arXiv subjects

Volodymyr Mazorchuk

Publications and source records attributed to Volodymyr Mazorchuk.

At least 127 records · Page 7Linked to original sources

arXiv2009

Blocks and modules for Whittaker pairs

Inspired by recent activities on Whittaker modules over various (Lie) algebras we describe some general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case we obtain a very general setup for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general setup and the general setup of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.

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arXiv2009

Some homological properties of the category $\mathcal{O}$, II

We show, in full generality, that Lusztig's $\mathbf{a}$-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category $\mathcal{O}$, 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 tilting modules. These complexes, in turn, can be interpreted as the images of simple modules under projective functors in the Koszul dual of the category $\mathcal{O}$. Finally, we describe the dominant projective module and also projective-injective modules in some subcategories of $\mathcal{O}$ and show how one can use categorification to decompose the regular representation of the Weyl group into a direct sum of cell modules, extending the results known for the symmetric group (type $A$).

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arXiv2009

Supports of weight modules over Witt algebras

In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras $W_n$, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a $\Z^n$-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\oplus_{α\in\Z^n}\mathfrak {g}_α$ with respect to the abelian subalgebra $\mathfrak {g}_0$, with the property $[\mathfrak{g}_α,\mathfrak {g}_β]= \mathfrak {g}_{α+β}$ for all $α,β\in\Z^n$, $α\neq β$ (this class contains the algebra $W_n$).

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arXiv2009

Classification of simple q_2-supermodules

We classify all simple supermodules over the queer Lie superalgebra $\mathfrak{q}_{2}$ up to classification of equivalence classes of irreducible elements in a certain Euclidean ring.

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arXiv2008

Symmetric quasi-hereditary envelopes

We show how any finite-dimensional algebra can be realized as an idempotent subquotient of some symmetric quasi-hereditary algebra. In the special case of rigid symmetric algebras we show that they can be realized as centralizer subalgebras of symmetric quasi-hereditary algebras. We also show that the infinite-dimensional symmetric quasi-hereditary algebras we construct admit quasi-hereditary structure with respect to two opposite orders, that they have strong exact Borel and $Δ$-subalgebras and the corresponding triangular decompositions.

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arXiv2008

Regular strongly typical blocks of $\mathcal{O}^{\mathfrak{q}}$

We use the technique of Harish-Chandra bimodules to prove that regular strongly typical blocks of the category $\mathcal{O}$ for the queer Lie superalgebra $\mathfrak{q}_n$ are equivalent to the corresponding blocks of the category $\mathcal{O}$ for the Lie algebra $\mathfrak{gl}_n$.

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arXiv2008

Koszul duality for stratified algebras I. Quasi-hereditary algebras

We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules.

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arXiv2008

Simple modules over factorpowers

In this paper we study complex representations of the factorpower $\fp(G,M)$ of a finite group $G$ acting on a finite set $M$. This includes the finite monoid $\FP$, which can be seen as a kind of a ``balanced'' generalization of the symmetric group $S_n$ inside the semigroup of all binary relations. We describe all irreducible representations of $\fp(G,M)$ and relate them to irreducible representations of certain inverse semigroups. In particular, irreducible representations of $\FP$ are related to irreducible representations of the maximal factorizable submonoid of the dual symmetric inverse monoid. We also show that in the latter cases irreducible representations lead to an interesting combinatorial problem in the representation theory of $S_n$, which, in particular, is related to Foulkes' conjecture. Finally, we show that all simple $\fp(G,M)$-modules are unitarizable and that tensor products of simple $\fp(G,M)$-modules are completely reducible.

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arXiv2008

Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

Inspired by the results of [R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand model, preprint math.RT arXiv:0709.3962], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the $q$-rook monoid algebra.

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arXiv2008

Categorification of Wedderburn's basis for \mathbb{C}[S_n]

M. Neunh{ö}ffer studies in \cite{Ne} a certain basis of $\mathbb{C}[S_n]$ with the origins in \cite{Lu} and shows that this basis is in fact Wedderburn's basis. In particular, in this basis the right regular representation of $S_n$ decomposes into a direct sum of irreducible representations (i.e. Specht or cell modules). In the present paper we rediscover essentially the same basis with a categorical origin coming from projective-injective modules in certain subcategories of the BGG-category $\mathcal{O}$. An important role in our arguments is played by the dominant projective module in each of these categories. As a biproduct of the study of this dominant projective module we show that {\it Kostant's problem} (\cite{Jo}) has a negative answer for some simple highest weight module over the Lie algebra $\mathfrak{sl}_4$, which disproves the general belief that Kostant's problem should have a positive answer for all simple highest weight modules in type $A$.

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arXiv2008

Quadratic duals, Koszul dual functors, and applications

The paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalises previous results in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality functor for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality of translation and Zuckerman functors for the classical category O in a quite elementary and explicit way. As applications we propose a definition of a "Koszul" dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category O.

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arXiv2007

On the irreducible representations of a finite semigroup

Work of Clifford, Munn and Ponizovski{\uı} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovski{\uı} result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.

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arXiv2007

A new approach to Kostant's problem

For every involution $\mathbf{w}$ of the symmetric group $S_n$ we establish, in terms ofa special canonical quotient of the dominant Verma module associated with $\mathbf{w}$, an effective criterion, which allows us to verify whether the universal enveloping algebra $U(\mathfrak{sl}_n)$ surjects onto the space of all ad-finite linear transformations of the simple highest weight module $L(\mathbf{w})$. An easy sufficient condition derived from this criterion admits a straightforward computational check for example using a computer. All this is applied to get some old and many new results, which answer the classical question of Kostant in special cases, in particular we give a complete answer for simple highest weight modules in the regular block of $\mathfrak{sl}_n$, $n\leq 5$.

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arXiv2007

On three approaches to conjugacy in semigroups

We compare three approaches to the notion of conjugacy for semigroups, the first one via the transitive closure of the $uv\sim vu$ relation, the second one via an action of inverse semigroups on themselves by partial transformations, and the third one via characters of finite-dimensional representations.

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arXiv2007

A combinatorial approach to functorial quantum sl(k) knot invariants

This paper contains a categorification of the sl(k) link invariant using parabolic singular blocks of category O. Our approach is intended to be as elementary as possible, providing combinatorial proofs of the main results of Sussan. We first construct an exact functor valued invariant of webs or 'special' trivalent graphs labelled with 1, 2, k-1, k satisfying the MOY relations. Afterwards we extend it to the sl(k)-invariant of links by passing to the derived categories. The approach using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the sl(k)-homology, based on foams and the Kapustin-Lie formula. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction.

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arXiv2007

Isolated subsemigroups in the variants of T_n

We classify all isolated, completely isolated, and convex subsemigroups in the semigroup T_n of all transformations of an n-element set, considered as the semigroup with respect to a sandwich operation.

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arXiv2007

A brief review of abelian categorifications

This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra sl(n). The review is intended to give non-experts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles.

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arXiv2007

On functors associated to a simple root

Associated to a simple root of a finite-dimensional complex semisimple Lie algebra, there are several endofunctors (defined by Arkhipov, Enright, Frenkel, Irving, Jantzen, Joseph, Mathieu, Vogan and Zuckerman) on the BGG category $\mathcal{O}_0$. We study their relations, compute cohomologies of their derived functors and describe the monoid generated by Arkhipov's and Joseph's functors and the monoid generated by Irving's functors. It turns out that the endomorphism rings of all elements in these monoids are isomorphic. We prove that the functors give rise to an action of the singular braid monoid on the bounded derived category of $\mathcal{O}_0$. We also use Arkhipov's, Joseph's and Irving's functors to produce new generalized tilting modules.

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