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

Abstract

Lie bialgebra structures on the extended affine Lie algebra sl 2 (C q ) are investigated.In particular, all Lie bialgebra structures on sl 2 (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 sl 2 (C q ) with coefficients in the tensor product of its adjoint module, namely, H 1 ( sl 2 (C q ), sl 2 (C q ) ⊗ sl 2 (C q )) = 0.

Quantizations of the extended affine Lie algebra $\widetilde{\frak{sl}_2(\mathbb{C}_q)}$

“…Although a general method for twisting both the product and coproduct of a bialgebra does not appear, it is possible to twist the corresponding coproduct in such a way that it remains compatible with its original multiplication, unit, and counit (see [18]). In this paper, we shall concentrate on the quantization being assort to the so-called Drinfel'd twist of the extended affine Lie algebra (EALA) sl 2 (C q ), whose Lie bialgebra structures were determined in [25]. The EALA sl 2 (C q ) was first introduced in [19] in the sense of quasisimple Lie algebras and systematically investigated in [2].…”