Differential Calculus on the Quantum Superspace and Deformation of Phase Space

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, $G{L}_q(m|n)$, 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 $G{L}_q(m|n)$ and the quantum superdeterminant. We apply these results for the $G{L}_q(m|n)$ to the deformed phase-space of supercoordinates and their momenta, from which we construct the $\hat{R}$-matrix of q-deformed orthosymplectic group $OS{p}_q(2n|2m)$ 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 $OS{p}_q(2|2)$.

17 pages, KUCP-47

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