On the Hopf structure of $U_{p,q}(gl(1|1))$ and the universal ${\cal T}$-matrix of $Fun_{p,q}(GL(1|1))$

TL;DR

This paper explores the Hopf algebra structure of the quantum superalgebra $U_{p,q}(gl(1|1))$ and derives its universal ${ m T}$-matrix, extending duality techniques to the superalgebra case.

Contribution

It extends the Hopf duality framework to the superalgebra $U_{p,q}(gl(1|1))$ and constructs its universal ${ m T}$-matrix, revealing the exponential relationship with $Fun_{p,q}(GL(1|1))$.

Findings

Derived the Hopf structure of $U_{p,q}(gl(1|1))$.

Constructed the universal ${ m T}$-matrix for $Fun_{p,q}(GL(1|1))$.

Established the exponential relationship between the algebra and its dual.

Abstract

Using the technique developed by Fronsdal and Galindo (Lett. Math. Phys. 27 (1993) 57) for studying the Hopf duality between the quantum algebras $Fun_{p,q}(GL(2))$ and $U_{p,q}(gl(2))$, the Hopf structure of $U_{p,q}(gl(1|1))$, dual to $Fun_{p,q}(GL(1|1))$, is derived and the corresponding universal ${\cal T}$-matrix of $Fun_{p,q}(GL(1|1))$, embodying the suitably modified exponential relationship $U_{p,q}(gl(1|1))$ $\rightarrow$ $Fun_{p,q}(GL(1|1))$, is obtained.