Semisimple Hopf algebras via geometric invariant theory

Meir, Ehud

doi:10.1016/j.aim.2017.02.020

“…The goal of the present paper is to study this classification problem, for a general finite dimensional Hopf algebra over an algebraically closed field of characteristic zero, from a geometric point of view. This will continue the study done in [Mei17] and [DKS03] where geometric invariant theory was applied to study finite dimensional semisimple Hopf algebras. We will show that for a given Hopf algebra H over an algebraically closed field K of characteristic 0 the set of all equivalence classes of cocycle deformations X H has a natural structure of an affine algebraic variety.…”

Section: Introductionmentioning

confidence: 57%

“…Notice that for the above theorem we do not require the Hopf algebra H to be semisimple, even though the semisimplicity was a necessary condition to apply geometric invariant theory to Hopf algebras in [DKS03] and [Mei17].…”

Section: Introductionmentioning

confidence: 99%

Hopf cocycle deformations and invariant theory

Meir

¹

2019

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For a given finite dimensional Hopf algebra H we describe the set of all equivalence classes of cocycle deformations of H as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the Universal Coefficients Theorem in the case of a group algebra, and we also give examples from other families of Hopf algebras, including dual group algebras and Bosonizations of Nichols algebras. In particular, we use the methods developed here to classify the cocycle deformations of a dual pointed Hopf algebra associated to the symmetric group on three letters. We also give an example of a cocycle deformation over a dual group algebra, which has only rational invariants, but which is not definable over the rational field. This differs from the case of group algebras, in which every 2-cocycle is equivalent to one which is definable by its invariants.

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“…Geometric invariant theory provides a natural tool to study such algebraic structures. This was carried out in [Me17] and [DKS03] for finite dimensional semisimple Hopf algebras. The idea is the following: by fixing some discrete invariants such as the dimension of the Hopf algebra in [DKS03] or the dimension of the irreducible representations in [Me17], and by fixing a basis for the Hopf algebra, the Hopf algebra can be described using structure constants.…”

Section: Introductionmentioning

confidence: 99%

“…This was carried out in [Me17] and [DKS03] for finite dimensional semisimple Hopf algebras. The idea is the following: by fixing some discrete invariants such as the dimension of the Hopf algebra in [DKS03] or the dimension of the irreducible representations in [Me17], and by fixing a basis for the Hopf algebra, the Hopf algebra can be described using structure constants. In this way a Hopf algebra can be seen as a point in a certain affine space A N (where N is the total number of structure constants involved).…”

Section: Introductionmentioning

confidence: 99%

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Invariant rings and representations of the symmetric groups

Meir¹,

Govc²

2019

Preprint

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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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Interpolations of monoidal categories and algebraic structures by invariant theory

Meir¹

2023

Sel. Math. New Ser.

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In this paper we give a general construction of symmetric monoidal categories that generalizes Deligne’s interpolated categories, the categories introduced by Knop, and the recent TQFT construction of Khovanov, Ostrik, and Kononov. The categories we will consider are generated by an algebraic structure. In a previous work by the author a universal ring of invariants $$\mathfrak {U}$$ U for algebraic structures of a specific type was constructed. It was shown that any algebraic structure of this type in $${\text {Vec}}_K$$ Vec K gives rise to a character $$\chi :\mathfrak {U}\rightarrow K$$ χ : U → K . In this paper we consider algebraic structure in general symmetric monoidal categories, not only in $${\text {Vec}}_K$$ Vec K , and general characters on $$\mathfrak {U}$$ U . From any character $$\chi :\mathfrak {U}\rightarrow K$$ χ : U → K we construct a symmetric monoidal category $${\mathcal {C}}_{\chi }$$ C χ , analogous to the universal construction from TQFT. We then give necessary and sufficient conditions for a given character to arise from a structure in an abelian category with finite dimensional hom-spaces. We call such characters good characters. We show that if $$\chi $$ χ is good then $${\mathcal {C}}_{\chi }$$ C χ is abelian and semisimple, and that the set of good characters forms a K-algebra. We also show that the categories $${\mathcal {C}}_{\chi }$$ C χ contain all categories of the form $${\text {Rep}}(G)$$ Rep ( G ) , where G is reductive. The construction of $${\mathcal {C}}_{\chi }$$ C χ gives a way to interpolate algebraic structures, and also symmetric monoidal categories, in a way that generalizes Deligne’s categories $${\text {Rep}}(S_t)$$ Rep ( S t ) , $${\text {Rep}}({\text {GL}}_t(K))$$ Rep ( GL t ( K ) ) , and $${\text {Rep}}(\text {O}_t)$$ Rep ( O t ) . We also explain how one can recover the recent construction of 2 dimensional TQFT of Khovanov, Ostrik, and Kononov, by the methods presented here. We give new examples, of interpolations of the categories $${\text {Rep}}({\text {Aut}}_{{\mathcal {O}}}(M))$$ Rep ( Aut O ( M ) ) where $${\mathcal {O}}$$ O is a discrete valuation ring with a finite residue field, and M is a finite module over it. We also generalize the construction of wreath products with $$S_t$$ S t , which was introduced by Knop.

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Semisimple Hopf algebras via geometric invariant theory

Cited by 9 publications

References 24 publications

Hopf cocycle deformations and invariant theory

Hopf cocycle deformations and invariant theory

Invariant rings and representations of the symmetric groups

Interpolations of monoidal categories and algebraic structures by invariant theory

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