 Weight (representation theory)

A weight of a representation is a simultaneous eigenvalue functional for a commuting Cartan subalgebra. In general algebra representations it can be expressed as a one-dimensional character of the acting commutative algebra. A weight vector is an eigenvector carrying that functional; a weight space is the corresponding eigenspace, and weight multiplicity is its dimension.

In the context of representation theory, particularly in the study of Lie algebras and Lie groups, the term "weight" refers to a specific type of character associated with a representation. ### Key Points about Weights: 1. **Representation of a Lie Algebra**: A representation of a Lie algebra \( \mathfrak{g} \) on a vector space \( V \) involves a linear action of the algebra on \( V \).