Lie algebras graded by the weight system $(\Theta_{n},sl_{n})$

Abstract

A Lie algebra $L$ is said to be $(\Theta_{n},sl_{n})$-graded if it contains a simple subalgebra $\mathfrak{g}$ isomorphic to $sl_{n}$ 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 $(\Theta_{n},sl_{n})$-graded Lie algebras for $n\ge5$, classify these Lie algebras and determine their central extensions.