Modified symplectic structures in cotangent bundles of Lie groups

TL;DR

This paper extends the concept of modified symplectic structures from affine spaces to cotangent bundles of Lie groups, allowing for non-commutative variables while maintaining mathematical consistency.

Contribution

It demonstrates that the extended symplectic structure with non-commutative variables can be consistently applied to cotangent bundles of Lie groups, generalizing previous affine space results.

Findings

Extension is mathematically consistent for Lie groups.

Non-commutative configuration and momentum variables are incorporated.

Provides a framework for non-commutative symplectic geometry on Lie groups.

Abstract

In earlier work (*) we studied an extension of the canonical symplectic structure in the cotangent bundle of an affine space ${\cal Q}={\bf R}^N$, by additional terms implying the Poisson non-commutativity of both configuration and momentum variables. In this article, we claim that such an extension can be done consistently when ${\cal Q}$ is a Lie group $G$. -- (*) : F.J.Vanhecke, C.Sigaud and A.R.da Silva, arXiv:math-phys/0502003 and Braz.J.Phys.{\bf 36},no IB,194(2006)