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N. Andruskiewitsch

Publications and source records attributed to N. Andruskiewitsch.

At least 19 recordsLinked to original sources
arXiv2026

On Noetherian pointed Hopf algebras

In arXiv:1405.4105 it was asked whether an affine Hopf algebra with finite Gelfand-Kirillov dimension is necessarily Noetherian. It is well-known that the converse is not true --take the group algebra of a polycyclic group which is not nilpotent-by-finite. However, it was conjectured in arXiv:2301.04428 that a pointed affine Noetherian Hopf algebra whose group of group-likes is nilpotent-by-finite necessarily has finite Gelfand-Kirillov dimension. In the present paper we conjecture that a post-Nichols algebra over a polycyclic-by-finite group is Noetherian if and only if it is affine and has finite Gelfand-Kirillov dimension. Partial results supporting this conjecture are presented; and their consequences for the preceding questions and conjectures is analyzed.

math.RA↗
arXiv2024

Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups

We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a consequence, all Nichols algebras over groups of odd order are of diagonal type, which allows us to describe all pointed Hopf algebras of odd dimension.

math.QA↗
arXiv2023

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(n,q) and PSp(2n,q)

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of grouplike elements isomorphic to PSL(n,q) (n greater than or equal to 4), PSL(3,q) (q greater than 2), or PSp(2n,q) (n greater than or equal to 3), is isomorphic to a group algebra, completing work in arXiv:1506.06794.

math.QA↗
arXiv2011

On a family of Hopf algebras of dimension 72

We investigate a family of Hopf algebras of dimension 72 whose coradical is isomorphic to the algebra of functions on S_3. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.

math.QA↗
arXiv2010

Pointed Hopf algebras over the sporadic simple groups

We show that every finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group is a group algebra, except for the Fischer group Fi22, the Baby Monster and the Monster. For these three groups, we give a short list of irreducible Yetter-Drinfeld modules whose Nichols algebra is not known to be finite-dimensional.

math.QA↗
arXiv2010

On Nichols algebras associated to simple racks

This is a report on the present state of the problem of determining the dimension of the Nichols algebra associated to a rack and a cocycle. This is relevant for the classification of finite-dimensional complex pointed Hopf algebras whose group of group-likes is non-abelian. We deal mainly with simple racks. We recall the notion of rack of type D, collect the known lists of simple racks of type D and include preliminary results for the open cases. This notion is important because the Nichols algebra associated to a rack of type D and any cocycle has infinite dimension. For those racks not of type D, the computation of the cohomology groups is needed. We discuss some techniques for this problem and compute explicitly the cohomology groups corresponding to some conjugacy classes in symmetric or alternating groups of low order.

math.QA↗
arXiv2010

On Twisted homogeneous racks of type D

We develop some techniques to check when a twisted homogeneous rack of class (L,t,θ) is of type D. Then we apply the obtained results to the cases L an alternating group on n letters, n\geq 5, or L a sporadic group.

math.QA↗
arXiv2010

On pointed Hopf algebras associated with the symmetric groups II

This is a sequel to arXiv:0807.2406. It is shown that the Nichols algebras over the symmetric groups S_m, m > 4, are all infinite-dimensional, except (maybe) those related to the transpositions considered by Fomin and Kirillov, resp. Milinski and Schneider, and the class of type (2,3) in S_5.

math.QA↗
arXiv2010

Pointed Hopf algebras over some sporadic simple groups

Any finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group, with the possible exception of the Fischer groups Fi22, the Baby Monster B and the Monster M, is a group algebra.

math.QA↗
arXiv2010

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

It is shown that Nichols algebras over alternating groups A_m, m>4, are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to A_m is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups S_m are all infinite-dimensional, except maybe those related to the transpositions considered in [FK], and the class of type (2,3) in S_5. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra of (X,q) is infinite dimensional, for q an arbitrary cocycle. arXiv:0904.3978 is included here.

math.QA↗
arXiv2009

The Nichols algebra of a semisimple Yetter-Drinfeld module

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of finite-dimensional pointed Hopf algebras over S_3, and of finite-dimensional Nichols algebras over S_4. The revised version contains an extended introduction with references to recent applications, and a simplified definition of the Weyl groupoid of a semisimple Yetter-Drinfeld module. Key words: Hopf algebras, quantum groups, Weyl groupoid

math.QA↗
arXiv2008

New techniques for pointed Hopf algebras

We present techniques that allow to decide that the dimension of some pointed Hopf algebras associated with non-abelian groups is infinite. These results are consequences of arXiv:0803.2430v1. We illustrate each technique with applications.

math.QA↗
arXiv2006

Irreducible representations of liftings of quantum planes

In this note, the irreducible representations of a lifting of a quantum plane are determined. Both authors thank Hans-Jürgen Schneider for pointing out a mistake in the published version of the paper, that is corrected here.

math.RT↗
arXiv2006

On the classification of finite-dimensional pointed Hopf algebras

We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order of $G(A)$ are $>7$. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.

math.QA↗
arXiv2004

A characterization of quantum groups

We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension, which are domains, whose groups of group-like elements are finitely generated and abelian, and whose infinitesimal braidings are positive.

math.QA↗
arXiv2001

Pointed Hopf algebras

This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras $A$ by first determining the graded Hopf algebra $\gr A$ associated to the coradical filtration of $A$. The $A_{0}$-coinvariants elements form a braided Hopf algebra $R$ in the category of Yetter-Drinfeld modules over the coradical $A_{0} = \ku Γ$, $Γ$ the group of group-like elements of $A$, and $\gr A \simeq R # A_{0}$. We call the braiding of the primitive elements of $R$ the infinitesimal braiding of $A$. If this braiding is of Cartan type \cite{AS2}, then it is often possible to determine $R$, to show that $R$ is generated as an algebra by its primitive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that $\gr A \simeq R # \ku Γ$. In the last Chapter, as a concrete illustration of the method, we describe explicitly all finite-dimensional pointed Hopf algebras $A$ with abelian group of group-likes $G(A)$ and infinitesimal braiding of type $A_{n}$ (up to some exceptional cases). In other words, we compute all the liftings of type $A_n$; this result is our main new contribution in this paper.

math.QA↗