Last update: 12 August 2013
Definition 1.1 An algebra is a vector space over the complex numbers with multiplication such that is an associative ring with identity and
The last condition merely says that the subalgebra is central, i.e. the scalars are scalars.
Homework Problem 1.2
|1.||Develop the following theory for algebras over other fields. Identify places where prime characteristic or lack of algebraic closure causes problems.|
|2.||Develop this theory for Lie algebras, Jordan algebras, or alternative algebras, etc.|
|3.||Develop a specific base theoretic cohomology for semisimple algebras.|
Example 1.3 An important example of an associative algebra with identity is the set of all matrices over the complex numbers with the usual matrix multiplication. We will write
for the identity element. Then the scalars are isomorphic to the subalgebra of scalar matrices
We will often work with particular bases of an algebra.
Definition 1.4 Let be a finite dimensional algebra and choose a basis The structure constants of are the scalars defined by the multiplication rules
Alternatively, one may define an algebra structure on a vector space by specifying a set of structure constants. Of course, the structure constants depend on the particular choice of basis.
For example, in one might choose the basis of matrix units
where is the matrix with a 1 in the row and column, and zeros elsewhere. If is a matrix, then we express in this basis. Matrix multiplication is merely the linear extension of the multiplication rule
and so the structure constants are given by
We turn next to defining our notions of representation and module.
Definition 1.5 An algebra homomorphism is a map such that
|1.||is an additive group homomorphism)|
Definition 1.6 An algebra homomorphism is called a representation. The number is called the dimension of the representation.
Group representations fit into this scheme as follows. One notion is that a representation of a group is a group homomorphism Equivalently, we may consider group representations as a special case of algebra representations as follows:
Definition 1.7 The group algebra (often written is the vector space of finite combinations of elements of with multiplication given as the linear extension of the group multiplication.
Example 1.8 If then
with multiplication defined as in the following example:
where the products are taken in the group
By definition, the elements of form a basis for
Definition 1.9 A group representation of a group is an algebra representation of the group algebra
Definition 1.10 A module for an algebra is a vector space over together with an that is a map denoted such that
We next explore the relationship between modules and representations. Let be a vector space of dimension and let be a basis of If is a linear transformation then in terms of this basis we may express for some scalars The map given by is an algebra isomorphism.
Now suppose that is an If then for some scalars We claim that defined by is a representation of The bilinearity of the action implies is a linear transformation. Clearly and it remains to check that is multiplicative. Let and write and Then
By definition, hence is a representation of
Thus, given an and a choice of basis, we can produce a representation of What happens if we choose a different basis of
Let be another basis of and let be the representation defined as above using this new basis.
Definition 1.11 Two representations and are equivalent provided there exists a matrix such that
If we take to be the change of basis matrix such that then
hence i.e. the representations are equivalent. We have shown that up to equivalence, a module corresponds to a unique representation.
On the other hand, if we have a representation then letting be the space of column vectors produces an by
Definition 1.12 Given modules and a linear transformation is a module homomorphism provided for all and We say and are isomorphic (and write provided is invertible.
Homework Problem 1.13
|1.||Show that two and are isomorphic if and only if their associated representations are equivalent.|
|2.||Given two equivalent representations and Show that and induce isomorphic structures on the space of column vectors|
Thus module and representation are equivalent notions. In the future we will not separate the two halves of the brain, but instead use these concepts simultaneously. Often, we will denote a module and its corresponding representation by a single letter, i.e. the module affords the representation defined by
Definition 1.14 A submodule is a vector subspace of that is invariant under the action of that is for all and
A representation is a subrepresentation of provided and is equivalent to a representation of the form
Definition 1.15 A module is simple provided has no nonzero proper submodules.
A representation irreducible provided has no nonzero proper subrepresentations.
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.