Classification of pointed rank one Hopf algebras

Scherotzke, Sarah

doi:10.1016/j.jalgebra.2008.01.028

Cited by 23 publications

(46 citation statements)

References 7 publications

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“…Now the statement holds by applying Proposition 2.3.1, (x⊗1+g⊗x) p = x p ⊗1+g p ⊗x p +(g)(ad x) p−1 ⊗x. Alternatively, one may also derive from the argument in [20,Corollary 4.10] and [24] to show the result.…”

Section: Van C Nguyen and Xingting Wangmentioning

confidence: 99%

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic

Nguyen

Wang

2018

Algebra Colloq.

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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) = ...

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Section: Van C Nguyen and Xingting Wangmentioning

confidence: 99%

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic

Nguyen

Wang

2018

Algebra Colloq.

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“…Hopf-Ore extensions were introduced and studied by Panov in [9]. Krop, Radford and Scherotzke studied finite dimensional rank one pointed Hopf algebras over an algebraically closed field in [5] and [10], respectively. They showed that a finite dimensional rank one pointed Hopf algebra over an algebraically closed field is isomorphic to a quotient of a Hopf-Ore extension of a group algebra.…”

Section: Introductionmentioning

confidence: 99%

Green ring of the category of weight modules over the Hopf–Ore extensions of group algebras

Sun

Chen

2019

Communications in Algebra

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In this paper, we continue our study of the tensor product structure of category W of weight modules over the Hopf-Ore extensions kG(χ −1 , a, 0) of group algebras kG, where k is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of χ and χ(a) are different. Then we describe the Green ring r(W) of the tensor category W. It is shown that r(W) is isomorphic to the polynomial algebra over the group ring ZĜ in one variable when |χ(a)| = |χ| = ∞, and that r(W) is isomorphic to the quotient ring of the polynomial algebra over the group ring ZĜ in two variables modulo a principle ideal when |χ(a)| < |χ| = ∞. When |χ(a)| |χ| < ∞, r(W) is isomorphic to the quotient ring of a skew group ring Z[X]♯Ĝ modulo some ideal, where Z[X] is a polynomial algebra over Z in infinitely many variables.

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“…They classified all finite dimensional pointed Hopf algebras of rank one over an algebraically field k of characteristic 0. Scherotzke classified such Hopf algebras for the case of char(k) = p > 0 in [23]. It was shown in [18,23] that a finite dimensional pointed Hopf algebra of rank one over an algebraically closed field is isomorphic to a quotient of a Hopf-Ore extension of its coradical.…”

Section: Introductionmentioning

confidence: 99%

“…Scherotzke classified such Hopf algebras for the case of char(k) = p > 0 in [23]. It was shown in [18,23] that a finite dimensional pointed Hopf algebra of rank one over an algebraically closed field is isomorphic to a quotient of a Hopf-Ore extension of its coradical. Wang, Li and Zhang [24,25] studied the representations of finite dimensional pointed Hopf algebras of rank one over an algebraically closed field of characteristic zero.…”

Section: Introductionmentioning

confidence: 99%

Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

Sun

Chen

2017

Communications in Algebra

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In this paper, we investigate the tensor structure of the category of finite dimensional weight modules over the Hopf-Ore extensions kG(χ −1 , a, 0) of group algebras kG. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that k is an algebraically closed field of characteristic zero, and the orders of χ and χ(a) are the same.2010 Mathematics Subject Classification. 16G30, 16T99.

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Classification of pointed rank one Hopf algebras

Abstract: In this paper we classify the finite-dimensional pointed rank one Hopf algebras which are generated as algebras by the first element of the coradical filtration over a field of prime characteristic.Comment: 25 page

Cited by 23 publications

References 7 publications

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic

Green ring of the category of weight modules over the Hopf–Ore extensions of group algebras

Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

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Classification of pointed rank one Hopf algebras

Abstract: In this paper we classify the finite-dimensional pointed rank one Hopf algebras which are generated as algebras by the first element of the coradical filtration over a field of prime characteristic.Comment: 25 page

Cited by 23 publications

References 7 publications

Pointed p3-Dimensional Hopf Algebras in Positive Characteristic

Pointed p3-Dimensional Hopf Algebras in Positive Characteristic

Green ring of the category of weight modules over the Hopf–Ore extensions of group algebras

Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

Contact Info

Product

Resources

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Pointed p³-Dimensional Hopf Algebras in Positive Characteristic

Pointed p³-Dimensional Hopf Algebras in Positive Characteristic