We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus action, which we call a “hypertoric enveloping algebra.” We define an analogue of BGG category O for this algebra, and identify it with a certain category of sheaves on a hypertoric variety. We prove that a regular block of this category is highest weight and Koszul, identify its Koszul dual, comput...

In this paper, we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by ...

We prove an equivalence of two A∞-functors, via Orlov’s Landau–Ginzburg/ Calabi–Yau (LG/CY) correspondence. One is the Polishchuk–Zaslow mirror symmetry functor elliptic curves, and other a localized from Fukaya category T2 to noncommutative matrix factorizations. As corollary, we that LMgrLt realizes homological for any t.

(P) where / is the complex /-homomorphism and , y denotes localization at p. Both J and ** are infinite loop maps, and it is natural to ask whether this result is infinitely deloopable; that is, whether J<ff = / as infinite loop maps. This is the Stable Adams Conjecture. We announce here two independent proofs of this conjecture. Details will appear in [2] and [6]. METHOD 1. Our proof is based ...

Let G be a reductive algebraic group defined over an algebraically closed field k. Let H be a closed connected subgroup of G containing a maximal torus T of G. In [13] it was shown (at least in characteristic zero) that the parabolic subgroups of G can be characterized among all such subgroups H by a certain finiteness property of the induction functor (-)Iz and its derived functors Lk,G(-). Th...

We study the category O for a general Coxeter system using a formulation of Fiebig. The translation functors, the Zuckerman functors and the twisting functors are defined. We prove the fundamental properties of these functors, the duality of Zuckerman functor and generalization of Verma’s result about homomorphisms between Verma modules.

Let C be an abelian or exact category with enough projectives and let P be the full subcategory of projective objects of C . We consider the stable category C/P modulo projectives, as a left triangulated category [14], [36]. Then there is a triangulated category S(C/P) associated to C/P, which is universal in the following sense. There exists an exact functor S : C/P -t S(C/P) such that any exa...