Weights and Weight Spaces

Weight theory provides a systematic method to decompose and classify representations of Lie algebras through simultaneous eigenspaces of abelian subalgebras.

DefinitionCartan Subalgebra

A Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ is a maximal abelian subalgebra consisting of semisimple elements (diagonalizable in the adjoint representation).

For $\mathfrak{sl}_n(\mathbb{C})$ , the standard Cartan subalgebra is the diagonal matrices: $\mathfrak{h} = \{\text{diag}(a_1, \ldots, a_n) : \sum a_i = 0\}$ .

DefinitionWeight

Let $V$ be a representation of $\mathfrak{g}$ and $\mathfrak{h}$ a Cartan subalgebra. A weight is a linear functional $\lambda \in \mathfrak{h}^*$ such that the weight space: $V_\lambda = \{v \in V : H \cdot v = \lambda(H) v \text{ for all } H \in \mathfrak{h}\}$ is non-zero.

An element $v \in V_\lambda$ is called a weight vector of weight $\lambda$ .

TheoremWeight Space Decomposition

For a finite-dimensional representation $V$ of a semisimple Lie algebra $\mathfrak{g}$ : $V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda$

The direct sum is finite, and the weights form a finite subset of $\mathfrak{h}^*$ .

ExampleWeights of $\mathfrak{sl}_2$ Representations

For the $(n+1)$ -dimensional irreducible representation $V_n$ of $\mathfrak{sl}_2$ , with Cartan element $H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ :

The weights are $\{n, n-2, n-4, \ldots, -n\}$ (integers differing by 2, from $n$ to $-n$ ).

Each weight space is one-dimensional: $\dim V_{n-2k} = 1$ for $k = 0, 1, \ldots, n$ .

DefinitionHighest Weight

A highest weight is a maximal weight with respect to a chosen ordering on $\mathfrak{h}^*$ . A highest weight vector $v_\lambda$ satisfies:

$H \cdot v_\lambda = \lambda(H) v_\lambda$ for all $H \in \mathfrak{h}$
$E_\alpha \cdot v_\lambda = 0$ for all positive root spaces $\mathfrak{g}_\alpha$

The highest weight module $V(\lambda)$ is generated by a highest weight vector.

Remark

Weight theory is fundamental because:

Classification: Irreducible representations are determined by highest weights
Character formulas: Weyl character formula expresses characters in terms of weights
Tensor products: Decomposition via weight multiplicities
Geometric interpretation: Weights correspond to torus eigenvalues, connecting to moment maps

For semisimple Lie algebras, the set of dominant integral weights parametrizes finite-dimensional irreducible representations, providing a complete classification.