Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$

TL;DR

This paper classifies Lie bialgebra structures on the extended affine Lie algebra sl_2(\u211d_q), showing they are all triangular coboundary, by proving the triviality of a specific first cohomology group.

Contribution

It demonstrates that all Lie bialgebra structures on sl_2(_q) are triangular coboundary, extending techniques to more general affine Lie algebras.

Findings

All Lie bialgebra structures are triangular coboundary.

Proves the triviality of the first cohomology group for the algebra.

Techniques may apply to broader classes of affine Lie algebras.

Abstract

Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Lie bialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to be triangular coboundary. This result is obtained by employing some techniques, which may also work for more general extended affine Lie algebras, to prove the triviality of the first cohomology group of $\widetilde{sl_2(\mathbb{C}_q)}$ with coefficients in the tensor product of its adjoint module, namely, $H^1(\widetilde{sl_2(\mathbb{C}_q)},\widetilde{sl_2(\mathbb{C}_q)}\otimes\widetilde{sl_2(\mathbb{C}_q)})=0$.