The Drinfeld double gl(n) \oplus t_n

TL;DR

This paper presents a new perspective on the quantum deformation of gl(n) by constructing it as a self-dual Drinfeld double using Borel subalgebras and a natural Lie bialgebra structure.

Contribution

It introduces a novel construction of gl(n) as a self-dual Drinfeld double, providing insights into its quantum deformation.

Findings

Constructed gl(n) as a self-dual Drinfeld double.

Identified a natural Lie bialgebra structure on gl(n).

Provided a new perspective on standard quantum deformation.

Abstract

The two isomorphic Borel subalgebras of gl(n), realized on upper and lower triangular matrices, allow us to consider the gl(n) \opus t_n algebra as a self-dual Drinfeld double. Compatibility conditions impose the choice of an orthonormal basis in the Cartan subalgebra and fix the basis of gl(n). A natural Lie bialgebra structure on gl(n) is obtained, that offers a new perspective for its standard quantum deformation.