The associative algebra (cf.
Associative rings and algebras)
over
whose elements are all possible finite sums of the type
,
,
,
the operations being defined by the formulas:
(The sum on the right-hand side of the second
formula is also finite.) This algebra is denoted by
;
the elements of
form a basis of this algebra; multiplication of basis elements
in the group algebra is induced by the group multiplication. The algebra
is isomorphic to the algebra of functions defined on
with values in
which assume only a finite number of non-zero values; in
this algebra multiplication is the convolution of these functions.
The same construction can also be considered for the case when
is an associative ring. One thus arrives at the concept of the
group ring of a group
over a ring
;
if
is commutative and has a unit element, the group ring is often called the
group algebra of the group over the ring
as well.
Group algebras were introduced by
G. Frobenius
and
I. Schur
[1]
in connection with the study of group
representations, since studying the representations of
over a field
is equivalent to studying modules over the group algebra
.
Thus,
Maschke's theorem
is formulated in the language of group algebras as follows: If
is a finite group and
is a field, then the group algebra
is semi-simple if and only if the order of
is not divisible by the characteristic of
.
In the early
1950s
group algebras of
infinite groups were studied in the context of integer group algebras
in algebraic topology, and for the investigation of the structure
of groups. This was also promoted by a number of problems on group algebras,
the best known of which is whether or not the group
algebra of a torsion-free group contains zero divisors
(Kaplansky's problem).
Some directions in studies on group rings and algebras.
Radicality and semi-simplicity.
A group ring has a non-zero nilpotent ideal if and only if
has a non-zero nilpotent ideal or if the order of some finite normal subgroup in
is divisible by the order of an element of the additive group of the ring
.
If
is a ring without nil ideals and if the order of each element of
is not divisible by the order of any element of the additive group of
,
then
has no nil ideals. The group algebra
over a field of characteristic zero is semi-simple, i.e. has vanishing
Jacobson radical,
if
contains a transcendental element over the field of rational numbers.
Imbedding of a group algebra into a skew-field.
The group algebra of an ordered group is imbeddable in a
skew-field
(the
Mal'tsev–von Neumann theorem).
It is believed that this is also true for any right-ordered group.
Connection between ring-theoretic properties of the group ring
with the structure of the group
and the ring
.
As an example,
is primary if and only if the ring
is primary and if the group
has no finite normal subgroups.
The
isomorphism problem:
If the group rings
and
are isomorphic as
-algebras,
what is the connection between the structures of the groups
and
.
In particular, when are
and
isomorphic? It was found that a solvable torsion group of
class two is uniquely determined by its group ring
over the ring of integers, and that a countable Abelian
-group
is uniquely determined by its group ring over a ring of characteristic
.
Different generalizations of the concept of a group
algebra have been considered. An example is the concept of the
cross product
of a group and a ring, which retains many properties of a group algebra.
See also the references to
Representation of a group.