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Wikipedia / Algebra representation

In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.

Examples

Linear complex structure

One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R which corresponds to . Then a representation of C is a real vector space V, together with an action of C on V). Concretely, this is just an action of  , as this generates the algebra, and the operator representing (the image of in End(V)) is denoted J to avoid confusion with the identity matrix I.

Polynomial algebras

Another important basic class of examples are representations of polynomial algebras, the free commutative algebras – these form a central object of study in commutative algebra and its geometric counterpart, algebraic geometry. A representation of a polynomial algebra in variables over the field K is concretely a K-vector space with

A basic result about such representations is that, over an algebraically closed field, the representing matrices are simultaneously triangularisable.

and is used in understanding the structure of a single linear operator on a finite-dimensional vector space. Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form.

In some approaches to noncommutative geometry, the free noncommutative algebra (polynomials in non-commuting variables) plays a similar role, but the analysis is much more difficult.

Weights

Eigenvalues and eigenvectors can be generalized to algebra representations.

The generalization of an eigenvalue (i.e., an algebra homomorphism from the algebra to its underlying ring: a linear functional that is also multiplicative). This is known as a weight, and the analog of an eigenvector and eigenspace are called weight vector and weight space.

is determined by which scalar it maps the generator T is bilinear, "which multiple" is an A-linear functional of A (an algebra map A → R – note that on the left, multiplication is the algebra action, while on the right, multiplication is scalar multiplication.

Because a weight is a map to a commutative ring (because in both cases it is just multiplication by scalars), so common eigenvectors of an algebra must be in the set on which the algebra acts commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the polynomial algebras in a set of commuting matrices, a weight vector of this algebra is a simultaneous eigenvector-space. These weights – in particularly their geometry – are of central importance in understanding the representation theory of Lie algebras, specifically the finite-dimensional representations of semisimple Lie algebras.

generators, it corresponds geometrically to an algebraic variety-dimensional space, and the weight must fall on the variety – i.e., it satisfies the defining equations for the variety. This generalizes the fact that eigenvalues satisfy the characteristic polynomial of a matrix in one variable.

See also

Notes

References