On the number of types of finite dimensional Hopf algebras

Beattie, Margaret; Dăscălescu, S.; Grunenfelder, L.

doi:10.1007/s002220050302

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Remark1

Conjecture the Hopf Algebras Of A Given Finite Dimension Ar1

Introduction1

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2000

2017

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Cited by 56 publications

(34 citation statements)

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“…The converse is, in general, false. We take from [4] [2] the following example : for a fixed primitive p-th root of unity λ and t ∈ K * , the family of Hopf algebra H t generated by c, x 1 , x 2 and subject to…”

Section: Remarkmentioning

confidence: 99%

Degeneration, Rigidity and Irreducible Components of Hopf algebras

Makhlouf

2005

Algebra Colloq.

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Section: Remarkmentioning

confidence: 99%

Degeneration, Rigidity and Irreducible Components of Hopf algebras

Makhlouf

2005

Algebra Colloq.

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“…3.3], Andruskiewitsch and Schneider [AS,Thm. 0.3], Beattie, Dȃscȃlescu and Grünen-felder [BDG,Thm. 1], and Müller [M,Thm.…”

Section: Conjecture the Hopf Algebras Of A Given Finite Dimension Armentioning

confidence: 99%

Defending the negated Kaplansky conjecture

Masuoka

2001

Proc. Amer. Math. Soc.

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“…The classification in the second focuses on pointed Hopf algebras over an algebraically closed field of characteristic 0; see [Andruskiewitsch and Schneider 2006;. Pointed Hopf algebras play an important role in [Andruskiewitsch and Schneider 1998;Beattie et al 1999;Gelaki 1998], where Kaplansky's 10th conjecture is refuted by constructing infinitely many nonisomorphic Hopf algebras of a given prime power dimension.…”

Section: Introductionmentioning

confidence: 99%

Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of typeG₂

Hu¹,

Wang²

2009

Pacific J. Math.

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From the 2-parameter quantum group U r,s (G 2 ) defined by Hu and Shi in 2007, we construct finite-dimensional pointed Hopf algebras u r,s (G 2 ) (that is, restricted 2-parameter quantum groups); these turn out to be Drinfel'd doubles. Crucial is a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G 2 in the 2-parameter quantum version. We exhibit the possible commutation relations among quantum root vectors. Then we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions, by determining their left and right integrals. We also determine all the Hopf algebra isomorphisms of u r,s (G 2 ) by describing its sets of left (right) skew-primitive elements.

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On the number of types of finite dimensional Hopf algebras

Cited by 56 publications

References 8 publications

Degeneration, Rigidity and Irreducible Components of Hopf algebras

Degeneration, Rigidity and Irreducible Components of Hopf algebras

Defending the negated Kaplansky conjecture

Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of typeG₂

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On the number of types of finite dimensional Hopf algebras

Cited by 56 publications

References 8 publications

Degeneration, Rigidity and Irreducible Components of Hopf algebras

Degeneration, Rigidity and Irreducible Components of Hopf algebras

Defending the negated Kaplansky conjecture

Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of typeG2

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Product

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Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of typeG₂