scholarly journals Representations of a non-pointed Hopf algebra

2021 ◽  
Vol 6 (10) ◽  
pp. 10523-10539
Author(s):  
Ruifang Yang ◽  
◽  
Shilin Yang
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Hopf Algebras ◽  
Indecomposable Modules ◽  
Representation Ring ◽  
Pointed Hopf Algebras ◽  
Dimension One ◽  
Gk Dimension

<abstract><p>In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations.</p></abstract>

REPRESENTATIONS OF SIMPLE POINTED HOPF ALGEBRAS

2004 ◽  
Vol 03 (01) ◽  
pp. 91-104 ◽  
Author(s):  
SHILIN YANG
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Hopf Algebras ◽  
Representation Type ◽  
Ground Field ◽  
Indecomposable Modules ◽  
Pointed Hopf Algebras

In this paper, the representation type of a class of pointed Hopf algebras is determined. As an application, all indecomposable modules of a simple-pointed Hopf algebra R(q,α) are classified. The Clebsch–Gordan-like formula for the decomposition of the tensor product, taken on the ground field, of two indecomposable modules of R(q,α) is obtained.

Ideals of Finite-Dimensional Pointed Hopf Algebras of Rank One

2021 ◽  
Vol 28 (02) ◽  
pp. 351-360
Author(s):  
Yu Wang ◽  
Zhihua Wang ◽  
Libin Li
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Adjoint Action ◽  
Prime Ideals ◽  
Closed Field ◽  
Finite Dimensional ◽  
Maximal Ideals ◽  
Primitive Ideals ◽  
Rank One ◽  
Pointed Hopf Algebras

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable [Formula: see text]-module is generated by one element. In particular, any indecomposable submodule of [Formula: see text] under the adjoint action is generated by a special element of [Formula: see text]. Using this result, we show that the Hopf algebra [Formula: see text] is a principal ideal ring, i.e., any two-sided ideal of [Formula: see text] is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras.

scholarly journals Classification of affine prime regular Hopf algebras of GK-dimension one

2016 ◽  
Vol 296 ◽  
pp. 1-54 ◽  
Author(s):  
Jinyong Wu ◽  
Gongxiang Liu ◽  
Nanqing Ding
Keyword(s):  
Hopf Algebras ◽  
Dimension One ◽  
Gk Dimension

scholarly journals McKay matrices for finite-dimensional Hopf algebras

2021 ◽  
pp. 1-46
Author(s):  
Georgia Benkart ◽  
Rekha Biswal ◽  
Ellen Kirkman ◽  
Van C. Nguyen ◽  
Jieru Zhu
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Eigenvalues And Eigenvectors ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
Generalized Eigenvectors ◽  
The Matrix ◽  
Dual Module ◽  
The Right ◽  
Finite Dimensional Hopf Algebra

Abstract For a finite-dimensional Hopf algebra $\mathsf {A}$ , the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$ -module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$ -modules with $\mathsf {V}$ . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$ . We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$ -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

The Grothendieck rings of Wu–Liu–Ding algebras and their Casimir numbers (I)

2021 ◽  
pp. 2250178
Author(s):  
Ruifang Yang ◽  
Shilin Yang
Keyword(s):  
Hopf Algebras ◽  
Commutative Rings ◽  
New Class ◽  
Grothendieck Rings ◽  
Dimension One ◽  
Gk Dimension

Wu–Liu–Ding algebras are a class of affine prime regular Hopf algebras of GK-dimension one, denoted by [Formula: see text]. In this paper, we consider their quotient algebras [Formula: see text] which are a new class of non-pointed semisimple Hopf algebras. We describe the Grothendieck rings of [Formula: see text] when [Formula: see text] is odd. It turns out that the Grothendieck rings are commutative rings generated by three elements subject to some relations. Then we compute the Casimir numbers of the Grothendieck rings for [Formula: see text] and [Formula: see text].

scholarly journals HIGHER CONJUGATION COHOMOLOGY IN COMMUTATIVE HOPF ALGEBRAS

2001 ◽  
Vol 44 (1) ◽  
pp. 19-26 ◽  
Author(s):  
M. D. Crossley ◽  
Sarah Whitehouse
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Symmetric Group ◽  
Group Action ◽  
Hopf Algebras ◽  
Homotopy Theory ◽  
Cohomology Ring ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Primary 16W30

AbstractLet $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

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-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 Prime regular Hopf algebras of GK-dimension one

2010 ◽  
Vol 101 (1) ◽  
pp. 260-302 ◽  
Author(s):  
K. A. Brown ◽  
J. J. Zhang
Keyword(s):  
Hopf Algebras ◽  
Dimension One ◽  
Gk Dimension
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