Differential Calculus on the Quantum Superspace and Deformation of Phase Space

TL;DR

This paper develops non-commutative differential calculus on quantum superspaces with bosonic and fermionic coordinates, studies multi-parametric quantum deformations of supergroups, and constructs related R-matrices for deformed phase spaces.

Contribution

It introduces explicit R-matrices and quantum determinants for quantum supergroups, extending deformation theory to supersymmetric phase spaces.

Findings

Derived quantum matrix relations for $GL_q(m|n)$

Constructed R-matrices for $OSp_q(2n|2m)$

Analyzed quantum super-Clifford algebras

Abstract

We investigate non-commutative differential calculus on the supersymmetric version of quantum space where the non-commuting super-coordinates consist of bosonic as well as fermionic (Grassmann) coordinates. Multi-parametric quantum deformation of the general linear supergroup, $GL_q(m|n)$, is studied and the explicit form for the ${\hat R}$-matrix, which is the solution of the Yang-Baxter equation, is presented. We derive the quantum-matrix commutation relation of $GL_q(m|n)$ and the quantum superdeterminant. We apply these results for the $GL_q(m|n)$ to the deformed phase-space of supercoordinates and their momenta, from which we construct the ${\hat R}$-matrix of q-deformed orthosymplectic group $OSp_q(2n|2m)$ and calculate its ${\hat R}$-matrix. Some detailed argument for quantum super-Clifford algebras and the explict expression of the ${\hat R}$-matrix will be presented for the case of $OSp_q(2|2)$.