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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. See also * Lie algebra representation * Weight (representation theory) In the mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weight (representation Theory)
In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n\times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Module
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space V together with a collection of operators on V satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
