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