arXiv subjects

Cristian Vay

Publications and source records attributed to Cristian Vay.

At least 19 recordsLinked to original sources

arXiv2025

Lifting in a non semisimple world

This is a contribution to the problem of classifying all deformations - a. k. a. liftings - of the bosonization of a Nichols algebra $\mathfrak{B}(V)$ over a cosemisimple and non-semisimple Hopf algebra $H$. Such a situation arises when the underlying field has positive characteristic or when $H$ is infinite-dimensional. Given an $H$-module $M$ that is an extension of $V$ by $H$, we first introduce an algebra $T(V)\#_MH$ which generalizes the usual bosonization $T(V)\#H$. Indeed, these two objects coincide when $M$ is a trivial extension. We provide necessary conditions for $T(V)\#_MH$ to be a Hopf algebra and a cocycle deformation of $T(V)\#H$. These conditions appear particularly natural when $H$ is a group algebra. We then prove that every lifting is a quotient of $T(V)\#_MH$ for some extension $M$. Echoing Archimedes, $T(V)\#_MH$ stands as a fulcrum over which we can pivot to lift the relations of the Nichols algebra. From this point, one can replicate the strategy proposed by Andruskiewitsch, Angiono, Garcia I., Masuoka and the second author to show that every lifting is a cocycle deformation of $\mathfrak{B}(V)\#H$. We illustrate this idea with two examples of different nature. We classify all pointed liftings of the Fomin-Kirillov algebra $\mathcal{FK}_3$ in characteristic $2$, and prove they are all cocycle deformations one another. We also prove that the Jordanian enveloping algebra of $\mathfrak{sl}(2)$ defined by Andruskiewitsch, Angiono and Heckenberger is a cocycle deformation of the bosonization of the Jordan plane over the infinite cyclic group.

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arXiv2025

Linkage principle for small quantum groups

We consider small quantum groups with root systems of Cartan, super and modular type, among others. These are constructed as Drinfeld doubles of finite-dimensional Nichols algebras of diagonal type. We prove a linkage principle for them by adapting techniques from the work of Andersen, Jantzen and Soergel in the context of small quantum groups at roots of unity. Consequently we characterize the blocks of the category of modules. We also find a notion of (a)typicality similar to the one in the representation theory of Lie superalgebras. The typical simple modules turn out to be the simple and projective Verma modules. Moreover, we deduce a character formula for 1-atypical simple modules.

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arXiv2024

Koszul duality for Coxeter groups

We construct a "Koszul duality" equivalence relating the (diagrammatic) Hecke category attached to a Coxeter system and a given realization to the Hecke category attached to the same Coxeter system and the dual realization. This extends a construction of Beilinson-Ginzburg-Soergel and Bezrukavnikov-Yun in a geometric context, and of the first author with Achar, Makisumi and Williamson. As an application, we show that the combinatorics of the "tilting perverse sheaves" considered in arXiv:1802.07651 is encoded in the combinatorics of the canonical basis of the Hecke algebra of $(W,S)$ attached to the dual realization.

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arXiv2024

Simple modules of small quantum groups at dihedral groups

Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups $\mathbb{D}_{4t}$ with $t\geq 3$. To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module.

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arXiv2023

The Green ring of a family of copointed Hopf algebras

The copointed liftings of the Fomin-Kirillov algebra $\mathcal{FK}_3$ over the algebra of functions on the symmetric group $\mathbb{S}_3$ were classified by Andruskiewitsch and the author. We demonstrate here that those associated to a generic parameter are Morita equivalent to the non-simple blocks of well-known Hopf algebras: the Drinfeld doubles of the Taft algebras and the small quantum groups $u_{q}(\mathfrak{sl}_2)$. The indecomposable modules over these were classified independently by Chen, Chari--Premet and Suter. Consequently, we obtain the indocomposable modules over the generic liftings of $\mathcal{FK}_3$. We decompose the tensor products between them into the direct sum of indecomposable modules. We then deduce a presentation by generators and relations of the Green ring.

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arXiv2020

On the Hopf algebra structure of the Lusztig quantum divided power algebras

We study the Hopf algebra structure of Lusztig's quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build them from the plus, minus and zero parts by means of suitable actions and coactions within the formalism presented by Sommerhauser to describe triangular decompositions.

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arXiv2020

On Hopf algebras with triangular decomposition

In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with triangular decomposition. Finally, we extend to these Hopf algebras the main results of arXiv:1612.09220 regarding projective modules over Drinfeld doubles of bosonizations of Nichols algebras and groups.

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arXiv2018

On the representation theory of the Drinfeld double of the Fomin-Kirillov algebra $\mathcal{FK}_3$

Let $\mathcal{D}$ be the Drinfeld double of $\mathcal{FK}_3\#\Bbbk{\mathbb S}_3$. The simple $\mathcal{D}$-modules were described in arXiv:1409.0438. In the present work, we describe the indecomposable summands of the tensor product between them. We classify the extensions of the simple modules and show that $\mathcal{D}$ is of wild representation type. We also investigate the projective modules and their tensor products.

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arXiv2018

Mixed perverse sheaves on flag varieties of Coxeter groups

In this paper we construct an abelian category of "mixed perverse sheaves" attached to any realization of a Coxeter group, in terms of the associated Elias-Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias-Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

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arXiv2017

Copointed Hopf algebras over $\mathbb{S}_4$

We study the realizations of certain braided vector spaces of rack type as Yetter-Drinfeld modules over a cosemisimple Hopf algebra $H$. We apply the strategy developed in arXiv:1212.5279 to compute their liftings and use these results to obtain the classification of finite-dimensional copointed Hopf algebras over $\mathbb{S}_4$.

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arXiv2017

On projective modules over finite quantum groups

Let $\mathcal{D}$ be the Drinfeld double of the bosonization ${\mathfrak B}(V)\#\Bbbk G$ of a finite-dimensional Nichols algebra ${\mathfrak B}(V)$ over a finite group $G$. It is known that the simple $\mathcal{D}$-modules are parametrized by the simple modules over $\mathcal{D}(G)$, the Drinfeld double of $G$. This parametrization can be obtained by considering the head $\mathsf{L}(λ)$ of the Verma module $\mathsf{M}(λ)$ for every simple $\mathcal{D}(G)$-module $λ$. In the present work, we show that the projective $\mathcal{D}$-modules are filtered by Verma modules and the BGG Reciprocity $[\mathsf{P}(μ):\mathsf{M}(λ)]=[\mathsf{M}(λ):\mathsf{L}(μ)]$ holds for the projective cover $\mathsf{P}(μ)$ of $\mathsf{L}(μ)$. We use graded characters to proof the BGG Reciprocity and obtain a graded version of it. Also, we show that a Verma module is simple if and only if it is projective. We also describe the tensor product between projective modules.

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arXiv2016

Verma and simple modules for quantum groups at non-abelian groups

The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S3 attached to the 12-dimensional Fomin-Kirillov algebra, computing all the simple modules and calculating their dimensions.

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arXiv2016

The cohomology ring of the 12-dimensional Fomin-Kirillov algebra

The $12$-dimensional Fomin-Kirillov algebra $FK_3$ is defined as the quadratic algebra with generators $a$, $b$ and $c$ which satisfy the relations $a^2=b^2=c^2=0$ and $ab+bc+ca=0=ba+cb+ac$. By a result of A. Milinski and H.-J. Schneider, this algebra is isomorphic to the Nichols algebra associated to the Yetter-Drinfeld module $V$, over the symmetric group $\mathbb{S}_3$, corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring $Ext_{FK_3}^*(\Bbbk,\Bbbk)$, showing that it is a polynomial ring $S[X]$ with coefficients in the symmetric braided algebra of $V$. As an application we also compute the cohomology rings of the bosonization $FK_3\#\Bbbk\mathbb{S}_3$ and of its dual, which are $72$-dimensional ordinary Hopf algebras.

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arXiv2015

Explicit coproduct formula for quantum group of the type $G_2$

We find a coproduct formula in the explicit form for PBW-generators of the two-parameter quantum group $U_q^+(\frak{g})$ where $\frak{g}$ is a simple Lie algebra of type $G_2$. The similar formulas for quantizations of simple Lie algebras of infinite series are already known.

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arXiv2013

Representations of copointed Hopf algebras arising from the tetrahedron rack

We study the copointed Hopf algebras attached to the Nichols algebra of the affine rack $\Aff(\F_4,ω)$, also known as tetrahedron rack, and the 2-cocycle -1. We investigate the so-called Verma modules and classify all the simple modules. We conclude that these algebras are of wild representation type and not quasitriangular, also we analyze when these are spherical.

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arXiv2013

Finite-dimensional Pointed or Copointed Hopf algebras over affine racks

We study the pointed or copointed liftings of Nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal YD-realization of these racks. In the copointed case we complete the classification for the six affine racks whose Nichols algebra is known to be of finite dimension. In the pointed case our method allows us to finish four of them. In all of the cases the Hopf algebras obtained turn out to be cocycle deformations of their associated graded Hopf algebras. All of them are new examples of finite-dimensional copointed or pointed Hopf algebras over non-abelian groups.

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arXiv2013

Lifting via cocycle deformation

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.

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arXiv2012

From Hopf algebras to tensor categories

This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the category of representations of a suitable Hopf algebra.

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