Hopf cocycle deformations and invariant theory

Meir, Ehud

doi:10.1007/s00209-019-02326-5

Cited by 10 publications

(12 citation statements)

References 30 publications

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“…In fact, H * is a cocycle deformation of the Radford biproduct B#k S 3 , where B is the Nichols algebra generated by a, b and c and the dual group algebra k S 3 is the maximal semisimple sub-Hopf-algebra. For more details about this Hopf algebra see [M18,AV12].…”

Section: A Hopf Algebra With the Chevalley Property Which Is Not Basicmentioning

confidence: 99%

On isotypic decompositions for non-semisimple Hopf algebras

Koppen¹,

Meir²,

Schweigert³

2019

Preprint

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In this paper we study the isotypic decomposition of the regular module of a finitedimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.

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Section: A Hopf Algebra With the Chevalley Property Which Is Not Basicmentioning

confidence: 99%

On isotypic decompositions for non-semisimple Hopf algebras

Koppen¹,

Meir²,

Schweigert³

2019

Preprint

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“…In this paper we will encounter several different reductive groups. We will use the following lemma frequently, following Section 4 in [Me19].…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…Understanding the relations between the generators arising from the Schur-Weyl duality is more difficult. In [Me19] a similar GIT quotient was studied for two-cocycles over an arbitrary finite dimensional Hopf algebra. A description of all relations among these generators was also given.…”

Section: Introductionmentioning

confidence: 99%

Invariant rings and representations of the symmetric groups

Meir¹,

Govc²

2019

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In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K[U ] Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where U = End(W ⊕k ) and Γ = GL(W ) and on the case where U = End(V ⊗ W ) and Γ = GL(V ) × GL(W ), though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series.c k,l = m, c i,j = 0 unless (i, j) ∈ {(3, 1), (3, 2), (4, 1), (4, 2)}

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“…In addition, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry (cf. [34,35,38,53]) and mathematical physics (see, e. g., [6,10,19,20,23,31,36,56] and ref. therein).…”

Section: Introductionmentioning

confidence: 99%

Lifting spectral triples to noncommutative principal bundles

Schwieger,

Wagner

2020

Preprint

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Hopf cocycle deformations and invariant theory

Cited by 10 publications

References 30 publications

On isotypic decompositions for non-semisimple Hopf algebras

On isotypic decompositions for non-semisimple Hopf algebras

Invariant rings and representations of the symmetric groups

Lifting spectral triples to noncommutative principal bundles

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