We prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector space X. The result implies in particular that the Kontsevich deformation quantizations of S(X∗) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.

We prove the simple fact that the factor ring of a Koszul algebra by a regular, normal, quadratic element is a Koszul algebra. This fact leads to a new construction of quadratic Artin-Schelter regular algebras. This construction generalizes the construction of Artin-Schelter regular Clifford algebras. 1991 Mathematics Subject Classification. 16W50, 14A22.

We develop the cohomology theory of color Lie superalgebras due to Scheunert–Zhang in a framework of nonhomogeneous quadratic Koszul algebras. In this approach, the Chevalley– Eilenberg complex of a color Lie algebra becomes a standard Koszul complex for its universal enveloping algebra. As an application, we calculate cohomologies with trivial coefficients of Zn 2 – graded 3–dimensional color ...

In this paper we use linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras studied in [MR1, MR2] to give a geometric realization of the Iwahori–Matsumoto involution of affine Hecke algebras. More generally we prove that linear Koszul duality is compatible with convolution in a general context related to convolution algebras.

We show, in full generality, that Lusztig’s a-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category O, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of...

Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each f. g. projective R/I-module V has an Rprojective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P1 → P0 for all n ≥ 1, if d = 1. This computation yields a new proof of the classical Adams-Riemann-R...

Let R be a standard graded commutative algebra over field k , let K its Koszul complex viewed as differential -algebra, and H the homology of . This paper studies interplay between homological properties three algebras In particular, we introduce two definitions Koszulness that extend familiar property originally introduced by Priddy: one which applies to (and, more generally, any connected -al...

We prove a new criterion for the homogeneous coordinate ring of a finite set of points in Pn to be Koszul. Like the well known criterion due to Kempf [7] it involves only incidence conditions on linear spans of subsets of a given set. We also give a sufficient condition for the Koszul property to be preserved when passing to a subset of a finite set of points in Pn.

One of our main results is a classification all the weakly symmetric radical cube zero finite dimensional algebras over an algebraically closed field having a theory of support via the Hochschild cohomology ring satisfying Dade’s Lemma. Along the way we give a characterization of when a finite dimensional Koszul algebra has such a theory of support in terms of the graded centre of the Koszul dual.