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 \( V \).
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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.