Hochschild Cohomology and Deformations of Clifford-Weyl Algebras

Musson, Ian M.; Pinczon, Georges; Ushirobira, Rosane

doi:10.3842/sigma.2009.028

Cited by 5 publications

(9 citation statements)

References 32 publications

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“…Note that this result is already given at the formal deformation level in older papers (see for instance [28], where the deformed algebra is isomorphic to a Clifford-Weyl algebra). If the parameter α goes to 0, the part R n in S(M ) R n ⊗ S(BM ) remains undeformed while the part S(BM ) is endowed with the Moyal product.…”

Section: Universal Deformation Formulamentioning

confidence: 53%

Deformation quantization for Heisenberg supergroup

Bieliavsky

Goursac

Tuynman

2012

Journal of Functional Analysis

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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.

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Section: Universal Deformation Formulamentioning

confidence: 53%

Deformation quantization for Heisenberg supergroup

Bieliavsky

Goursac

Tuynman

2012

Journal of Functional Analysis

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“…We would like to remark that the Clifford-Weyl super algebra Cliff q (k) ⊗ A p (k) appearing in the previous theorem has the Z/2Z-grading given by usual grading of the Clifford (super) algebra Cliff q (k) and by considering the Weyl algebra A p (k) to be concentrated in degree zero (see [19], Example 1.2). This differs from the grading of the "Clifford-Weyl algebras" C(q, 2p) considered in [27], since in that case the Weyl algebra has also a nontrivial homogeneous component of odd degree.…”

Section: Introductionmentioning

confidence: 77%

Representations of Super Yang-Mills Algebras

Herscovich¹

2012

Commun. Math. Phys.

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We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are a generalization of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in [11], but in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras Cliffq(k) ⊗ Ap(k), for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.

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“…We investigate their symplectic structures and the deformation of their algebras of functions by the Moyal product, which leads to Clifford algebras. This is well-known, and develop for example in [18,36].…”

Section: Symplectic Supermanifolds Over One Pointmentioning

confidence: 90%

“…This leads to the Poisson bracket {ξ i , ξ j } = − i g ij , which is given by the bivector π = i 2 g ij ∂ ξ i ⊗ ∂ ξ j . Following [36], we introduce m ⋆ t = m ∧ • exp(t 2i π), the deformation of the exterior product m ∧ on ΛV * in the direction of π. As ΛV * is finite dimensional, m ⋆ 1 is well-defined, the exponential reducing to a finite sum.…”

Section: Symplectic Supermanifolds Over One Pointmentioning

confidence: 99%

Conformal Geometry of the Supercotangent and Spinor Bundles

Michel

2012

Commun. Math. Phys.

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We study the actions of local conformal vector elds X ∈ conf(M, g) on the spinor bundle of (M, g) and on its classical counterpart: the supercotangent bundle M of (M, g). We rst deal with the classical framework and determine the Hamiltonian lift of conf(M, g) to M. We then perform the geometric quantization of the supercotangent bundle of (M, g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map. The quantum and classical actions of conf(M, g) turn, respectively, the space of di erential operators acting on spinor densities and the space of their symbols into conf(M, g)-modules. They are ltered and admit a common associated graded module. In the conformally at case, the latter helps us determine the conformal invariants of both conf(M, g)-modules, in particular the conformally odd powers of the Dirac operator.Keywords: onform l geometryD symple ti superm nifoldD spin geometryD geometri qu nE tiz tionD onform lly inv ri nt di'erenti l oper torsF

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Hochschild Cohomology and Deformations of Clifford-Weyl Algebras

Cited by 5 publications

References 32 publications

Deformation quantization for Heisenberg supergroup

Deformation quantization for Heisenberg supergroup

Representations of Super Yang-Mills Algebras

Conformal Geometry of the Supercotangent and Spinor Bundles

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