classification of finite-dimensional representations of semi-simple Lie algebras

If $𝔤$ 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 $𝔤$ with highest weight $\lambda$ for any dominant weight $\lambda \in {\mathrm{\Lambda }}_W$, where ${\mathrm{\Lambda }}_W$ is the weight lattice of $𝔤$, and every irreducible representation of $𝔤$ is of this type.

Title	classification of finite-dimensional representations of semi-simple Lie algebras
Canonical name	ClassificationOfFinitedimensionalRepresentationsOfSemisimpleLieAlgebras
Date of creation	2013-03-22 13:11:40
Last modified on	2013-03-22 13:11:40
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	5
Author	bwebste (988)
Entry type	Definition
Classification	msc 17B20
Defines	highest weight
Defines	highest vector
Defines	vector of highest weight
Defines	highest weight representation