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

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

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 $n\geq 2$ 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

Human peptide transporter deficiency: importance of HLA-B in the presentation of TAP-independent EBV antigens.

Abstract Two siblings with a peptide TAP deficiency were recently described. Despite poor cell surface expression of HLA class I molecules, these patients were not unusually susceptible to viral infections. The majority of the cell surface-expressed class I molecules were HLA-B products as assessed by cytofluorometry and biochemical analysis. Analysis of two peptides eluted from the class I molecules expressed by TAP-deficient EBV B lymphoblastoid cell lines indicated that both were derived from cytosolic proteins and presented by HLA-B molecules. Peripheral alphabeta CD8+ T cells were present and their TCR repertoire was polyclonal. Most of the alphabeta CD8+ T cell clones studied (21 of 22) were nonreactive against cells expressing normal levels of the same HLA alleles as those of the TAP-deficient patients. However, it was possible to isolate one cytotoxic CD8+ alphabeta T cell clone recognizing the EBV protein LMP2 presented by HLA-B molecules on TAP-deficient cells. These observations suggest that in the TAP-deficient patients, CD8+ alphabeta T cells could mature and be recruited in immune responses to mediate HLA class I-restricted cytotoxic defense against viral infections. They also strengthen the physiologic importance of a TAP-independent processing pathway of the LMP2 protein, which was previously shown to contain several other TAP-independent epitopes.</jats:p