22 research outputs found
On Two Theorems About Symplectic Reflection Algebras
We give a new proof and an improvement of two Theorems of J. Alev, M.A.
Farinati, T. Lambre and A.L. Solotar : the first one about Hochschild
cohomology spaces of some twisted bimodules of the Weyl algebra W and the
second one about Hochschild cohomology spaces of the smash product G * W (G a
finite subgroup of SP(2n)), and as an application, we then give a new proof of
a Theorem of P. Etingof and V. Ginzburg, which shows that the Symplectic
Reflection Algebras are deformations of G * W (and, in fact, all possible
ones).Comment: corrected typo
Hochschild Cohomology and Deformations of Clifford-Weyl Algebras
We give a complete study of the Clifford-Weyl algebra C(n,2k) from Bose-Fermi statistics, including Hochschild cohomology (with coefficients in itself). We show that C(n,2k) is rigid when n is even or when k ≠ 1. We find all non-trivial deformations of C(2n+1,2) and study their representations
Projectively equivariant quantizations over the superspace
We investigate the concept of projectively equivariant quantization in the
framework of super projective geometry. When the projective superalgebra
pgl(p+1|q) is simple, our result is similar to the classical one in the purely
even case: we prove the existence and uniqueness of the quantization except in
some critical situations. When the projective superalgebra is not simple (i.e.
in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a
one-parameter family of equivariant quantizations. We also provide explicit
formulas in terms of a generalized divergence operator acting on supersymmetric
tensor fields.Comment: 19 page
Topics on n-ary algebras
We describe the basic properties of two n-ary algebras, the Generalized Lie
Algebras (GLAs) and, particularly, the Filippov (or n-Lie) algebras (FAs), and
comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and
Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology
relevant for the central extensions and infinitesimal deformations of FAs. It
is seen that semisimple FAs do not admit central extensions and, moreover, that
they are rigid. This extends the familiar Whitehead's lemma to all
FAs, n=2 being the standard Lie algebra case. When the n-bracket of the FAs is
no longer required to be fully skewsymmetric one is led to the n-Leibniz (or
Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz
algebras, those with an anticommutative n-bracket, we study the class of
n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first
n-1 entires of the n-Leibniz bracket.Comment: 11 page
Closedness of star products and cohomologies
We first review the introduction of star products in connection with
deformations of Poisson brackets and the various cohomologies that are related
to them. Then we concentrate on what we have called ``closed star products" and
their relations with cyclic cohomology and index theorems. Finally we shall
explain how quantum groups, especially in their recent topological form, are in
essence examples of star products.Comment: 16 page
Supertrace and superquadratic Lie structure on the Weyl algebra, and applications to formal inverse Weyl transform
Abstract. Using the Moyal -product and orthosymplectic supersymmetry, we construct a natural nontrivial supertrace and an associated nondegenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra W. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples
The Hidden Group Structure Of Quantum Groups: Strong Duality, Rigidity And Preferred Deformations.
: A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyal-product-like deformations are naturally found for all FRT-models on coefficients and C 1 -functions. Strong rigidity (H 2 bi = f0g) under deformations in the category of bialgebras is proved and consequences are deduced. AMS classification: Primary 17B37, 16W30, 22C05 46H99, 81R50. Running title : Topological quantum groups. (In press in Communications in Mathematical Physics, end of 1993) 1 Universit'e de Bourgogne - Laboratoire de Physique Math'ematique B.P. 138, 21004 DIJON Cedex - FRANCE, e-mail: [email protected] 2 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 191046395 U.S.A. e-mail: [email protected] and murray@math..