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Let <math>V</math> be a ] of a ] <math>\mathfrak{g}</math> and assume that Let <math>V</math> be a ] of a ] <math>\mathfrak{g}</math> and assume that
a ] <math>\mathfrak{h}</math> of <math>\mathfrak{g}</math> is chosen. A weight space <math>V_\mu\subset V</math> of ] <math>\mu\in\mathfrak{h}^*</math> is defined by a ] <math>\mathfrak{h}</math> of <math>\mathfrak{g}</math> is chosen. A weight space <math>V_\mu\subset V</math> of ] <math>\mu\in\mathfrak{h}^*</math> is defined by

Revision as of 04:14, 1 November 2007

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

V μ := { v ∈ V ; ∀ h ∈ h h ⋅ v = μ ( h ) v } {\displaystyle 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 {\displaystyle V_{\mu }} 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 {\displaystyle \mu } .

Elements of the weight spaces are called weight vectors.

See also

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