Descent, fields of invariants, and generic forms via symmetric monoidal categories

Meir, Ehud

doi:10.1016/j.jpaa.2015.10.016

Cited by 8 publications

(12 citation statements)

References 17 publications

(34 reference statements)

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“…Since the Hopf algebra H is already defined over k = K G , γ W is again a cocycle deformation of H. The structure constants for the multiplication and the coaction are given by applying γ to the structure constants of W . The fact that the Hopf algebra H is already defined over k is crucial here, since otherwise we would get that γ W is a comodule algebra over γ H. For more on Galois twisting of algebraic structures, see Section 8 of [Mei16]. It is easy to show that since f ∈ H * k and h(i) ∈ H k it holds that cγ W (l, σ, f, h(1), .…”

Section: The Algebras K[amentioning

confidence: 99%

“…The comultiplication in this algebra is given on the generators by the rules ∆(g) = g ⊗ g and ∆(x) = x ⊗ 1 + g ⊗ x. The classification of 2-cocycles for this Hopf algebras are known (see [Mas94] and the examples in [Mei16]). We will follow now some of the ideas of [Mei16] and describe this problem as a problem in invariant theory.…”

Section: Taft Hopf Algebrasmentioning

confidence: 99%

“…The classification of 2-cocycles for this Hopf algebras are known (see [Mas94] and the examples in [Mei16]). We will follow now some of the ideas of [Mei16] and describe this problem as a problem in invariant theory. To do so, we will begin by taking a cocycle deformation of H n , and analyze its structure.…”

Section: Taft Hopf Algebrasmentioning

confidence: 99%

“…We begin by analysing W similar to the way we proceeded in the previous examples. In the language of [Mei16], we study the fundamental category of W . Whenever it will be convenient for us, we will use the isomorphism W ∼ = H of right H-comodules.…”

Section: 4mentioning

confidence: 99%

“…Carrying out the analysis of the algebra of invariants for a given Hopf algebra H can be quite difficult (the presentation which we will give here has in general infinitely many generators and infinitely many relations). In order to overcome this problem we will combine certain results from [Mei16], where certain classification results for 2-cocycles were also obtained by using a categorical construction, and we will show some alternative ways to describe the variety X H and the ring K[X H ], which will help us to simplify the calculations. We shall explain how, in many examples, we can find a subvariety Y ′ ⊆ Y H and a subgroup N ⊆ GL(W ) which acts on Y ′ such that the natural map Y ′ /N → Y /GL(W ) is an isomorphism of varieties, or at least a bijection.…”

Section: Introductionmentioning

confidence: 99%

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Hopf cocycle deformations and invariant theory

Meir

2019

Math. Z.

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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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Section: The Algebras K[amentioning

confidence: 99%

Section: Taft Hopf Algebrasmentioning

confidence: 99%

Section: Taft Hopf Algebrasmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hopf cocycle deformations and invariant theory

Meir

2019

Math. Z.

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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

Meir

2017

Advances in Mathematics

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Abstract. We study Hopf algebras via tools from geometric invariant theory. We show that all the invariants we get can be constructed using the integrals of the Hopf algebra and its dual together with the multiplication and the comultiplication, and that these invariants determine the isomorphism class of the Hopf algebra. We then define certain canonical subspaces Inv i,j of tensor powers of H and H * , and use the invariant theory to prove that these subspaces satisfy a certain non-degeneracy condition. Using this non-degeneracy condition together with results on symmetric monoidal categories, we prove that the spaces Inv i,j can also be described as (A , where A is the group of Hopf automorphisms of H. As a result we prove that the number of possible Hopf orders of any semisimple Hopf algebra over a given number ring is finite. we give some examples of these invariants arising from the theory of Frobenius-Schur Indicators, and from Reshetikhin-Turaev invariants of three manifolds. We give a complete description of the invariants for a group algebra, proving that they all encode the number of homomorphisms from some finitely presented group to the group. We also show that if all the invariants are algebraic integers, then the Hopf algebra satisfies Kaplansky's sixth conjecture: the dimensions of the irreducible representations of H divide the dimension of H.

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Descent, fields of invariants, and generic forms via symmetric monoidal categories

Cited by 8 publications

References 17 publications

Hopf cocycle deformations and invariant theory

Hopf cocycle deformations and invariant theory

Interpolations of monoidal categories and algebraic structures by invariant theory

Semisimple Hopf algebras via geometric invariant theory

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