Last update: 27 August 2013
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Definition 3.1 If is an algebra, let denote the matrices with entries in
Definition 3.2 Let be a subset of The centralizer or commutant of is the set Note that although the inclusion may be proper.
Proposition 3.3 For any subset is an algebra over
The set inherits a ring structure from since for all and all Moreover, and so must lie in the center of Hence acts as scalars on which gives the vector space and algebra structures.
Proposition 3.4 If are subsets then
Since the previous proposition implies that But
Let be a representation of an algebra so The image of is the algebra of the representation is a subalgebra of We are interested in the centralizer algebra of the representation
From the module perspective, suppose is an and let be the set of all linear transformations of Then
Example 3.7 Let be an arbitrary algebra and let be an irreducible representation of We claim
Let Then for all and so by Schur’s lemma, for some
Example 3.8 Let be an (associative) algebra. Recall the left regular representation given by What is
The associative law shows that the algebra of transformations given by right multiplication by elements of
Let and let be defined by Then for all Hence acts on as right multiplication by and so
We are interested in the centralizer algebras of arbitrary semisimple algebras, so we will develop some notation for block matrix algebras.
Definition 3.9 Let be a matrix and be a matrix, both over the same field (or algebra). The tensor product of and is the matrix with entries in described by the block structure For example, is zero other than the block in the position, where it agrees with
Problem 3.10 Suppose and are and matrices, respectively, and suppose also that and respectively, are and (so that we may form the products and
Let be a subalgebra of We defined to be matrices with entries in so elements of are of the form where the are matrices in Thus, is a subalgebra of Let us compute the centralizer which also lies inside
Suppose and write where each is a complex matrix. Then commutes with all elements of i.e. for all choices of matrices
If we multiply this out, the left hand side becomes Performing the same calculation on the right hand side and setting it equal to the previous computation gives Comparing coefficients of we see that for all choices of elements for each
Since is a subalgebra, we know 0 and lie in Fix indices and choose elements given by and all others zero. Then
Make a second choice of elements, with where and all other Equation 3.11 becomes
Hence is block diagonal. Moreover, 3.11 becomes and so Thus hence (where we mean the set of matrices of the form for The reverse inclusion is left as an exercise, and we claim
Proposition 3.12 If is a subalgebra of then
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.