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Isotypic component : ウィキペディア英語版
Isotypic component
The Isotypic component of weight \lambda of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight \lambda.
==Definition==

* A finite-dimensional module V of a reductive Lie algebra \mathfrak (or of the corresponding Lie group) can be decomposed into irreducible submodules
: V = \oplus_^N V_i .
* Each finite-dimensional irreducible representation of \mathfrak is uniquely identified (up to isomorphism) by its highest weight
:\forall i \in \ \exists \lambda \in P(\mathfrak) : V_i \simeq M_\lambda, where M_\lambda denotes the highest weight module with highest weight \lambda.
* In the decomposition of V , a certain isomorphism class might appear more than once, hence
: V \simeq \oplus_ ( \oplus_^ M_) .
This defines the isotypic component of weight \lambda of V:
\lambda(V) := \oplus_^ V_i \simeq \mathbb^ \otimes M_ where d_\lambda is maximal.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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