Abstracts of the talks at
Hopf Algebras Conference in Tsukuba
Nicol´as Andruskiewitsch, Universidad Nacional de C´ordoba A finite-dimensional Lie algebra arising from a Nichols algebra of di- agonal type
Abstract: Let B be a finite-dimensional Nichols algebra of diagonal type over an algebraically closed field of characteristic 0. The distin- guished pre-Nichols algebra of B, introduced and studied in [3], has several nice properties including finite GK-dimension and action of the Weyl groupoid. Its graded dual, called the Lusztig algebra of B, was subsequently introduced and studied in [1]. We will outline these con- structions. Then we will present the Lusztig algebra as an extension (as braided Hopf algebras) of B by the universal enveloping algebra of a graded nilpotent Lie algebra. We will exhibit explicitly this Lie algebra in several cases, including the all B of rank 2.
References
[1] N. Andruskiewitsch , I. Angiono and F. Rossi Bertone, The divided powers algebra of a finite-dimensional Nichols algebra of diagonal type, Math. Res. Lett., to appear.
[2] N. Andruskiewitsch , I. Angiono and F. Rossi Bertone, A finite- dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2), arXiv:1603.09387.
[3] I. Angiono, Distinguished Pre-Nichols algebras, Transf. Groups 21 (2016), 1-33.
Iv´an Ezequiel Angiono, Universidad Nacional de C´ordoba A quantum version of the algebra of distributions of SL2
Abstract: We construct a family of Galois extensions of the small quantum group of sl2, which mimic a family of (finite-dimensional) subalgebras of the algebras of distributions of SL2 over a field of pos- itive characteristic. We prove a kind of Steinberg tensor product de- composition for the simple modules of these algebras.
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Hideto Asashiba, Shizuoka University
Cohen-Montgomery duality for bimodules and its applications
Abstract: We fix a commutative ring k and a group G. To include infinite coverings of k-algebras into consideration we usually regard k- algebras as locally bounded k-categories with finite objects, so we will work with small k-categories. For small k-categories R and S with G-actions we introduce G-invariant S-R-bimodules and their category denoted by S-ModG-R, and denote by R/G the orbit category of R by G, which is a small G-graded k-category. For small G-graded k- categories A and B we introduce G-graded B-A-bimodules and their category denoted by B-ModG-A, and denote by A#G the smash prod- uct of A and G, which is a small k-category with G-action. Then the Cohen-Montgomery duality theorem [2, 1] says that we have equiv- alences (R/G)#G ≃ R and (A#G)/G ≃ A, by which we identify these pairs. In the talk we introduce functors (−)/G : S-ModG-R → (S/G)-ModG(R/G) and (−)#G : B-ModG-A → (B#G)-ModG-(A#G), and show that they are equivalences and quasi-inverses to each other (by applying A := R/G, R := A#G, etc.), and have good properties with tensor products and projectivity. We apply this to equivalences of Morita type to obtain the following.
Theorem.(1) There exists a “G-invariant stable equivalence of Morita type” between R and S if and only if there exists a “G-graded stable equivalence of Morita type” between R/G and S/G.
(2) There exists a G-graded stable equivalence of Morita type” be- tween A and B if and only if there exists a “G-invariant stable equiv- alence of Morita type” between A#G and B#G.
Here we note that a G-invariant (resp. G-graded) stable equivalence of Morita type is defined to be a usual stable equivalence of Morita type with additional properties, and does not mean an equivalence be- tween stable categories of G-invariant (resp. G-graded) modules. The corresponding results for standard derived equivalences and singular equivalences of Morita type hold as well.
References
[1] Asashiba, H., A generalization of Gabriel’s Galois covering functors II: 2-categorical Cohen- Montgomery duality, to appear in Applied Cat- egorical Structures, DOI: 10.1007/s10485-015-9416-9. (preprint arXiv: 0905.3884)
[2] Cohen, M. and Montgomery, S., Group-graded rings, smash prod- ucts, and group actions, Trans. Amer. Math. Soc. 282 (1984), 237– 258.
Takahiro Hayashi, Nagoya University
Group-theoretical categories and representation theory of weak Hopf al- gebras
Abstract: In this talk, we consider two classes of weak Hopf algebras whose module categories give realizations of (some of) group-theoretical categories. Using A-B torsors (Hopf biGalois extensions), we show the equivalence of module categories of these two classes of weak Hopf algebras, without assuming the semisimplicity. Also, we give an equiv- alence of character theories of these two classes. As an application, we compute 2nd Frobenius-Schur indicators of group-theoretical cate- gories corresponding to symmetric groups and their Young subgroups, which generalize results of Montgomery school and Schauenburg. Also, we give a combinatorial formula for the number of involutions in sym- metric groups.
Christian Kassel, CNRS and Universite de Strasbourg Distinguishing simple algebras using polynomial identities
Abstract: Thanks to the celebrated Amitsur-Levitzki theorem it can easily be checked that any simple finite-dimensional algebra over the complex numbers is determined up to isomorphism by its polynomial identities. Given a finite group G, Aljadeff and Haile recently proved that any simple G-graded algebra is determined (up to isomorphism) by its G-graded polynomial identities. In this lecture we enlarge the framework and raise the following question: given a finite-dimensional Hopf algebra H over an algebraically closed field of characteristic zero, are the simple H-comodule algebras determined up to isomorphism by their polynomial H-identities. After defining what a polynomial H-identity for an H-comodule algebra is, we exhibit finite sets of poly- nomial H-identities which distinguish the Galois objects over H in the case when H is a monomial Hopf algebra or H = E(n).
Takuya Matsumoto, Nagoya University
An example of the generalized quantum groups in the conformal field theory
Abstract: The conformal field theory is a remarkable example of the quantum field theory which allows mathematical rigorous treatment in terms of the Virasoro vertex operator algebra. In this talk, we address the case that the Virasoro central charge is the positive rational level, where the situation drastically differs from the generic case. This theory is referred as to the extended W-algebra of type sl2. After reviewing the theory, we explain how the quantum groups of type sl2
with q at roots of unity arise and its Drinfeld double construction from the viewpoint of the conformal field theory side.
This is based on the collaboration work with Prof. Yoshitake Hashimoto and Prof. Akihiro Tsuchiya.
Susan Montgomery, University of Southern California Hopf automorphisms and twisted extensions
Abstract: We give some applications of a Hopf algebra K constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnar’s smash coproduct Hopf algebra.
We find connections between the exponent and Frobenius-Schur in- dicators of K and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A.
This is joint work with Maria Vega and Sarah Witherspoon. Sonia Natale, Universidad Nacional de C´ordoba
Examples of tensor categories arising from matched pairs of groups Abstract: Let (G, Γ) be a matched pair of finite groups. We introduce the notion of (G, Γ)-crossed action on a tensor category. We show that every tensor category C endowed with a (G, Γ)-crossed action gives rise to a tensor category C(G,Γ) that fits into an exact sequence of tensor categories Rep G −→ C(G,Γ) −→ C. We also define the notion of a (G, Γ)-braiding in a tensor category endowed with a (G, Γ)-crossed ac- tion, which extends the notion of a G-crossed braided tensor category of Turaev. We show that if C is equipped with a (G, Γ)-braiding, then the tensor category C(G,Γ) is a braided tensor category in a canonical way.
Katsunori Saito, Nagoya University Toward quantization of Galois theory
Abstract: Picard-Vessiot theory that is a Galois theory of linear dif- ferential equations established in the 19th century. The Picard-Vessiot Galois groups are linear algebraic groups. Picard-Vessiot theory gen- eralized for linear difference equations. The Galios groups for linear difference equations are also linear algebraic groups. Takeuchi, Ma- suoka and Amano tried to extend Picard-Vessiot theory in the frame- work of Hopf algebra. Since they considered mainly commutative or co-commutative Hopf algebra operating on commutative rings of func- tions, Galois groups in their theories are however commutative or co- commutative Hopf algebras. It is quite natural to wonder if we could quantize Picard-Vessiot theory so that Hopf algebras that are neither commutative nor co-commutative appear as Galois group. We suc- ceeded in constructing a general Hopf Galois theory for linear equations with constant coefficients, in which Galois groups are neither commu- tative nor co-commutative Hopf algebra in general.
This talk is based on a joint work with Akira Masuoka and Hiroshi Umemura.
Yoichi Shibukawa, Hokkaido University
Construction of Hopf algebroids by means of parallelepipeds
Abstract: In the Euclidean 3-space, six parallelograms make a paral- lelepiped. Through the flip map, this system gives birth to the coordi- nate ring of the general linear group, which is a Hopf algebra. The aim of this talk is to introduce a generalization of this construction. We produce Hopf algebroids by means of dynamical Yang-Baxter maps related to parallelepipeds. This is a generalization of the Faddeev- Reshetikhin-Takhtajan construction.
Kenichi Shimizu, Shibaura Institute of Technology
Integrals for Hopf algebras from a tensor-categorical viewpoint
Abstract: A Haar measure on a locally compact group induces a linear functional on the space of integrable functions on the group. By the definition of a Haar measure, the functional has the property called the translation invariance. Moss Sweedler introduced the notion of integrals for Hopf algebras based on this observation. Integrals are used to establish many results on Hopf algebras and it is no doubt that they are an essential tool in studying Hopf algebras.
Our aim is to understand the Hopf algebra theory from a tensor- categorical viewpoint. Although there have been many pioneering works in this direction, there still are basic notions for Hopf algebras that are not understood in the context of tensor categories. In this talk, I will introduce integrals for finite tensor categories and explain that some formulas involving integrals for Hopf algebras can make sense for general finite tensor categories. As a demonstration of this machinery, I generalize indicators of finite-dimensional Hopf algebras introduced by Kashina, Montgomery and Ng to general finite tensor categories. Noriyuki Suwa, Chuo University
Group algebras and normal basis problem
Abstract: We discuss the inverse Hopf-Galois problem with normal basis, generalizing a method presented by Serre ⟨Groupes algebriques et corps de classes⟩. The unit group scheme of the group algebra of a finite flat group scheme plays a key role for cleft Hopf-Galois extensions.
Blas Torrecillas, Universidad de Almel´ıa
Galois theory and cleft extensions for monoidal cowreaths
Abstract: The Galois theory for monoidal cowreaths is developed. Cleft cowreaths are introduced in this context and its relation with the normal basis property investigated. The connection of this class of cowreath with some wreath algebra structures is obtained. Finally, several applications to quasi-Hopf algebras will be discussed.
This is a joint work with D. Bulacu. Hiroyuki Yamane, Toyama University Nichols Topology
Abstract: In the talk, I explain a toplogy which is weaker than Zariski topology and is essential to study of representation theory of Drinfeld double of Nichols algebras of diagonal-type.
