Contraction of a representation
onto an invariant subspace
The contraction of a representation of a group
onto a subgroup
(of an algebra
onto a subalgebra
) is the representation
of the group (algebra)
defined by the formula
for all
. A contraction of a representation
is also called a restriction (or reduction) of the representation
onto an invariant subspace or onto a subgroup or subalgebra. If
is a continuous representation, then a contraction of
is also continuous.
Comments
More usually one speaks of restriction of a representation and reduction of a representation. More precisely, if is a representation of a group, algebra
and
is a subgroup, subalgebra
then
can be restricted to
. Secondly, if
is an invariant subspace of
, then
induces a representation of
(and
) in
yielding a subrepresentation of the given representation; this is sometimes called reduction of a representation to an invariant subspace. The phrase contraction of a representation refers more often to the following situation. Consider a representation
over a ring
of an algebra
(or group) in an
-module
,
. Suppose that there is a representation
of
over
,
such that
(i.e.
for all
). Here
denotes the module of power series in
with coefficients in
. Then the representation
is said to be a contraction of
and
is called a deformation of
. Intuitively
is an (infinitesimal) family of representations parametrized by
. More generally one also considers the situation where the algebra
is also deformed at the same time, so that one has an algebra
over
such that
and a representation
of
over
. Cf. also Deformation.
Contraction of a representation. A.I. Shtern (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contraction_of_a_representation&oldid=17456