Finite Weyl groupoids

Cuntz, Michael; Heckenberger, I.

doi:10.1515/crelle-2013-0033

Cited by 33 publications

(103 citation statements)

References 20 publications

(52 reference statements)

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“…(b) Later, the classification of the finite generalized root systems was obtained in [36]. There are finite GRS that do not arise from arithmetic Nichols algebras; at least one of them arises from a finite dimensional Nichols algebras of diagonal type in positive characteristic.…”

Section: Example 230mentioning

confidence: 99%

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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Section: Example 230mentioning

confidence: 99%

On finite dimensional Nichols algebras of diagonal type

2017

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“…The proof for type F uses stronger facts than the proof for type D, as it relies on the classification from [13]. By this reason, our order of preference for application of these criteria is first type D, then F.…”

Section: Remark 25mentioning

confidence: 99%

“…• If G is a sporadic simple group, then G collapses, except for the groups G = F i 22 Table 1] and improved in [14,Appendix], of examples not known to be finite-dimensional. • If G = SL 2 (q), GL 2 (q), PGL 2 (q) or PSL 2 (q), all irreducible Yetter-Drinfeld modules M (O, ρ) have infinite dimensional Nichols algebra, except for a list of examples not known to be finite-dimensional given in [15,16].…”

Section: Introductionmentioning

confidence: 99%

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes inPSLn(q)

2015

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“…In a series of papers I. Heckenberger and the first author investigate a class of objects called finite Weyl groupoids, a generalization of Weyl groups. Their work results in a complete classification of these objects, [CH15]. Since Weyl groupoids are in one to one correspondence with crystallographic arrangements [Cun11a], and these constitute a large subclass of the known simplicial arrangements, this explains a large subset of the arrangements in Grünbaum's list.…”

Section: Introductionmentioning

confidence: 99%

Supersolvable simplicial arrangements

Cuntz

Mücksch

2019

Advances in Applied Mathematics

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Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irreducible simplicial arrangements of rank greater than three, our result shows that supersolvability imposes a strong integrality property; such an arrangement is called crystallographic. Furthermore we introduce Coxeter graphs for simplicial arrangements which serve as our main tool of investigation.2010 Mathematics Subject Classification. 20F55, 52C35, 14N20.

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Finite Weyl groupoids

Cited by 33 publications

References 20 publications

On finite dimensional Nichols algebras of diagonal type

On finite dimensional Nichols algebras of diagonal type

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes inPSLn(q)

Supersolvable simplicial arrangements

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