Standard Complex for Quantum Lie Algebras

Abstract

For a quantum Lie algebra $Γ$, let $Γ^\wedge$ be its exterior extension (the algebra $Γ^\wedge$ is canonically defined). We introduce a differential on the exterior extension algebra $Γ^\wedge$ which provides the structure of a complex on $Γ^{\wedge}$. In the situation when $Γ$ is a usual Lie algebra this complex coincides with the "standard complex". The differential is realized as a commutator with a (BRST) operator $Q$ in a larger algebra $Γ^\wedge[Ω]$, with extra generators canonically conjugated to the exterior generators of $Γ^{\wedge}$. A recurrent relation which defines uniquely the operator $Q$ is given.