Quantization of the external algebra on a Poisson-Lie group

TL;DR

This paper classifies the graded Poisson-Lie structures on the external algebra of $GL(N)$, showing only two such structures exist, which correspond to the classical limits of quantum differential calculi on $GL_q(N).

Contribution

It explicitly describes the two graded Poisson-Lie structures on the external algebra of $GL(N)$ and links them to quantum group calculus.

Findings

Only two graded Poisson-Lie structures exist on the external algebra.

These structures are the classical limits of quantum differential calculi.

The explicit forms of these structures are provided.

Abstract

We show that the external algebra $\cal M$ on $GL(N)$ can be equipped with the graded Poisson brackets compatible with the group action. We prove that there are only two graded Poisson-Lie structures (brackets) on $\cal M$ and we obtain their explicit description. We realize that just these two structures appear as the quasiclassical limit of the bicovariant differential calculi on the quantum linear group $GL_q (N)$.