Deformation quantization for Heisenberg supergroup

Abstract

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R d . However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R m|n as a byproduct of Weyl ordered Kirillov's orbit method adapted to the graded setting. To do so, we have to introduce the notion of C*-superalgebra, which is compatible with the deformation, and which can be seen as corresponding to noncommutative superspaces. We also use this construction to interpret the renormalizability of a noncommutative Quantum Field Theory.

Superunitary Representations of Heisenberg Supergroups

“…, 3. Conversely, a fundamental symmetry J defines a fundamental decomposition given by The above definition of Hilbert superspace is slightly more general that the one provided in [16], where a fundamental symmetry J is fixed as part of the structure. This freedom in the choice of J will be important in the following sections.…”

“…Definition 3.7 ( [16]) A superunitary operator between H (1) and H (2) is a homogeneous operator Φ ∈ B(H (1) , H (2) ) of degree 0 satisfying:…”

“…Their natural superalgebraic structures induce topology, in a similar fashion as for Krein spaces. Building on [16], we introduce the following definition of Hilbert superspace. Definition 1.3 A Hilbert superspace of parity σ ∈ Z 2 is a complex Z 2 -graded vector space H = H 0 ⊕ H 1 endowed with a homogeneous superhermitian inner product −, − of degree σ, such that there exists a fundamental symmetry J, i.e., an endomorphism J of H of degree σ satisfying, ∀x, y ∈ H,…”

Noncommutative supergeometry and quantum supergroups

“…Some work in the direction of noncommutative Q-manifolds was also performed in [9,10]. However, at the analytic level, NCSG was really initiated in [11] with the definition of its basic objects: Hilbert superspaces and C*-superalgebras that endow noncommutative superspaces with topological properties. In this paper, we review these analytic notions, as well as the corresponding notion of symmetry -quantum supergroup -defined and studied in the Fréchet setting [12].…”

Developments and Retrospectives in Lie Theory

“…Moreover the resulting family of Fréchet algebras {(S(G), θ )} θ∈R deforms the commutative Fréchet algebra structure on S(G) given by the pointwise multiplication of functions corresponding to the value θ = 0 of the deformation parameter. Note that such a program was achieved in [17] for abelian Lie groups and in [8] for abelian Lie supergroups.…”