Drinfeld double of deformed quantum algebras

Abstract

We provide a deformation, ${\mathfrak{f}}_{\beta }$, of Lusztig algebra $\mathbf{f}$. Various quantum algebras in literatures, including half parts of two-parameter quantum algebras, quantum superalgebras, and multi-parameter quantum algebras/superalgebras, are all specializations of ${\mathfrak{f}}_{\beta }$. Moreover, ${\mathfrak{f}}_{\beta }$ is isomorphic to Lusztig algebra $\mathbf{f}$ up to a twist. As a consequence, half parts of those quantum algebras are isomorphic to Lusztig algebra $\mathbf{f}$ over a big enough ground field up to certain twists. We further construct the entire algebra ${\mathbf{U}}_{\beta ,\xi }$ by Drinfeld double construction. As special cases, above quantum algebras all admit a Drinfeld double construction under certain assumptions.

23 pages

References in corpus (1)

• Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{{\mathfrak{sl}}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis