Lie algebras graded by the weight system $({\Theta }_n,s{l}_n)$

Abstract

A Lie algebra $L$ is said to be $({\Theta }_n,s{l}_n)$-graded if it contains a simple subalgebra $\mathfrak{g}$ isomorphic to $s{l}_n$ such that the $\mathfrak{g}$-module $L$ decomposes into copies of the adjoint module, the trivial module, the natural module $V$, its symmetric and exterior squares $S^{2}V$ and ${\wedge }^{2}V$ and their duals. We describe the multiplicative structures and the coordinate algebras of $({\Theta }_n,s{l}_n)$-graded Lie algebras for $n\ge 5$, classify these Lie algebras and determine their central extensions.

to appear in J.Algebra