Noncommutative supergeometry and quantum supergroups

Abstract

This is a review of concepts of noncommutative supergeometry -namely Hilbert superspace, C*-superalgebra, quantum supergroup -and corresponding results. In particular, we present applications of noncommutative supergeometry in harmonic analysis of Lie supergroups, non-formal deformation quantization of supermanifolds, quantum field theory on noncommutative spaces; and we give explicit examples as deformation of flat superspaces, noncommutative supertori, solvable topological quantum supergroups.

Double-graded quantum superplane

“…For example, R-R field backgrounds lead to 'θ − θ' deformations and gravitino backgrounds lead to 'x − θ' deformations (see [16,34]). It is probably fair to say that the mathematics literature on 'noncommutative superspaces' is not so developed (the reader may consult [17] for an overview).…”

On a ℤ2n-Graded Version of Supersymmetry

“…However, as almost all of the differential geometry of supermanifolds and Z n 2 -manifolds can be treated an interpreted using the methods from classical differential geometry and algebraic geometry. Thus, one should view Z n 2 -manifolds as a starting place for more general noncommutative supergeometries where more abstract algebraic methods are needed (see for example de Goursac [26], Grosse & Reiter [30] and Schwarz [48]).…”