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Volodymyr Mazorchuk

Publications and source records attributed to Volodymyr Mazorchuk.

At least 73 records · Page 4Linked to original sources
arXiv2018

Simple supermodules over Lie superalgebras

We show that, for many Lie superalgebras admitting a compatible $\mathbb{Z}$-grading, Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra. This reduces the classification problem for the former to the one for the latter. Our result applies to all classical Lie superalgebra of type $I$, in particular, to the general linear Lie superalgebra $\gl(m|n)$. In the latter case we also show that the rough structure of simple $\gl(m|n)$-supermodules and also that of Kac supermodules depends only on the annihilator of the $\mf{gl}(m)\oplus \mf{gl}(n)$-input and hence can be computed using the combinatorics of BGG category $\mathcal{O}$.

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arXiv2018

Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations

The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes of faithful simple transitive $2$-representations of the fiat $2$-subquotient of $\cC$ associated with a diagonal $\mathcal{H}$-cell in $\mathcal{J}$. As an application, we classify simple transitive $2$-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_3$ and $B_4$.

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arXiv2017

Simple transitive 2-representations of small quotients of Soergel bimodules

In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-rep\-re\-sen\-ta\-ti\-ons for the quotient of the $2$-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $2$-representations are exhausted by cell $2$-representations. However, in Coxeter types $I_2(2k)$, where $k\geq 3$, there exist simple transitive $2$-representations which are not equivalent to cell $2$-representations.

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arXiv2017

The G-centre and gradable derived equivalences

We propose a generalisation for the notion of the centre of an algebra in the setup of algebras graded by an arbitrary abelian group G. Our generalisation, which we call the G-centre, is designed to control the endomorphism category of the grading shift functors. We show that the G-centre is preserved by gradable derived equivalences given by tilting modules. We also discuss links with existing notions in superalgebra theory and apply our results to derived equivalences of superalgebras.

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arXiv2017

Indecomposable manipulations with simple modules in category O

We study the problem of indecomposability of translations of simple modules in the principal block of BGG category $\mathcal{O}$ for $\mathfrak{sl}_n$, as conjectured in \cite{KiM}. We describe some general techniques and prove a few general results which may be applied to study various special cases of this problem. We apply our results to verify indecomposability for $n\leq 6$. We also study the problem of indecomposability of shufflings and twistings of simple modules and obtain some partial results.

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arXiv2017

Pyramids and 2-representations

We describe a diagrammatic procedure which lifts strict monoidal actions from additive categories to categories of complexes avoiding any use of direct sums. As an application, we prove that every simple transitive $2$-representation of the $2$-category of projective bimodules over a finite dimensional algebra is equivalent to a cell $2$-representation.

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arXiv2017

Classification problems in 2-representation theory

This article surveys recent advances and future challenges in the $2$-representation theory of finitary $2$-categories with a particular emphasis on problems related to classification of various classes of $2$-representations.

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arXiv2017

Fiat categorification of the symmetric inverse semigroup IS_n and the semigroup F^*_n

Starting from the symmetric group $S_n$, we construct two fiat $2$-categories. One of them can be viewed as the fiat "extension" of the natural $2$-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This $2$-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other $2$-category can be viewed as the fiat "extension" of the $2$-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This $2$-category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.

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arXiv2017

Bimodules over uniformly oriented A_n quivers with radical square zero

We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, determine existing adjunctions between its $1$-morphisms and find a minimal generating set with respect to the tensor structure. We also prove that, for the algebras mentioned above, every simple transitive $2$-representation of the $2$-category of projective bimodules is equivalent to a cell $2$-representation.

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arXiv2017

Characterisation and applications of $\Bbbk$-split bimodules

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk$-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk$-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk$-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\Bbbk[x,y]/(x^2,y^2,xy)$.

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arXiv2016

Graded simple Lie algebras and graded simple representations

For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional $Q$-graded simple Lie algebras over any algebraically closed field of characteristic $0$ based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-graded Lie algebra (not necessarily simple) to classification of gradings on simple modules. For finite-dimensional $Q$-graded semisimple algebras we obtain a graded analogue of the Weyl Theorem.

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arXiv2016

Parabolic projective functors in type A

We classify projective functors on the regular block of Rocha-Caridi's parabolic version of the BGG category $\mathcal{O}$ in type $A$. In fact, we show that, in type $A$, the restriction of an indecomposable projective functor from $\mathcal{O}$ to the parabolic category is either indecomposable or zero. As a consequence, we obtain that projective functors on the parabolic category $\mathcal{O}$ in type $A$ are completely determined, up to isomorphism, by the linear transformations they induce on the level of the Grothendieck group, which was conjectured by Stroppel in \cite{St}.

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arXiv2016

Simple transitive 2-representations for some 2-subcategories of Soergel bimodules

We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that simple transitive $2$-representations are exhausted by cell $2$-representations. In the $B_2$ case we show that, apart from cell $2$-representations, there is a unique, up to equivalence, additional simple transitive $2$-representation and we give an explicit construction of this $2$-representation.

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arXiv2016

Dualities and derived equivalences for category O

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated to a reductive finite dimensional Lie algebra. In particular we find that, contrary to the original category O and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category O as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category O. It seems that only some special cases were known thus far, where the proof relied purely on a geometric approach to category O. We subsequently classify all derived equivalence classes of blocks of category O in type A which preserve the Koszul grading.

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arXiv2016

Special modules over positively based algebras

We use the Perron-Frobenius Theorem to define, study and, in some sense, classify special simple modules over arbitrary finite dimensional positively based algebras. For group algebras of finite Weyl groups with respect to the Kazhdan-Lusztig basis, this agrees with Lusztig's notion of a special module introduced in [Lu1].

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arXiv2015

Transitive 2-representations of finitary 2-categories

In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-Hölder Theorem where the weak composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-representations. Finally, we show that this large class contains finitary quotients of $2$-Kac-Moody algebras.

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