Connected Hopf algebras of dimension<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>

Wang, Xingting

doi:10.1016/j.jalgebra.2013.06.008

Cited by 30 publications

(34 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then,

H

is a restricted universal enveloping algebra as in cases (2) and (3) (cf. [, Theorem 7.1]).

□

…”

Section: Resultsmentioning

confidence: 98%

See 1 more Smart Citation

Hopf algebras of prime dimension in positive characteristic

Wang

2019

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We prove that a Hopf algebra of prime dimension p over an algebraically closed field, whose characteristic is equal to p, is either a group algebra or a restricted universal enveloping algebra. Moreover, we show that any Hopf algebra of prime dimension p over a field of characteristic q>0 is commutative and cocommutative when q=2 or p<4q. This problem remains open in positive characteristic when 2

show abstract

“…Then,

H

is a restricted universal enveloping algebra as in cases (2) and (3) (cf. [, Theorem 7.1]).

□

…”

Section: Resultsmentioning

confidence: 98%

“…So,

H ≅ {End}_{H} false(H false) = {End}_{H} false(M false)

is local and

H^{*}

is connected. It follows from the classification of all connected

p

‐dimensional Hopf algebras [, Theorem 7.1] that

H^{*}

is commutative. Finally, since

H

is always cocommutative and

double-struckk

is algebraically closed, we know

H

is pointed (cf.…”

Section: Resultsmentioning

confidence: 99%

Hopf algebras of prime dimension in positive characteristic

Wang

2019

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this case, the primitive space P (H) = k x ⊕ k y is indeed a two-dimensional restricted Lie algebra. We can apply the classifications in [25,Theorem 7.4 (1)- (5)] and obtain the following five classes of H:…”

Section: Claimmentioning

confidence: 99%

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic

Nguyen

Wang

2018

Algebra Colloq.

Self Cite

View full text Add to dashboard Cite

We classify pointed p 3 -dimensional Hopf algebras H over any algebraically closed field k of prime characteristic p > 0. In particular, we focus on the cases when the group G(H) of group-like elements is of order p or p 2 , that is, when H is pointed but is not connected nor a group algebra. This work provides many new examples of (parametrized) non-commutative and non-cocommutative finite-dimensional Hopf algebras in positive characteristic. G G YD of either diagonal type or Joran type; hence, further cases occur where the structures arise. Interested readers may refer to corresponding section(s) for detailed classification results.At last, we emphasize that the principle proposed by Andruskiewitsch and Schneider in [2] to study pointed Hopf algebras in characteristic zero is generally applicable 2010 Mathematics Subject Classification. 16T05, 17B60.We obtain the following classes of pointed p 3 -dimensional Hopf algebras in characteristic p > 0:Case A1. Liftings from gr H = k a, b, g /(g p = 1, a p = b p = 0, ab = ba, ga = ag, gb = bg),When u = 0, there are 1 infinite parametric family and 10 finite classes of H having structured lifted from case (A1). When u = 0, there are 2(p − 1) infinite parametric families and 6(p − 1) finite classes of H.Case A2. Liftings from gr H = k a, b, g /(g p = 1, a p = b p = 0, ab = ba, ga = ag, gb = bg),There are 5 finite classes of H having structured lifted from case (A2).Case A3. (p = 2) Liftings from gr H = k a, b, g /(g p = 1, a p = b p = 0, ab = ba, ga = bg, gb = ag), with ∆There are 5 finite classes of H having structured lifted from case (A3). POINTED p 3 -DIMENSIONAL HOPF ALGEBRAS IN POSITIVE CHARACTERISTIC 3Case B. (p > 2) Liftings from gr H = k a, b, g /(g p = 1, a p = b p = 0, ab − ba = 1 2 a 2 , ga = ag, gb = (a + b)g), with ∆(g) = g ⊗ g, ∆(a) = a ⊗ 1 + g ⊗ a,Due to complicated computations in characteristic p, the lifting in case (B) is not clear in general; however, we show in Section 4.2 the case when p = 3 for illustration and make a conjecture for the lifting for p > 3.Case C. Liftings from gr H = k a, b, g /(g p = 1, a p = b p = 0, ab = ba, ga = ag, gb = bg), with ∆(g) = g ⊗ g, ∆(a) = a ⊗ 1 + g ǫ ⊗ a,When ǫ = 0, we get 1 infinite parametric family and 2 finite classes of H. When ǫ = 1, due to complicated computations in characteristic p, the lifting in this case (Cb) is not clear in general. We show in Section 4.3 the cases (C) when p = 2, and when p > 2 with additional assumption gx = xg, where x is the lifting of a from grH to H. Case D1. Liftings from gr H = k a, g /(g p 2 = 1, a p = 0, ga = ag), with ∆(g) = g ⊗ g, ∆(a) = a ⊗ 1 + g ǫ ⊗ a, ε(g) = 1, ε(a) = 0, S(g) = g −1 , S(a) = −ag −ǫ , for ǫ ∈ {0, 1, p}.When ǫ = 0, we get 2 finite classes of H. When ǫ = 1, there are 1 infinite parametric family and 2 finite classes of H. When ǫ = p, there are 2 finite classes of H. Case D2. Liftings from gr H = k a, g 1 , g 2 /(g p 1 = g p 2 = 1, a p = 0, g i a = ag i ), with ∆(g i ) = g i ⊗ g i , ∆(a) = a ⊗ 1 + g ǫ 1 ⊗ a, ε(g i ) = 1, ε(a) = 0, S(g i ) = g −1 i , S(a) = ...

show abstract

“…The proof of these two theorems is dependent on the information of these primitive cohomologies. Since a few years ago, the idea of the primitive cohomology has been used in an essential way in several projects, see [Zh,WZZ3,W1,W2,WW,NWW].…”

Section: Introductionmentioning

confidence: 99%

Primitive cohomology of Hopf algebras

2016

View full text Add to dashboard Cite

Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non locally PI, pointed Hopf algebra domains of Gelfand-Kirillov dimension two; and secondly we classify all pointed Hopf algebras of rank one. The first classification extends some results of Brown, Goodearl and others in an ongoing project to understand all Hopf algebras of low Gelfand-Kirillov dimension. The second generalizes results of Krop-Radford and Wang-You-Chen which classified Hopf algebras of rank one under extra hypothesis. Properties and algebraic structures of the primitive cohomology are discussed.

show abstract

Connected Hopf algebras of dimensionp2

Cited by 30 publications

References 5 publications

Hopf algebras of prime dimension in positive characteristic

Hopf algebras of prime dimension in positive characteristic

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic

Primitive cohomology of Hopf algebras

Contact Info

Product

Resources

About

Connected Hopf algebras of dimensionp2

Cited by 30 publications

References 5 publications

Hopf algebras of prime dimension in positive characteristic

Hopf algebras of prime dimension in positive characteristic

Pointed p3-Dimensional Hopf Algebras in Positive Characteristic

Primitive cohomology of Hopf algebras

Contact Info

Product

Resources

About

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic