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

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.

Title classification of finite-dimensional representations of semi-simple Lie algebras ClassificationOfFinitedimensionalRepresentationsOfSemisimpleLieAlgebras 2013-03-22 13:11:40 2013-03-22 13:11:40 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 17B20 highest weight highest vector vector of highest weight highest weight representation