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classification of finite-dimensional representations of semi-simple Lie algebras (Definition)

If $ \mathfrak{g}$ is a semi-simple Lie algebra, then we say that an representation $ V$ has highest weight $ \lambda$, if there is a vector $ v\in V_\lambda$, the weight space of $ \lambda$, such that $ Xv=0$ for $ X$ in any positive root space, and $ v$ is called a highest vector, or vector of highest weight.

There is a unique (up to isomorphism) irreducible finite dimensional representation of $ \mathfrak{g}$ with highest weight $ \lambda$ for any dominant weight $ \lambda\in\Lambda_W$, where $ \Lambda_W$ is the weight lattice of $ \mathfrak{g}$, and every irreducible representation of $ \mathfrak{g}$ is of this type.



"classification of finite-dimensional representations of semi-simple Lie algebras" is owned by bwebste.
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Also defines:  highest weight, highest vector, vector of highest weight, highest weight representation
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Cross-references: type, weight lattice, dominant weight, finite dimensional, irreducible, isomorphism, positive root, weight space, vector, representation, semi-simple Lie algebra
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This is version 2 of classification of finite-dimensional representations of semi-simple Lie algebras, born on 2002-12-04, modified 2007-03-02.
Object id is 3653, canonical name is ClassificationOfFiniteDimensionalRepresentationsOfSemiSimpleLieAlgebras.
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AMS MSC: 17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive )
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