Finite dimensional Nichols algebras over certainp-groups

2016 ◽  
Vol 45 (9) ◽  
pp. 3691-3702
Author(s):  
Yibo Yang ◽  
Shenglin Zhu
Keyword(s):  
Finite Dimensional ◽  
Nichols Algebras

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

scholarly journals Finite-dimensional Nichols algebras over dual Radford algebras

2020 ◽  
pp. 2140001
Author(s):  
O. Márquez ◽  
D. Bagio ◽  
J. M. J. Giraldi ◽  
G. A. García
Keyword(s):  
Jorge Luis Borges ◽  
Drinfeld Modules ◽  
Indecomposable Modules ◽  
Simple Modules ◽  
Generators And Relations ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Nichols Algebras ◽  
The Way

For [Formula: see text], let [Formula: see text] be the dual of the Radford algebra of dimension [Formula: see text]. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over [Formula: see text]. Along the way, we describe the simple objects in [Formula: see text] and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case [Formula: see text]. There are 18 possible cases. We present by generators and relations, the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for [Formula: see text] and [Formula: see text], [Formula: see text], which recovers some results of the second and third author in the former case, and of Xiong in the latter. Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es. Jorge Luis Borges

scholarly journals POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250066
Author(s):  
SHOUCHUAN ZHANG ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Groups ◽  
Drinfeld Modules ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Necessary And Sufficient

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by 𝕊nare infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by A to be finite dimensional.

scholarly journals Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Chen Qian ◽  
Jing Wang
Keyword(s):  
Root System ◽  
Hopf Algebras ◽  
Enveloping Algebras ◽  
Finite Dimensional ◽  
Quantized Enveloping Algebras ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Simply Connected

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.

scholarly journals On the restricted Jordan plane in odd characteristic

2020 ◽  
pp. 2140012
Author(s):  
Nicolás Andruskiewitsch ◽  
Héctor Peña Pollastri
Keyword(s):  
Hopf Algebra ◽  
Positive Characteristic ◽  
Drinfeld Double ◽  
Nichols Algebra ◽  
Simple Modules ◽  
Restricted Version ◽  
Defining Relations ◽  
Finite Dimensional ◽  
Nichols Algebras

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given.

scholarly journals Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II: Unipotent classes in symplectic groups

2016 ◽  
Vol 18 (04) ◽  
pp. 1550053 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Giovanna Carnovale ◽  
Gastón Andrés García
Keyword(s):  
Finite Field ◽  
Hopf Algebras ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Type Ii ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Regular Classes ◽  
Groups Of Lie Type

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups.

scholarly journals ON NICHOLS ALGEBRAS OVER PGL(2, q) AND PSL(2, q)

2010 ◽  
Vol 09 (02) ◽  
pp. 195-208 ◽  
Author(s):  
SEBASTIÁN FREYRE ◽  
MATÍAS GRAÑA ◽  
LEANDRO VENDRAMIN
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Drinfeld Modules ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Mathieu Groups ◽  
Nichols Algebras

We compute necessary conditions on Yetter–Drinfeld modules over the groups PGL(2, q) = PGL(2, 𝔽q) and PSL(2, q) = PSL(2, 𝔽q) to generate finite-dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that any finite-dimensional pointed Hopf algebra over the Mathieu groups M20 or M21 = PSL(3, 4) is the group algebra.

scholarly journals Quantum function algebras from finite-dimensional Nichols algebras

2020 ◽  
Vol 14 (3) ◽  
pp. 879-911
Author(s):  
Marco Andrés Farinati ◽  
Gastón Andrés García
Keyword(s):  
Finite Dimensional ◽  
Function Algebras ◽  
Nichols Algebras ◽  
Quantum Function
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