Related papers: Graded super duality for general linear Lie supera…
We study a version of the BGG category O for Dynkin Borel subalgebras of root-reductive Lie algebras g, such as gl(\infty). We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In…
We show that if $\Lambda$ is a $n$-Koszul algebra and $E=E(\Lambda)$ is its Yoneda algebra, then there is a full subcategory $\mathcal{L}_E$ of the category $Gr_E$ of graded $E$-modules, which contains all the graded $E$-modules presented…
The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When…
We explain how Lie superalgebras of types gl and osp provide a natural framework generalizing the classical Schur and Howe dualities. This exposition includes a discussion of super duality, which connects the parabolic categories O between…
We construct categorifications of tensor products of arbitrary finite-dimensional irreducible representations of $\mathfrak{sl}_k$ with subquotient categories of the BGG category $\mathcal{O}$, generalizing previous work of Sussan and…
We establish a maximal parabolic version of the Kazhdan-Lusztig conjecture \cite[Conjecture 5.10]{CKW} for the BGG category $\mathcal{O}_{k,\zeta}$ of $\mathfrak{q}(n)$-modules of "$\pm \zeta$-weights", where $k\leq n$ and…
We use a unified elementary approach to prove the second part of classical, mixed, super, and mixed super Schur-Weyl dualities for general linear groups and supergroups over an infinite ground field of arbitrary characteristic. These…
Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there…
We formulate and establish a super duality which connects parabolic categories $O$ between the ortho-symplectic Lie superalgebras and classical Lie algebras of $BCD$ types. This provides a complete and conceptual solution of the irreducible…
An purely algebraic proof of the PBW theorem of U_q(gl(m,n)) is given. Lusztig's conjecture is extended to the super case. The Lusztig's tensor product theorem is established.
We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the…
Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with gr U = A is Koszul dual to differential graded algebra (A!,d). We relate the module categories of this dual pair by a tensor-Hom adjunction. This descends to…
Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in…
This paper provides a new class of examples for the Koszul dualities established in~\cite{5}. We study quadratic monomial algebras from the perspective of Koszul duality, with particular emphasis on finitely presented and finitely…
Let $A$ be an augmented differential graded algebra over a field $k$ of characteristic zero, and let $A^!=\mathbf{R}\mathrm{Hom}_A(k,k)$ be its Koszul dual algebra. Blumberg and Mandell showed that, under some finiteness conditions of $A$,…
In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the…
A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…
We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version…
It was conjectured by Bennett, Chari, and Manning that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We…
We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group $G$ is strongly-graded-equivalent to the skew group algebra by a product partial action of $G$. As to a…