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.
Categories: math.RA
