Agustin Garcia Iglesias

National University of Cordoba, Argentina | UNC · Department of Mathematics and Physics

Let $(V,c)$ be a finite-dimensional braided vector space of diagonal type. We show that the Gelfand Kirillov dimension of the Nichols algebra $\mathfrak{B}(V)$ is finite if and only if the corresponding root system is finite, that is $\mathfrak{B}(V)$ admits a PBW basis with a finite number of generators. This had been conjectured in arXiv:1606.025...

This paper contributes to the proof of the conjecture posed in arXiv:1606.02521, stating that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 3 or that the braiding is of Cartan type.

This article serves a two-fold purpose. On the one hand, it is asurvey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construc...

Let $V$ be a braided vector space of diagonal type with a principal realization in the category of Yetter-Drinfeld modules of a cosemisimple Hopf algebra $H$ and such that the Nichols algebra $\mathfrak{B}(V)$ is finitely presented. We show that every lifting of $V$ is a cocycle deformation of $\mathfrak{B}(V)\#H$. In particular, it follows that ev...

This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to constru...

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 [1] to compute their liftings and use these results to obtain the classification of finite-dimensional copointed Hopf algebras over S4.

We use the inflation-restriction sequence and a result of Etingof and Gra\~na on the rack cohomology to give a explicit description of 2-cocycles of finite indecomposable quandles with values in an abelian group. Several applications are given.

We complete the classification of Hopf algebras whose infinitesimal braiding is a principal Yetter-Drinfeld realization of a braided vector space of Cartan type $G_2$ over a cosemisimple Hopf algebra. We develop a general formula for a class of liftings in which the quantum Serre relations hold. We give a detailed explanation of the procedure for f...

Let p be an odd prime, m ∈ ℕ and set q = pm, G = PSLn(q). Let θ be a standard graph automorphism of G, d be a diagonal automorphism and Frq be the Frobenius endomorphism of PSLn(double-struck Fq). We show that every (d ○ θ)-conjugacy class of a (d ○ θ, p)-regular element in G is represented in some Frq-stable maximal torus of PSLn(double-struck Fq)...

Let $H$ be a Hopf algebra. A finite-dimensional lifting of $V\in{}^{H}_{H}\mathcal{YD}$ arising a cocycle deformation of $A=\mathfrak{B}(V)\#H$ naturally defines a twist in the dual Hopf algebra $A^*$. We follow this recipe to write down explicit examples and show that this construction extends known techniques for defining twists. We also contribu...

After the classification of the finite-dimensional Nichols algebras of diagonal type [17, 18], the determination of its defining relations [7, 6], and the verification of the generation in degree-s1 conjecture [6], there is still one step missing in the classification of complex finite-dimensional Hopf algebras with abelian group, without restricti...

Let $p$ be an odd prime, $m\in {\mathbb N}$ and set $q=p^m$, $G=\operatorname{PSL}_n(q)$. Let $\theta$ be a standard graph automorphism of $G$, $d$ be a diagonal automorphism and $\operatorname{Fr}_q$ be the Frobenius endomorphism of $\operatorname{PSL}_n(\overline{{\mathbb F}_q})$. We show that every $(d\circ \theta)$-conjugacy class of a $(d\circ...

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.

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 meth...

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.

Let K be an algebraically closed field of characteristic 0. We conclude the classification of finite-dimensional pointed Hopf algebras whose group of group-likes is $\mathbb{S}_4$ . We also describe all pointed Hopf algebras over $\mathbb{S}_5$ whose infinitesimal braiding is associated to the rack of transpositions.

We classify exact indecomposable module categories over the representation category of all non-trivial Hopf algebras with coradical S_3 and S_4. As a byproduct, we compute all its Hopf-Galois extensions and we show that these Hopf algebras are cocycle deformations of their graded versions.

We show that any finite-dimensional pointed Hopf algebra over an abelian group $\Gamma$ such that its infinitesimal braiding is of standard type is generated by group-like and skew-primitive elements. This fact agrees with the long-standing conjecture by Andruskiewitsch and Schneider. We also show that the quantum Serre relations hold in any coradi...

The classification of finite-dimensional pointed Hopf algebras with group S 3 was finished by N. Andruskiewitsch, I. Heckenberger, and H.-J. Schneider [in Am. J. Math. 132, No. 6, 1493-1547 (2010; Zbl 1214.16024)]: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all s...

The classification of finite-dimensional pointed Hopf algebras with group S_3 was finished in "The Nichols algebra of a semisimple Yetter-Drinfeld module", arXiv:0803.2430v1 [math.QA], by Andruskiewitsch, Heckenberger and Schneider: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here...

Let k be an algebraically closed field of characteristic 0. We conclude the classification of finite dimensional pointed Hopf algebras whose group of group-likes is S_4. We also describe all pointed Hopf algebras over S_5 whose infinitesimal braiding is associated to the rack of transpositions. Comment: 22 pages. Some results extended