Last update: 6 November 2012
An algebra is a vector space over with a multiplication such that is a ring with identity and such that for all and
More precisely, an algebra is a vector space over with a multiplication that is associative, distributive, has an identity, and satisfies (1.1). Suppose that is a basis of and that are constants in such that
It follows from (1.1) and the distributive property that the equations (1.2) for completely determine the multiplication in The are called structure constants. The center of an algebra is the subalgebra
A nonzero element such that is called an idempotent. Two idempotents are orthogonal if A minimal idempotent is an idempotent that cannot be written as a sum of orthogonal idempotents
For each positive integer we denote the algebra of matrices with entries from and ordinary matrix multiplication by We denote the identity matrix in by For a general algebra denotes matrices with entries in We denote the algebra of matrices of the form
by Note that as algebras. The trace, of a matrix is the sum of the diagonal entries of
An algebra homomorphism of an algebra into an algebra is a map such that for all
A representation of an algebra is an algebra homomorphism
The dimension of the representation is The image of the representation is a finite dimensional algebra of matrices which we call the algebra of the representation It is a subalgebra of A faithful representation is a representation which is injective. In this case the algebra is called a faithful realization of and The character of the representation of is the function given by
An anti-representation of an algebra is a map such that for all
As before the dimension of the anti-representation is and the image, of the anti-representation is an algebra of matrices called the algebra of the anti-representation.
The group algebra of a group is the algebra of formal finite linear combinations of elements of where the multiplication is given by the linear extension of the multiplication in The elements of constitute a basis of A representation of the group is a representation of its group algebra.
Let be an algebra. An is a vector space with an action such that for all and
An homomorphism is a map between and such that for all and
An isomorphism is a bijective homomorphism.
By condition 3 of (1.5) the action of on is a linear transformation of If we specify a basis of then the linear transformation can be written as a matrix, where In this way we associate to every element of a matrix. This gives a representation of which we shall also denote by
Conversely, if is a dimensional representation of and is a dimensional vector space with basis then we can define the action of an element in by the action of the linear transformation on determined by the matrix so that for all
In this way becomes an Thus the notion of A is equivalent to the notion of representation. When we view the we are focusing on the vector space and when we view the representation we are focusing on the linear transformations (matrices).
Let be an with basis and let be another basis of and denote the change of basis matrix by Let and let be the matrices, with respect to the bases and respectively, of the linear transformation on induced by Then by elementary linear algebra we have that
This leads us to the following definition. Two dimensional representations and of an algebra are equivalent if there exists an invertible matrix such that (1.7) holds for all Isomorphic modules define equivalent representations.
The direct sum of two and is the of all pairs and with the action given by
for all The direct sum of two representations and of is the representation of given by
Direct sums of representations or are defined analogously. We denote factors, by Note that the algebra of the representation is
An subspace of an is a subspace of such that
An subspace of is just a submodule of Note that the intersection of any two invariant subspaces of is also an invariant subspace of
An with no submodules is a simple module. An irreducible representation is a representation that is not equivalent to a representation of the form
where is also representation of If are invariant subspaces of a representation and is irreducible then is either equal to 0 or A completely decomposable representation is a representation that is equivalent to a direct sum of irreducible representations. An algebra is called completely decomposable if every representation of is completely decomposable.
The centralizer of an algebra of matrices is the algebra of matrices such that for all matrices
The centralizer of a representation of an algebra is the algebra
1. Let be an algebra of matrices. Since all matrices in commute with all elements of
2. Schur's lemma. Let and be irreducible representations of of dimensions and If is a matrix such that
determines a linear transformation Since for all we have that
for all and Thus is an homomorphism. and are submodules of and respectively and are therefore either 0 or equal to or respectively. If or then In the remaining case is a bijection, and thus an isomorphism between and In this case we have that Thus the matrix is square and invertible.
Now suppose that and let be an eigenvalue of Then the matrix is such that for all The argument in the preceding paragraph shows that is either invertible or 0. But if is an eigenvalue of then Thus
3. Suppose that is a completely decomposable representation of an algebra and that where the are nonisomorphic irreducible representations of Schur's lemma shows that the -homomorphisms from to form a vector space The multiplicity of the irreducible representation un is
4. Suppose that is a completely decomposable representation of an algebra and that where the are nonisomorphic irreducible representations of and let Then If we view elements of as block diagonal matrices with blocks of size for each , then by using Ex 1 and Schur's lemma we get that
5. Let be an -module and let be an idempotent of Then is a subspace of and the action of on is a projection from to If are orthogonal idempotents of then and are mutually orthogonal subspaces of since if for some then So
6. Let be an idempotent in and suppose that for every for some constant If is not minimal then where are idempotents such that Then for some constant This implies that giving that either or So is minimal.
7. Let be a finite dimensional algebra and suppose that is an idempotent of If is not minimal then where and are orthogonal idempotents of If any idempotent in this sum is not minimal we can decompose it into a sum of orthogonal idempotents. We continue this process until we have decomposed as a sum of minimal orthogonal idempotents. At any particular stage in this process is expresed as a sum of orthogonal idempotents, So None of the spaces is 0 since and the spacers are all mutually orthogonal. Thus, since is finite dimensional it will only take a finite number of steps to decompose into minimal idempotents. A partition of unity is a decomposition of 1 into minimal orthogonal idempotents.
This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.