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 Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups

2020 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Giovanna Carnovale ◽  
Gastón Andrés García
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Mixed Classes ◽  
Type V ◽  
Groups Of Lie Type

Abstract We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from $$\mathbf {PSL}_n(q)$$ PSL n ( q ) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is $$\mathbf {PSp}_{2n}(q)$$ PSp 2 n ( q ) , $$\mathbf {P}{\varvec{\Omega }}^+_{4n}(q)$$ P Ω 4 n + ( q ) , $$\mathbf {P}{\varvec{\Omega }}^-_{4n}(q)$$ P Ω 4 n - ( q ) , $$^3D_4(q)$$ 3 D 4 ( q ) , $$E_7(q)$$ E 7 ( q ) , $$E_8(q)$$ E 8 ( q ) , $$F_4(q)$$ F 4 ( q ) , or $$G_2(q)$$ G 2 ( q ) with q even is the group algebra.

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 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 ON POINTED HOPF ALGEBRAS ASSOCIATED WITH THE MATHIEU SIMPLE GROUPS

2009 ◽  
Vol 08 (05) ◽  
pp. 633-672 ◽  
Author(s):  
FERNANDO FANTINO
Keyword(s):  
Irreducible Representation ◽  
Hopf Algebra ◽  
Conjugacy Class ◽  
Simple Group ◽  
Hopf Algebras ◽  
Simple Groups ◽  
Drinfeld Module ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras

Let G be a Mathieu simple group, s ∈ G, [Formula: see text] the conjugacy class of s and ρ an irreducible representation of the centralizer of s. We prove that either the Nichols algebra [Formula: see text] is infinite-dimensional or the braiding of the Yetter–Drinfeld module [Formula: see text] is negative. We also show that if G = M22 or M24, then the group algebra of G is the only (up to isomorphisms) finite-dimensional complex pointed Hopf algebra with group-likes isomorphic to G.

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.

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