Lifting of Nichols algebras of typeB 2

Beattie, Margaret; Dăscălescu, S.; Raianu, Serban

doi:10.1007/bf02784503

Cited by 23 publications

(54 citation statements)

References 14 publications

(24 reference statements)

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“…Definition 2.9. In what follows, denotes the largest Hopf ideal in G X (2) . The ideal is homogeneous in each x i ∈ X (see [11,Lemma 2.2]).…”

Section: Shuffle Representationmentioning

confidence: 99%

“…If the kernel of ξ defined in (2.15) is contained in the ideal G X (2) generated by x i x j , i, j ∈ I, then there exists a Hopf algebra projection π :…”

Section: Shuffle Representationmentioning

confidence: 99%

“…Moreover, if X is finite, then ∩ k X (as well as any differential ideal in k X ) is generated as a left ideal by constants from k X (2) (see [10,Corollary 7.8]). Thus, we may formulate the following criterion, which is useful for checking relations.…”

Section: Ms-criterionmentioning

confidence: 99%

“…In Section 9, we consider some important examples, including the right coideal subalgebra generated by S (k, m) with regular S. We also analyze in detail the simplest (but not trivial [2]) case, n = 2.…”

Section: Introductionmentioning

confidence: 99%

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Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Kharchenko¹

2011

J. Eur. Math. Soc.

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Abstract. We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group U + q (so 2n+1 ), provided that q is not a root of 1. If q has a finite multiplicative order t > 4, this classification remains valid for homogeneous right coideal subalgebras of the Frobenius-Lusztig kernel u + q (so 2n+1 ). In particular, the total number of right coideal subalgebras that contain the coradical equals (2n)!!, the order of the Weyl group defined by the root system of type B n .Keywords. Coideal subalgebra, Hopf algebra, PBW-basis IntroductionIn the present paper, we continue the classification of right coideal subalgebras in quantised enveloping algebras begun in [13]. We offer a complete classification of right coideal subalgebras that contain all grouplike elements for the multiparameter version of the quantum group U + q (so 2n+1 ), provided that the main parameter q is not a root of 1. If q has a finite multiplicative order t > 4, this classification remains valid for homogeneous right coideal subalgebras of the multiparameter version of the Frobenius-Lusztig kernel u + q (so 2n+1 ). The main result of the paper is the establishment of a bijection between all sequences (θ 1 , . . . , θ n ) such that 0 ≤ θ k ≤ 2n − 2k + 1, 1 ≤ k ≤ n, and the set of all (homogeneous if q t = 1, t > 4) right coideal subalgebras of U + q (so 2n+1 ), q t = 1 (respectively of u + q (so 2n+1 )) that contain the coradical. (Recall that in a pointed Hopf algebra, the grouplike elements span the coradical.) In particular, there are (2n)!! different right coideal subalgebras that contain the coradical. Interestingly, this number coincides with the order of the Weyl group for the root system of type B n . In [13], we proved that the number of different right coideal subalgebras that contain the coradical of U + q (sl n+1 ) equals (n + 1)!, the order of the Weyl group for the root system of type A n . Recently, B. Pogorelsky [16] proved that the quantum Borel algebra U + q (g) for the simple Lie algebra of type G 2 has 12 different right coideal subalgebras over the coradical. This Conjecture. Let g be a simple Lie algebra defined by a finite root system R. The number of different right coideal subalgebras that contain the coradical in a quantum Borel algebra U + q (g) equals the order of the Weyl group defined by the root system R, provided that q is not a root of 1. 1In Section 2, following [13], we introduce the main concepts of the paper and we formulate known results that are useful for classification. In the third section, we prove auxiliary relations in a multiparameter version of U + q (so 2n+1 ). In the fourth section, we note that the Weyl basisin place of the Lie operation is a set of PBW-generators for U + q (so 2n+1 ) and u + q (so 2n+1 ). By means of the shuffle representation, in Theorem 4.3, we prove an explicit formula for the coproduct of these PBW-generators, which is the key result for further considerations:where τ i = 1 with only one exception, τ n = q, while g ki are su...

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“…Definition 2.9. In what follows, denotes the largest Hopf ideal in G X (2) . The ideal is homogeneous in each x i ∈ X (see [11,Lemma 2.2]).…”

Section: Shuffle Representationmentioning

confidence: 99%

“…If the kernel of ξ defined in (2.15) is contained in the ideal G X (2) generated by x i x j , i, j ∈ I, then there exists a Hopf algebra projection π :…”

Section: Shuffle Representationmentioning

confidence: 99%

Section: Ms-criterionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Kharchenko¹

2011

J. Eur. Math. Soc.

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“…For Cartan type A θ , there are expositions from scratch in [82] for generic q, in [12] for q = −1, and in [20] for q ∈ G N , N ≥ 3. Other Nichols algebras of rank 2 were presented in [27,45,53]. Standard type appeared in [22]; this paper contains a self-contained proof of the defining relations of the Nichols algebras of Cartan type at a generic parameter, i.e.…”

Section: Attributionmentioning

confidence: 97%

On finite dimensional Nichols algebras of diagonal type

2017

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This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand-Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59-124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164: 175-188, 2006; Heckenberger and Yamane in Math Z 259:255-276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) Communicated by Efim Zelmanov.

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Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

Heckenberger

Lochmann

Vendramin

2012

Transformation Groups

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We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.

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Lifting of Nichols algebras of typeB 2

Cited by 23 publications

References 14 publications

Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

Right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$

On finite dimensional Nichols algebras of diagonal type

Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

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