ? (K ? M + Z(M, L), K + Z(M, L)) ? (K ? M + Z(M, L), K1 ) = A.
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The above isoclinics imply the following isomorphism
? : L/Z(M, L) ?? K1 /Z(A).
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Thus K2 ? K1 and Z(A) ? K2 , which imply that ?(K/Z(M, L)) = K2 /Z(A). Hence Z(M, L) ?
Z(A) ? K2 ? K1 and consequently K2 = K1 if and only if K1 = L. Continuing in this way, we
Z(A) ? и и и ? K3 ? K2 ? K1 ,
such that Ki = Ki+1 if and only if L = K1 . Now since K1 /Z(A) ?
= K/Z(K ? M, K) and K/Z(K ?
M, K) satisfies the descending chain condition on subalgebras, there must exists a non-negative
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integer n such that Kn = Kn+1 so that K1 = L.
(ii) Put L = L/N, M = M/N, L? = L/N ? [M, L, . . . , L] and M? = M/N ? [M, L, . . . , L]. We show
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obtain a descending chain of subalgebras of K1 /Z(M, L) as follows:
that (M, L) ? (M?, L?). We define the maps ? and ? as follows:
?:
L
Z(N,L)
??
L?
Z(N?,L?)
l + Z(M, L) 7 ?? l? + Z(M, L)
and
? : [M, L, . . . , L] ?? [M?, L?, . . . , L?]
x 7 ?? x?.
One can easily see that the maps ? and ? are isomorphisms. Clearly, if N ? [M, L, . . . , L] = 0 and
by assuming M? ?
= L, then we obtain the isoclinism (M, L) ? (M, L).
= M and L? ?
Conversely, assume that J = N ? [M, L, . . . , L]. Then (M, L) ? (M/N, L/N) ? (M/J, L/J). Let
? : [M, L, . . . , L] ?? [K/J, L/J, . . . , L/J] be the corresponding automorphism arising from the
12
isoclinism (M, L) ? (M/J, L/J) and J1 be the ideal of [M, L, . . . , L] satisfying ?(J) = J1 /J. One
can see that J = 0 if and only if J1 = J so that ? : [M, L, . . . , L]/J ?? [M, L, . . . , L]/J1 is
an isomorphism. Similarly, we may consider the ideal J2 of [M, L, . . . , L] satisfying J ? J1 ?
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J2 ? [M, L, . . . , L] and ?(J2 ) = J2 /J1 . Then J = 0 if and only if J1 = J if and only if J2 = J1 .
Continuing this procedure, we obtain an increasing chain of ideals of [M, L, . . . , L] as follows:
such that Ji = Ji+1 if and only if J = 0. Hence, there must exists a number n > 0 such that
Jn = Jn+1 and consequently J = 0.
[M, L, . . . , L].
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Definition. The pair (M, L) of n-Lie algebras is called a stem pair of n-Lie algebras if Z(M, L) ?
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J ? J1 ? J2 ? и и и ? [M, L, . . . , L],
In what follows, we shall prove that in every isoclinism family of pairs of n-Lie algebras, there
always exists a stem pair, say (N, K), in such a way that K has minimum dimension.
Theorem 2.3. Let C be an isoclinism family of pairs of n-Lie algebras. Then
(i) C contains at least one stem pair of n-Lie algebras.
(ii) If (N, K) belongs to C, in which K has finite dimension, then (N, K) is a stem pair of n-Lie
algebras if and only if dim K = min{dim L|(M, L) ? C}.
13
Proof.
(i) Let (M, L) be an arbitrary pair of n-Lie algebras in C and S be the complement of Z(M, L) ?
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[M, L, . . . , L] in Z(M, L), that is, Z(M, L) = Z(M, L)?[M, L, . . . , L]?S. Clearly, S?[M, L, . . . , L] =
0. By Theorem 2.2, (M/S, L/S) ? (M, L). Put N = M/S and K = L/S. We observe that (N, K) ? C
Z(M, L)
M L
?
=
,
Z(N, K) = Z
S S
S
=
Z(M, L) ? [M, L, . . . , L] ? S
S
[M, L, . . . , L] ? S
S
M L
L
=
, ,...,
= [N, K, . . . , K].
S S
S
M
?
Thus (N, K) is a stem pair of n-Lie algebras in C, as required.
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and
(ii) Assume (N, K) is a stem pair of n-Lie algebras of finite dimension in C, which is isoclinic with
every pair (M, L) of C of finite dimension, that is, (N, K) ? (M, L). Since (N, K) is a stem pair, we
14
observe that
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[M, L, . . . , L]
Z(M, L) + [M, L, . . . , L]
=
[M, L, . . . , L] ? Z(M, L)
Z(M, L)
L
L
M
,
,...,
=
Z(M, L) Z(M, L)
Z(M, L)
N
K
K
?
,
,...,
=
Z(N, K) Z(N, K)
Z(N, K)
[N, K, . . . , K]
.
Z(N, K)
Now from the isomorphism [M, L, . . . , L] ?
= [N, K, . . . , K] we conclude that
dim Z(N, K) = dim[M, L, . . . , L] ? Z(M, L) ? dim Z(M, L).
M
On the other hand, K/Z(N, K) ?
= L/Z(M, L), which implies that dim K ? dim L.
Conversely, assume that (N, K) ? C with K of minimum dimension. We show that (N, K) is a
stem pair of n-Lie algebras. From Section 1, we know that there exists an ideal J of K such that
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=
J ? Z(N, K) and (N, K) ? (N/J, K/J) in such a way that Z(N, K) = [N, K, . . . , K] ? Z(N, K) ? J.
Since K has minimum dimension, we must have J = 0 and hence the proof is complete.
Theorem 2.4. Let (N1 , K1 ) and (N2 , K2 ) be isoclinic stem pairs of n-Lie algebras. Then
Z(N1 , K1 ) ?
= Z(N2 , K2 ).
Proof. Assume (?, ?) is the corresponding isoclinism pair between (N1 , K1 ) and (N2 , K2 ). From the
definition, it follows that Z(Ni , Ki ) ? [Ni , Ki , . . . , Ki ] for i = 1, 2. We consider the isomorphism
15
? : [N1 , K1 , . . . , K1 ] ?? [N2 , K2 , . . . , K2 ]. For each k1i ? K1 , k2i ? ?(k1i + Z(N1 , K1 )) and
z1 ? Z(N1 , K1 ), we have
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0 = ?(0) = ?([z1 , k12 , . . . , k1n ]) = [?(z1 ), k22 , . . . , k2n ].
Thus ?(z1 ) ? Z(N2 , K2 ) and consequently ?(Z(N1 , K1 )) ? Z(N2 , K2 ). The reverse inclusion
3. SOME PROPERTIES OF THE FACTOR SET ON A PAIR OF N-LIE ALGEBRAS
In this section, we introduce factor sets on pairs of n-Lie algebras and apply to obtain some results
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on stem pairs of n-Lie algebras of finite dimension.
Definition. Let (M, L) be a pair of n-Lie algebras and put M = M/Z(M, L). A factor set on the
pair of n-Lie algebras (M, L) is defined as an n-linear map
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follows by considering ? ?1 .
f :
M
M
M
О
О иии О
?? Z(M, L),
Z(M, L) Z(M, L)
Z(M, L)
such that for all x1 , . . . , xn , y1 , . . . , yn ? M, the following conditions are satisfied:
(1) f (x1 , . . . , xi , . . . , xj , . . . , xn ) = 0 if and only if xi = xj ;
(2) f ([x1 , . . . , xn ], y2 , . . . , yn ) = f ([x1 , y2 , . . . , yn ], x2 , . . . , xn ) + f (x1 , [x2 , y2 , . . . , yn ], x3 , . . . , xn ) +
f (x1 , x2 , [x3 , y2 , . . . , yn ], x4 , . . . , xn )+и и и+f (x1 , . . . , xn?1 , [xn , y2 , . . . , yn ]) =
16
Pn
i=1 f (x1 , . . . , xi?1 , [xi , y2 , . . . , yn
Taking the above notation, we have the following
an
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Lemma 3.1. Let f be a factor set on the pair of n-Lie algebras (M, L). Then
(i) Mf = Z(M, L) О M = {(zi , mi ) : zi ? Z(M, L), mi ? M, i = 1, . . . , n} is a stem n-Lie algebra
[(z1 , m1 ), . . . , (zn , mn )] = (f (m1 , . . . , mn ), [m1 , . . . , mn ]),
for all zi ? Z(M, L) and mi ? M (i = 1, . . . , n).
M
(ii) ZMf = {(z, 0) ? Mf : z ? Z(M, L)} ?
= Z(M, L).
Proof. We show that Mf is an n-Lie algebra under the given actions. The first condition is simply
satisfied, as
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under the component-wise addition, and the multiplication defined as follows:
[(z1 , m1 ), . . . , (zi , mi ), . . . , (zj , mj ), . . . , (zn , mn )]
= (f (m1 , . . . mi , . . . , mj , . . . , mn ), [m1 , . . . mi , . . . , mj , . . . , mn ]) = (0, 0),
for all zk ? Z(M, L), mk ? M, and 1 ? k ? n with (zi , mi ) = (zj , mj ).
17
Now we investigate the Jacobi identity. We have
[[(z11 , m11 ), . . . , (z1n , m1n )], (z21 , m21 ), . . . , (zn1 , mn1 )]
an
us
cr
ip
t
= [(f (m11 , . . . , m1n ), [m11 , . . . , m1n ]), (z21 , m21 ), . . . , (zn1 , mn1 )]
= (f ([m11 , . . . , m1n ], m21 , . . . , m2n ), [[m11 , . . . , m1n ], m21 , . . . , m2n ])
=
n
X
=
i=1
n
X
(f (m11 , . . . , [m1i , m21 , . . . , m2n ], . . . , m1n ), [m11 , . . . , [m1i , m21 , . . . , m2n ], . . . , m1n ])
n
X
[(z11 , m11 ), . . . , (f (m1i , m21 , . . . , m2n ), [m1i , m21 , . . . , m2n ]), . . . , (z1n , m1n )]
n
X
[(z11 , m11 , . . . , [(z1i , m1i ), (z21 , m21 ), . . . , (z2n , m2n )], . . . , (z1n , m1n )],
i=1
=
=
i=1
as required.
M
i=1
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i=1
n
X
[m11 , . . . , [m1i , m21 , . . . , m2n ], . . . , m1n ]
f (m11 , . . . , [m1i , m21 , . . . , m2n ], . . . , m1n ),
The following lemma is needed to prove our next result.
Lemma 3.2. Let (M, L) be a pair of n-Lie algebras. Then there exists a factor set f on (M, L) such
that M ?
= Z(M, L) О M = Mf .
Proof. Let K be a complement of the vector subspace Z(M, L) in M, that is, M = K ? Z(M, L).
Let g be a map defined as follows:
M
= M ?? M = Z(M, L) + M
g : Z(M,L)
m = k + Z(M, L) 7 ?? k (k ? M).
18
!
On the other hand, since g(m) = k, we have g(m) = k = m. Moreover,
an
us
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ip
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[g(m1 ), . . . , g(mn )] = g[m1 , . . . , mn ],
for
[g(m1 ), . . . , g(mn )] = [g(m1 ), . . . , g(mn )] + Z(M, L)
= [g(m1 ), . . . , g(mn )] = [m1 , . . . , mn ]
= [m1 , . . . , mn ] = g[m1 , . . . , mn ].
M
Thus
[g(m1 ) + Z(M, L), . . . , g(mn ) + Z(M, L)] = g[m1 , . . . , mn ] + Z(M, L),
and consequently
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= [g(m1 ) + Z(M, L), . . . , g(mn ) + Z(M, L)]
[g(m1 ) + Z(M, L), . . . , g(mn ) + Z(M, L)] ? g[m1 , . . . , mn ] ? Z(M, L).
Now we define the factor set f as follows:
f : M О и и и О M ?? Z(M, L)
(m1 , . . . , mn ) 7 ?? [g(m1 ) + Z(M, L), . . . , g(mn ) + Z(M, L)] ? g[m1 , . . . , mn ].
19
Obviously, the first condition of the factor set for f is satisfied. Hence it is enough to check the
second one. We have
an
us
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ip
t
f ([x1 , . . . , xn ], y2 , . . . , yn )
= [g([x1 , x2 , . . . , xn ]), g(y2 ), . . . , g(yn )] ? g[[x1 , x2 , . . . , xn ], y2 , . . . , yn ]
=
n
X
[g(x1 ), . . . , [g(xi ), g(y2 ), . . . , g(yn )], . . . , g(xn )] ? g
n
X
([g(x1 ), . . . , [g(xi ), g(y2 ), . . . , g(yn )], . . . , g(xn )] ? g[x1 , . . . , [xi , y2 , . . . , yn ], . . . , xn ])
n
X
f (x1 , . . . , [xi , y2 , . . . , yn ], . . . , xn ).
i=1
=
i=1
=
n
X
i=1
M
Now consider the map ? defined as follows:
? : (Z(M, L), M, f ) ?? M = K ? Z(M, L)
(z, m) 7 ?? z + g(m) = z + k,
for all z ? Z(M, L) and m ? M/Z(M, L). Clearly, ? is well-defined map. The map ? is injective,
for
?(z1 , m1 ) = ?(z2 , m2 ) ? z1 + g(m1 ) = z2 + g(m2 )
? z1 + k1 = z2 + k2 ? z1 ? z2 = k2 ? k1 .
20
!
[x1 , . . . , [xi , y2 , . . . , yn ], . . . , xn ]
i=1
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= [[g(x1 ), g(x2 ), . . . , g(xn )], g(y2 ), . . . , g(yn )] ? g[[x1 , x2 , . . . , xn ], y2 , . . . , yn ]
Since z1 ? z2 ? Z(M, L), k2 ? k1 ? K and K ? Z(M, L) = 0, we have that (z1 , m1 ) = (z2 , m2 ). Also,
? is surjective as
an
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ip
t
?[(z1 , m1 ), . . . , (zn , mn )] = ?(f (m1 , . . . , mn ), [m1 , . . . , mn ])
= f (m1 , . . . , mn ) + g[m1 , . . . , mn ]
= [z1 + g(m1 ), . . . , zn + g(mn )]
= [?(z1 , m1 ), . . . , ?(zn , mn )].
Hence the proof is complete.
M
The following result establishes a relationship between stem pairs and factor sets in n-Lie algebras.
Theorem 3.3. Let C be a family of isoclinic pairs of n-Lie algebras and (M, L) be a stem pair in
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= [g(m1 ), . . . , g(mn )]
C. Then for every stem pair (J, K) of C, there exists a factor set f on (M, L) satisfying
J?
= Mf = Z(M, L) О M ?
= M.
= Z(J, K) О J = Jg ?
Proof. Let (?, ?) be the isoclinism pair between stem pairs of n-Lie algebras (M, L) and (J, K). By
Theorem 2.4, we have Z(J, K) ?
= Z(M, L) and the above lemma yields a factor set g on (J, K) such
that Jg = Z(J, K) О J. Since ?(Z(M, L)) = Z(J, K), we may define the following factor set
f : M О и и и О M ?? Z(M, L)
(m1 , . . . , mn ) 7 ?? ? ?1 (g(?(m1 ), . . . , ?(mn ))).
21
Now define the map ? : M ?
= Z(M, L) О M ?? Z(J, K) О J ?
= Jg given by ?(z, m1 ) =
(?(z), ?(m1 )). Clearly, ? is a well-defined bijection. To prove ? is a homomorphism, we observe
an
us
cr
ip
t
that
?[(z1 , m1 ), . . . , (zn , mn )] = ?(f (m1 , . . . , mn ), [m1 , . . . , mn ])
= (?(f (m1 , . . . , mn )), ?[m1 , . . . , mn ])
= [(?(z1 ), ?(m1 )) . . . , (?(zn ), ?(mn ))]
= [?(z1 , m1 ), . . . , ?(zn , mn )],
M
as required.
Lemma 3.4. Let f and g be two factor sets on the pair of n-Lie algebras (M, L). If h : Mf ?? Mg
is an isomorphism satisfying h(ZMf ) = ZMg , then h induces the automorphisms h1 and h2 on n-Lie
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= (g(?(m1 ), . . . , ?(mn )), ?[m1 , . . . , mn ])
algebras M and Z(M, L), respectively.
Proof. Clearly, h induces a homomorphism h : Mf /ZMf ?? Mg /ZMg defined by h((z, m)+ZMf ) =
h(z, m) + ZMg . Now we define the map h1 : M/Z(M, L) ?? M/Z(M, L) as follows:
h((0, m) + ZMf ) = h(0, m) + ZMg = (0, h1 (m)) + ZMg ,
?m?
M
.
Z(M, L)
Also, we may define the map h2 : Z(M, L) ?? Z(M, L) given by
h(z, 0) = (h2 (z), 0),
? z ? Z(M, L).
22
A simple verification shows that h1 and h2 are the required automorphisms.
Theorem 3.5. The isomorphism h : Mf ?? Mg satisfying h(ZMf ) = ZMg induces the
an
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automorphisms h1 of M/Z(M, L) and h2 of Z(M, L) if and only if there exists a linear map
h2 (g(m1 , . . . , mn ) + ? [m1 , . . . , mn ]) = g(h1 (m1 ), . . . , h1 (mn )),
for all m1 , . . . , mn ? M/Z(M, L).
h(z, 0) = (h2 (z), 0),
M
Proof. Let m ? M/Z(M, L) and z ? Z(M, L). By Lemma 3.4, we have
h(0, m) + ZMg = (0, h1 (m)) + ZMg ,
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? : M/Z(M, L) ?? Z(M, L) such that
h(0, m) = (0, h1 (m)) + (zm , 0).
Now we define the linear map ? : M ?? Z(M, L) given by ? (m) = zm . Then
h(z, m) = h(z, 0) + h(0, m) = (h2 (z), 0) + (0, h1 (m)) + (zm , 0)
= (h2 (z, 0) + (0, h1 (m)) + (? (m), 0) = (h2 (z) + ? (m), h1 (m)).
23
Hence
h([(0, m1 ), . . . , (0, mn )]) = [h(0, m1 ), . . . , h(0, mn )]
an
us
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t
= [(0, h1 (m1 )) + (? (m1 ), 0), . . . , (0, h1 (mn )) + (? (mn ), 0)]
= [(? (m1 ), h1 (m1 )), . . . , (? (mn ), h1 (mn ))]
On the other hand,
h[(0, m1 ), . . . , (0, mn )]
= h(f (m1 , . . . , mn ), [m1 , . . . , mn ])
(?)
(??)
M
= h(h2 (f (m1 , . . . , mn )) + ? ([m1 , . . . , mn ]), h1 ([m1 , . . . , mn ])).
Now the result follows from the equalities of (?) and (??).
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= (g(h1 (m1 ), . . . , h1 (mn )), [h1 (m1 ), . . . , h1 (mn )]).
Conversely, we show that h induces the automorphisms h1 and h2 . Define the map h : Mf ?? Mg
given by h(z, m) = (h2 (z) + ? (m), h1 (m)), for all z ? Z(M, L) and m ? M/Z(M, L). Clearly, h is
surjective. Moreover,
h(z, m) = (h2 (z), 0) + (0, h1 (m)) + (? (m), 0)
= (h2 (z) + ? (m), h1 (m)),
24
and
h[(0, m1 ), . . . , (0, mn )] = h(f (m1 , . . . , mn ), [m1 , . . . , mn ])
an
us
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t
= (h2 (f (m1 , . . . , mn )) + ? ([m1 , . . . , mn ]), h1 ([m1 , . . . , mn ]))
= (g(h1 (m1 ), . . . , h1 (mn )), h1 ([m1 , . . . , mn ])).
On the other hand,
(I)
= (g(h1 (m1 ), . . . , h1 (mn )), [h1 (m1 ), . . . , h1 (mn )]).
(II)
The relations (I) and (II) show that h is a homomorphism. Also
and
M
h(z, 0) = (h2 (z) + ? (0), h1 (0)) = (h2 (z), 0),
h(0, m) + ZMg = (? (m), h1 (m)) + ZMg
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[h(0, m1 ), . . . , h(0, mn )] = [(? (m1 ), h1 (m1 )), . . . , (? (mn ), h1 (mn ))]
= (0, h1 (m)) + (? (m), 0) + ZMg = (0, h1 (m)) + ZMg ,
for all (0, h1 (m)) ? Mg . The proof is complete.
Definition. The pairs of n-Lie algebras (M, L) and (J, K) are isomorphic if M ?
= K.
= J and L ?
In the following, we shall prove our main results of this section.
25
Theorem 3.6. Let (M, L) and (J, K) be stem pairs of n-Lie algebras of finite dimension. Then
(M, L) ? (J, K) if and only of (M, L) ?
= (J, K).
an
us
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ip
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Proof. Clearly, (M, L) ? (J, K) whenever (M, L) ?
= (J, K). To prove the converse, assume that
(M, L) ? (J, K). By Theorem 3.4 there exist factor sets f and g such that
and J ?
= Z(J, K) О J = Jg .
Let (?, ?) be the corresponding isoclinisms between (M, L) and (J, K). Then Z(Mf ) ?
=
= Z(M, L) ?
ZMf . Clearly, ZMf is contained in Z(Mf ), which implies that Z(Mf ) = ZMf . Now, consider the
automorphisms ? of M and ? of Z(M, L) defined as follows:
?(z, 0) = (? (z), 0).
M
?((0, m) + ZMf ) = (0, ?(m)) + ZMg ,
Then we have the following commutative diagram
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M?
= Z(M, L) О M = Mf
M О M О иии О M
? О? ОиииО?
/ M О M О иии О M
?1
Mf О Mf О и и и О Mf
?О?ОиииО?
?2
/ Jg О Jg О и и и О Jg
P2
P1
[M f , M f , . . . , M f ]
?
/ [Jg , Jg , . . . , Jg ],
26
in which M f = Mf /ZMf , J g = Jg /ZJg , and ?1 , ?2 are defined as follows:
?1 (m1 , . . . , mn ) = ((0, m1 ) + ZMf , . . . , (0, mn ) + ZMf ),
an
us
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?2 (m1 , . . . , mn ) = ((0, m1 ) + ZMg , . . . , (0, mn ) + ZMg ).
The commutativity of diagram yields
?[(0, m1 ), . . . , (0, mn )] = [(0, ?(m1 )), . . . , (0, ?(mn ))]
On the other hand,
(1)
?[(0, m1 ), . . . , (0, mn )] = ?(f (m1 , . . . , mn ), [m1 , . . . , mn ])
= ? (f (m1 , . . . , mn ), 0) + ?(0, [m1 , . . . , mn ]).
(2)
conjunction with (1) and (2):
M
If d[m1 , . . . , mn ] denotes the first term of ?(0, [m1 , . . . , mn ]), then the following equality arises in
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= (g(?(m1 ), . . . , ?(mn )), [?(m1 ), . . . , ?(mn )]).
? (f (m1 , . . . , mn ), 0) + d[m1 , . . . , mn ] = g(?(m1 ), . . . , ?(mn )).
Now we extend d to M by assigning it the zero element on the complement of [M, . . . , M] in
M. By Theorem 3.3, Mf ?
= K. Therefore
= J. Similarly, we can show that L ?
= Jg so that M ?
(M, L) ?
= (J, K).
ACKNOWLEDGMENT
The authors would like to thank the referee for careful reading of the manuscript and valuable
suggestions.
27
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us
cr
ip
t
2. Eshrati, M., Moghaddam, M. R. R., Saeedi, F. (2016). Some properties of isoclinism in n-Lie
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4. Kasymov, Sh. M. (1987). On a theory of n-Lie algebras. Algebra Logika 26(3):277?297.
5. Moghaddam, M. R. R., Parvaneh, F. (2009). On the isoclinism of a pair of Lie algebras and
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6. Moneyhun, K. (1994). Isoclinism in Lie algebras. Algebras, Groups and Geometries 11(1):9?
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22.
7. Parvaneh, F., Moghaddam, M. R. R. (2011). Some properties of n-isoclinism in Lie algebras.
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3. Filippov, V. T. (1987). n-Lie algebras. Sib. Mat. Zh. 26(6):126?140.
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9. Salemkar, A. R., Mirzaie, F. (2010). Characterizing n-isoclinism classes of Lie algebras.
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