Weight space

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In mathematics, especially in the fields of representation theory of Lie groups and Lie algebras, a weight space is a generalization of eigenspace of an operator. The generalization of eigenvalue in this case is called a weight.

[edit] Definition

Let $V$ be a representation of a Lie algebra $\mathfrak{g}$ and assume that a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is chosen. A weight space $V_\mu\subset V$ of weight $\mu\in\mathfrak{h}^*$ is defined by

$V_\mu:=\{v\in V; \forall h\in\mathfrak{h}\quad h\cdot v=\mu(h)v\}$

Similarly, we can define a weight space $V μ$ for representation of a Lie group resp. an associative algebra as the subspace of eigenvectors of some maximal commutative subgroup resp. subalgebra of the eigenvalue $μ$ .

Elements of the weight spaces are called weight vectors.

[edit] References

Fulton W., Harris J., Representation theory: A first course, Springer, 1991
Goodmann R., Wallach N. R., Representations and Invariants of the Classical Groups, Cambridge University Press, Cambridge 1998.
Humphreys J., Introduction to Lie Algebras and Representation Theory, Springer Verlag, 1980.
Knapp A. W., Lie Groups Beyond an introduction, Second Edition, (2002)
Roggenkamp K., Stefanescu M., Algebra - Representation Theory, Springer, 2002.