On Poisson-Lie structure on the external algebra of the classical Lie groups

TL;DR

This paper presents a universal formula for the bicovariant bracket on the external algebra of Poisson-Lie groups, generalizing previous specific cases for $GL(N)$ and $SL(N)$ to any matrix Lie group.

Contribution

It provides a universal expression for the bicovariant bracket applicable to all matrix Lie groups, extending prior specific results.

Findings

Derived a general formula for the bicovariant bracket on external algebra

Unified previous results for $GL(N)$ and $SL(N)$ as special cases

Applicable to any matrix Lie group

Abstract

The general expression for the bicovariant bracket for odd generators of the external algebra on a Poisson-Lie group is given. It is shown that the graded Poisson-Lie structures derived before for $GL(N)$ and $SL(N)$ are the special cases of this bracket. The formula is the universal one and can be applied to the case of any matrix Lie group.