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 in which quatum supergroups are realized. Multiparametric quantum deformation of the general linear supergroup, GL q (m|n), is studied and the explicit form for theR-matrix is presented. We apply these results to the quantum phase-space construction of OSp q (2n|2m) and calculate theirR-matrices.

Covariant noncommutative differential geometry

“…To include super-spaces in the R-matrix approach of constructing the NCDG, one should carefully cake into account the sign-rule connected with the Z2-grading (cf., e.g., [22,23]). The quantum linear superspace Mq(rnln) corresponding to the quantum super-group Fq(GL(rnlr~)) is generated by even elements (xl,... , x,~), p(xi) = 0, and odd elements ((1,... , (~), P((~) = 1.…”

Duality on the quantum space(3) with two parameters

“…We also obtain the following anti-commutative relations between two the Cartan-Maurer generators, using the relations in (18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32):…”

“…In this direction, many efforts have been accomplished in order to develop noncommutative differential structures on quantum spaces(groups) [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Among them, as a fundamental work, the noncommutative differential calculus on quantum groups is introduced by Woronowicz [18].…”

Quantum deformedsu(m/n) algebra and superconformal algebra on quantum superspace

“…Namely, the quantum group transforms covariantly commutation relations of the quantum space. These commutation relations between coordinates Z I (x i , θ α ) (i = 1 ∼ m, α = 1 ∼ n) and derivatives ∂ I (∂ i , ∂ α ) of the quantum superspace are obtained in terms of a R-matrix for GL q (m|n) [11,15] as follows,…”

“…Also a quantum space has been studied intensively as a non-commutative space representing the quantum group [7,9]. Differential calculus on the quantum space is very intriguing as an application of the quantum group and useful to show interesting aspects of the quantum groups [10][11][12][13][14][15].…”