Intertwining operator
A continuous linear operator such that
, where
and
are mappings of a set
into two topological vector spaces
and
and
. This concept is especially fruitful in the case when
is a group or an algebra and
are representations of
. The set of intertwining operators forms the space
, which is a subspace of the space of all continuous linear mappings from
to
. If
and
, then
and
are called disjoint representations. If
contains an operator that defines an isomorphism of
and
, then
and
are equivalent. If
are locally convex spaces, if
and
are their adjoints, and if
and
are the representations contragredient to
and
, respectively (cf. Contragredient representation), then for any
, the operator
is contained in
. If
and
are finite-dimensional or unitary representations and
is irreducible, then
admits a subrepresentation equivalent to
if and only if
. See also Intertwining number.
References
[1] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) |
[2] | A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian) |
Intertwining operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intertwining_operator&oldid=47402