Drinfeld double of deformed quantum algebras

TL;DR

The paper introduces a deformation of Lusztig algebra that unifies various quantum algebras and constructs their Drinfeld doubles, revealing deep structural connections and isomorphisms.

Contribution

It presents a new deformation algebra encompassing multiple quantum algebra types and constructs their Drinfeld doubles, establishing isomorphisms and unifying frameworks.

Findings

Various quantum algebras are specializations of the deformed algebra.

The deformed algebra is isomorphic to Lusztig algebra up to a twist.

Constructed the Drinfeld double of the deformed algebra.

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.