Contemporary Mathematics
Volume 420, 2006
Group Gradings on Associative Superalgebras.
Y. A. Bahturin and I. P. Shestakov
ABSTRACT. In this paper we describe all group gradings by a finite abelian
group
G
of any simple associative superalgebra over an algebraically closed
field F. Some restrictions on the characteristic of F apply.
1. Introduction
Let
T
be an abelian group,
F
a field. An associative algebra
A
is called a
T-
superalgebra if
A
is equipped with a grading by T, that is,
A=
ffitET
At
where each
At
is a vector subspace of
A
and
At As CAts.
A subspace (subalgebra, ideal)
B
of
A
is called graded if
B
=
ffitET(B
nAt).
A superalgebra
A
is called simple if
A
has
no proper nonzero graded ideals. Using other terminology, a simple superalgebra
is a graded simple algebra.
Before we start our discussion we introduce two types of gradings by groups on
the matrix algebras
[4].
If
A= Mn(F)
the any n-tuple
(g1, ...
,gn)
of elements of
G defines an elementary G-grading of A if we define A
9
= Span{Eij
I gi
1g1 = g}.
Here
Eij
is usual matrix unit. Any grading obtained from this by an automorphism
of
A
is also called elementary.
A grading of
A= Mn(F)
by
G
~
Zn
x
Zn
is called an c-grading, where cis a
primitive nth root of 1, if
A
9
=Span {X9
},
for any
g E
G.
If
a,
bare the generators
of
G
and
g
=
aib1
then
X
9
= X~Xl. Finally,
(1) Xo
~
[
T
0
: ] , x.
~
0
1
0 0
n
cn-2
0 0 1 0
0
0 0 0 0
1 0 0 0
The mapping a::
G
x
G---- F*
given by
a(aib1,akb1)
=
c1k
is a multiplicative
bicharacter on
G
and
X
9
Xh
=
a(g, h)Xgh·
The ratio
f3(g,
h) =
a(g, h)ja(h,
g) is
2000
Mathematics Subject Classification.
Primary 16W20, 16W22, 16W50, 16W55, 17A70,
17B70, 17C70.
Key words and phrases.
Graded algebra, simple associative superalgebra, matrix algebra.
The research of the first author was partially supported by NSERC grant 227060-04 and
FAPESP, grant 02-01-00219.
The research of the second author was partially supported by CNPq, grant 304633/03-8 and
FAPESP, grant 05/54063-7.
@2006 American Mathematical Society
http://dx.doi.org/10.1090/conm/420/07964
Previous Page Next Page