Weight space

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V defines a linear functional on the subalgebra U of End(V) generated by the set of endomorphisms S; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This map is also multiplicative, and sends the identity to 1; thus it is an algebra homomorphism from U to the base field. This "generalized eigenvalue" is a prototype for the notion of a weight.

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