Isotypic component - Alchetron, The Free Social Encyclopedia

The Isotypic component of weight λ of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight λ .

Definition

A finite-dimensional module V of a reductive Lie algebra g (or of the corresponding Lie group) can be decomposed into irreducible submodules

V = ⊕ i = 1 N V i .

Each finite-dimensional irreducible representation of g is uniquely identified (up to isomorphism) by its highest weight

∀ i ∈ { 1 , … , N } ∃ λ ∈ P ( g ) : V i ≃ M λ , where M λ denotes the highest weight module with highest weight λ .

In the decomposition of V , a certain isomorphism class might appear more than once, hence

V ≃ ⊕ λ ∈ P ( g ) ( ⊕ i = 1 d λ M λ ) .

This defines the isotypic component of weight λ of V: λ ( V ) := ⊕ i = 1 d λ V i ≃ C d λ ⊗ M λ where d λ is maximal.

References

Isotypic component Wikipedia

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