Irreducible Lie algebras of infinite type

Abstract:

Let V be a finite dimensional vector space over an algebraically closed field of characteristic $\ne 2,3,5$. It is shown that if $L \subseteq {\text {gl}}(V)$ is an irreducible Lie algebra of infinite type then either $V = 2r \geqq 4$ and $L = {\text {sp}}(V),\dim V = 2r \geqq 4$ and $L = {\text {csp}}(V)$, or there exists $A \in L$ such that $A \ne 0 = {({\text {ad}}\;A)^2}$. As a corollary we obtain E. Cartan’s classification of the irreducible Lie algebras of infinite type over C.