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.

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

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 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.

Stably thick subcategories of modules over Hopf algebras

2001 ◽  
Vol 130 (3) ◽  
pp. 441-474 ◽  
Author(s):  
MARK HOVEY ◽  
JOHN H. PALMIERI
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Closure ◽  
Steenrod Algebra ◽  
Closed Field ◽  
Finite Dimensional ◽  
A Finite Group ◽  
Similar Classification ◽  
General Method

We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional co-commutative Hopf algebra B. Our method is based on that of Benson–Carlson–Rickard [BCR1], who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

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 Plane Curves Which are Quantum Homogeneous Spaces

2021 ◽  
Author(s):  
Ken Brown ◽  
Angela Ankomaah Tabiri
Keyword(s):  
Hopf Algebras ◽  
Final Section ◽  
Homogeneous Spaces ◽  
Plane Curve ◽  
Plane Curves ◽  
Closed Field ◽  
Quantum Homogeneous Space ◽  
Central Elements ◽  
Pointed Hopf Algebras ◽  
Future Work

AbstractLet $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

scholarly journals A CLASS OF QUASITRIANGULAR GROUP-COGRADED MULTIPLIER HOPF ALGEBRAS

2018 ◽  
Vol 62 (1) ◽  
pp. 43-57
Author(s):  
TAO YANG ◽  
XUAN ZHOU ◽  
HAIXING ZHU
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Finite Dimensional ◽  
Special Cases ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Classical Construction

AbstractFor a multiplier Hopf algebra pairing 〈A,B〉, we construct a class of group-cograded multiplier Hopf algebras D(A,B), generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai [Isr. J. Math. 158 (2007), 349–365]. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in M(B⊗A) we show the existence of quasitriangular structure on D(A,B). As an application, some special cases and examples are provided.

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