# Cartan theorem

Cartan's theorem on the highest weight vector. Let $\mathfrak g$ be a complex semi-simple Lie algebra, let $e _{i} ,\ f _{i} ,\ h _{i}$ , $i = 1 \dots r$ , be canonical generators of it, that is, linearly-independent generators for which the following relations hold: $$[ e _{i} ,\ f _{j} ] = \delta _{ij} h _{i} , [ h _{i} ,\ e _{j} ] = a _{ij} e _{j} , [ h _{i} ,\ f _{j} ] = - a _{ij} f _{j} ,$$ $$[ h _{i} ,\ h _{j} ] = 0 ,$$ where $a _{ii} = 2$ , $a _{ij}$ are non-positive integers when $i \neq j$ , $i ,\ j = 1 \dots r$ , $a _{ij} = 0$ implies $a _{ji} = 0$ , and let $\mathfrak t$ be the Cartan subalgebra of $\mathfrak g$ which is the linear span of $h _{1} \dots h _{r}$ . Also let $\rho$ be a linear representation of $\mathfrak g$ in a complex finite-dimensional space $V$ . Then there exists a non-zero vector $v \in V$ for which $$\rho ( e _{i} ) v = 0 , \rho ( h _{i} ) v = k _{i} v , i = 1 \dots r ,$$ where the $k _{i}$ are certain numbers. This theorem was established by E. Cartan . The vector $v$ is called the highest weight vector of the representation $\rho$ and the linear function $\Lambda$ on $\mathfrak t$ defined by the condition $\Lambda ( h _{i} ) = k _{i}$ , $i = 1 \dots r$ , is called the highest weight of the representation $\rho$ corresponding to $v$ . The ordered set $( k _{1} \dots k _{r} )$ is called the set of numerical marks of the highest weight $\Lambda$ . Cartan's theorem gives a complete classification of irreducible finite-dimensional linear representations of a complex semi-simple finite-dimensional Lie algebra. It asserts that each finite-dimensional complex irreducible representation of $\mathfrak g$ has a unique highest weight vector (up to proportionality), and that the numerical marks of the corresponding highest weight are non-negative integers. Two finite-dimensional irreducible representations are equivalent if and only if the corresponding highest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensional complex irreducible representation.