Applying recent results by Lowen-Van den Bergh we show that Hochschild cohomology is preserved under Koszul-Moore duality as a Gerstenhaber algebra. More precisely, the corresponding complexes are linked quasi-isomorphism of B-infinity-algebras.

Let Z/n denote the integers mod n and let % denote the finite Fourier transform on L\Z/n). We let ©2$), » F operate on ©2L2(Z/n) and show that ©2L2(Z/n) can be given a graded algebra structure (with no zero divisors) such that f(/g) ■" ^U)^(g). We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on 02L2(...

Following E. Kirchberg, [3], we call a bifunctor (A,B) → A⊗α B a C -algebraic tensor product functor if it is obtained by completing of the algebraic tensor product A ⊙ B of C-algebras in a functional way with respect to a suitable C-norm ‖ · ‖α. We call such a functor symmetric if the standard isomorphism A ⊙ B ∼= B ⊙ A extends to an isomorphism A⊗αB ∼= B⊗αA. Similarly, we call it associative ...

We analyze the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik’s commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equatio...

We investigate conditions on a graph C∗-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth (1,∞)-summable semfinite spectral triple. The local index theorem allows us to compute the pairing with K-theory. This produces invariants in the K-theory of the fixed point algebra, and these ...

We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands d...

Given a reductive Lie algebra g and choice of Cartan subalgebra h, the HarishChandra map gives an isomorphism between the center z of the universal enveloping algebra U(g) of g and the algebra of Weyl group invariant symmetric tensors of h, denoted S(h) . We define these objects and give a sketch of the proof, using sl(2,C) as a motivating example. The proofs and examples mirror those found in ...

We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of ...

In this paper we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex which is generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology of an infinite-dimensional Lie algebra of symplectic vector fields.

We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.