Dmitri Nikshych, Leonid Vainerman
We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.
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We study coactions of finite quantum groupoids on unital $C^*$-algebras and obtain the Tannaka-Krein reconstruction theorem for them.
In this work we give a generalization of matched pairs of (finite) groups to describe a general class of depth two inclusions of factor von Neumann algebras and the C*-quantum groupoids associated with, using double groupoids.
We propose a definition of compact quantum groupoids in the setting of C*-algebras, associate to such a quantum groupoid a regular C*-pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C*-bimodule and to pass to a measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and to étale groupoids and study principal compact C*-quantum groupoids.
We show how the C*-algebras of quantum complex projective spaces (standard or nonstandard) are related to groupoids.
We show that the C*-algebra of a quantum sphere $C(S_{q}^{2n+1})$ can be realized as a groupoid C*-algebra of a groupoid which is explicitly identified and is independent of the parameter $q$.
We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak $C^*$-Hopf algebra introduced by G. Bohm, F. Nill and K. Szlachanyi (with an involutive antipode), and a Kac bimodule -- an algebraic version of a Hopf bimodule, the notion introduced by J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.
We establish a Galois correspondence for finite quantum groupoid actions on II_1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such a subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain C^*-algebras related to it.
Let N_0 \subset N_1 a depth 2, finite index inclusion of type II1 factors and N_0 \subset N_1 \subset N_2 \subset N_3 ... the corresponding Jones tower. D. Nikshych et L. Vainerman built dual structures of quantum C*-groupoid on the relative commutants N'_0 \cap N_2 et N'_1 \cap N_3. Here I define a new duality which allows a symetric construction without changing the involution. So the Temperley-Lieb algebras are selfdual quantum C*-groupoids and the quantum C*-groupoids associated to a finite depth finite index inclusion can be choosen selfdual. I show that every finite-dimensional connexe q…
In this article, when G is a locally compact quantum group, we associate to a braided-commutative G-Yetter-Drinfel'd algebra $(N,a,\hat{a})$ equipped with a normal faithful semi-finite weight verifying some appropriate condition, a structure of a measured quantum groupoid. The dual structure is then given by $(N,\hat{a},a)$. Examples are given, especially the situation of a quotient type co-ideal of a compact quantum group. This construction generalizes the standard construction of a transformation groupoid. Most of the results were announced by the second author in 2011, at a conference in Wa…
Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang-Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II_1 factors. In this article, we associate to suitable dynamical quantum groups, which are a purely algebraic objects, Hopf C*-bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of C*-al…
A fundamental feature of quantum groups is that many come in pairs of mutually dual objects, like finite-dimensional Hopf algebras and their duals, or quantisations of function algebras and of universal enveloping algebras of Poisson-Lie groups. The same phenomenon was studied for quantum groupoids in various settings. In the purely algebraic setup, the construction of a dual object was given by Schauenburg and by Kadison and Szlachányi, but required the quantum groupoid to be finite with respect to the base. A sophisticated duality for measured quantum groupoids was developed by Enock, Lesieu…
Algebraic quantum transformation groupoids of compact type
本文引入了紧型代数量子变换广群及其离散型对偶，恢复了紧与离散Van Daele代数量子群间的对偶，并提供了源于Fell丛和量子商空间的实例。
We describe a finite quantum groupoid associated to any finite Vect_k enriched herd.
In this work we prove, in full details, that any finite dimensional $C^*$-quantum groupoid can be deformed in order that the square of the antipode is the identity on the base. We also prove that for any $C^*$-quantum groupoid with non abelian base, there is uncountably many $C^*$-quantum groupoids with the same underlying algebra structure but which are not isomorphic to it. In fact, the $C^*$-quantum groupoids are closed in an analog of the procedure presented by D.Nikshych ([N] 3.7) in a more general situation.
Construction of a $C^*$-algebraic quantum groupoid from a weak multiplier Hopf algebra
本文从弱乘子Hopf代数出发，通过引入准不变性条件，利用左积分构造Hilbert空间上的算子代数，证明可构建可分型C*-代数量子广群。
In this work we investigate the notion of action or coaction of a finite quantum groupoid in von Neumann algebras context. In particular we prove a double crossed product theorem and prove the existence of an universal von Neumann algebra on which any finite groupoids acts outerly. In previous works, N. Andruskiewitsch and S.Natale define for any match pair of groupoids two $C^*$-quantum groupoids in duality, we give here an interpretation of them in terms of crossed products of groupoids using a multiplicative partial isometry which gives a complete description of these structures. In a next …
Bialgebroids, separable bialgebroids, and weak Hopf algebras are compared from a categorical point of view. Then properties of weak Hopf algebras and their applications to finite index and finite depth inclusions of von Neumann algebras are shortly reviewed. A hint is given at a duality between bialgebroid actions and abstract inclusions in 2-categories.
In a former article, in collaboration with Jean-Michel Vallin, we have constructed two "quantum groupo\"ıds" dual to each other, from a depth 2 inclusion of von Neumann algebras $M_0\subset M_1$, in such a way that the canonical Jones'tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures. We are now investigating in greater details these structures in the presence of an appropriate modular theory on the basis $M'_0\cap M_1$, and we show how these examples fit with Lesieur's "measured quantum groupo\"ıds".
The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3-dimensional topological quantum field theory. In particular the representation …
We prove a duality theorem for quantum groupoid (weak Hopf algebra) actions that extends the well-known result for usual Hopf algebras.
We use categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.
Proposing a certain category of bialgebroid maps we show that the balanced depth 2 extensions appear as they were the finitary Galois extensions in the context of quantum groupoid actions, i.e., actions by finite bialgebroids, weak bialgebras or weak Hopf algebras. We comment on deformation of weak bialgebras, on half grouplike elements, on uniqueness of weak Hopf algebra reconstructions and discuss the example of separable field extensions.
Multiplicative partial isometries, manageability, and C*-algebraic quantum groupoids
本文提出乘积部分等距的概念，并通过可管理性条件构造C*-代数量子群体结构。
We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now well-known construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the notion of double groupoids, as introduced by Ehresmann.
In this article, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups before - faithfulness, uniqueness up to scaling, existence of a modular element and existence of a modular automorphism - for algebraic quantum groupoids under reasonable assumptions. The approach to integration developed in this article forms the basis for the extension of Pontrjagin duality to algebraic quantum groupoids, and for the passage from algebraic quantum…
We study Hamilton circuits of the Cayley graphs of the Weyl groupoids of the generalized quantum groups, or the quantum double of the Nichols algebras of diagonal-type, with finite root systems. We prove the existence of a Hamilton circuit for any of them and explicitly draw one of them for rank 3 and 4 cases.
We give a detailed comparison between the notion of a weak Hopf algebra (also called a quantum groupoid by Nikshych and Vainerman), and that of a $\times_R$-bialgebra due to Takeuchi (and also called a bialgebroid or quantum (semi)groupoid by Lu and Xu). A weak bialgebra is the same thing as a $\times_R$-bialgebra in which $R$ is Frobenius-separable. We extend the comparison to cover module and comodule theory, duality, and the question when a bialgebroid should be called a Hopf algebroid.
We continue our study of Cartan schemes and their Weyl groupoids. The results in this paper provide an algorithm to determine connected simply connected Cartan schemes of rank three, where the real roots form a finite irreducible root system. The algorithm terminates: Up to equivalence there are exactly 55 such Cartan schemes, and the number of corresponding real roots varies between 6 and 37. We identify those Weyl groupoids which appear in the classification of Nichols algebras of diagonal type.
Quantum groupoids are a joint generalization of groupoids and quantum groups. We propose a definition of a compact quantum groupoid that is based on the theory of C*-algebras and Hilbert bimodules. The essential point is that whenever one has a tensor product over the complex numbers in the theory of quantum groups, one now uses a certain tensor product over the base algebra of the quantum groupoid.
We extend the classification of finite Weyl groupoids of rank two. Then we generalize these Weyl groupoids to `reflection groupoids' by admitting non-integral entries of the Cartan matrices. This leads to the unexpected observation that the spectrum of the cluster algebra of type $A_{n-3}$ completely describes the set of finite reflection groupoids of rank two with $2n$ objects.
Like quantum groups, quantum groupoids frequently appear in pairs of mutually dual objects. We develop a general Pontrjagin duality theory for quantum groupoids in the algebraic setting that extends Van Daele's duality theory for multiplier Hopf algebras and overcomes the finiteness restrictions of the approach of Kadison, Szlachányi, Böhm and Schauenburg. Our construction is based on the integration theory for multiplier Hopf algebroids and yields, as a corollary, a duality theory for weak multiplier Hopf algebras with integrals. We compute the duals in several examples and introduce morphism…
We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum groupoids", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of "quantum groupoids". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of "quantum groupoids". On the other hand, we develop a suitable version of a "quantum du…
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitl…
Description of the smallest quantum groupoid associated with the A2 diagram. From the talk: ``Quantum groupoids and Ocneanu bialgebras for Coxeter-Dynkin systems'' given at the XV Colloquio Latinoamericano de Algebra, Cocoyoc, Mexico, July 20th-26th, 2003.
Hopf algebroids and quantum groupoids
我们引入了Hopf代数胚的概念，其中的全代数和基代数都不要求是交换的。当$R$ -矩阵适当作用时，给出了一类与Hopf代数的Drinfeld双模代数相关联的Hopf代数胚.当这种构造应用于量子群时，我们得到量子群胚的例子，它们是泊松群胚的半经典极限。详细计算了量子$sl(2)$的实例。
A new quantization of groupoids under the name of \times-Hopf coalgebras is introduced. We develop a Hopf cyclic theory with coefficients in stable-anti-Yetter-Drinfeld modules for \times-Hopf coalgebras. We use \times-Hopf coalgebras to study coextensions of coalgebras. Finally, equivariant \times-Hopf coalgebra Galois coextensions are defined and applied as functors between categories of stable anti-Yetter-Drinfeld modules over \times-Hopf coalgebras involved in the coextension.
Let $L$ be a quantum semigroupoid, more precisely a $\times_R$-bialgebra in the sense of Takeuchi. We describe a procedure replacing the algebra $R$ by any Morita equivalent, or in fact more generally any $\sqrt{\text{Morita}}$ equivalent (in the sense of Takeuchi) algebra $S$ to obtain a $\times_S$-bialgebra $\widetilde H$ with the same monoidal representation category.
We classify finite-dimensional irreducible highest weight modules of generalized quantum groups whose positive part is infinite dimensional and has a Kharchenko's PBW basis with an irreducible finite positive root system.
The purpose of this Note is to unify quantum groups and star-products under a general umbrella: quantum groupoids. It is shown that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e., the quantization problem, is posed. In particular, any regular triangular Lie bialgebroid is shown quantizable. For the Lie bialgebroid of a Poisson manifold, its quantization is equivalent to a star-product.
By using twist construction, we obtain a quantum groupoid $\cald\ot_{q}\uqg$ for any simple Lie algebra $\frakg$. The underlying Hopf algebroid structure encodes all the information of the corresponding elliptic quantum group-the quasi-Hopf algebras as obtained by Fronsdal, Arnaudon et. al. and Jimbo et. al..
Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang-Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a "solution" for short. Results of Etingof-Schedler-Soloviev, Lu-Yan-Zhu and Takeuchi on the set-theoretical quantum Yang-Baxter equation are generalized to the context of quivers, with groupoids playing the rôle of groups. The notion of "braided groupoid" is introduced. Braided groupoids are solutions and are characterized in terms of bije…
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitl…
We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra $D^{(1)}(2,1;x)$. Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck's analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid.
We establish a generalization of Kitaev models based on unitary quantum groupoids. In particular, when inputting a Kitaev-Kong quantum groupoid $H_\mathcal{C}$, we show that the ground state manifold of the generalized model is canonical isomorphic to that of the Levin-Wen model based on a unitary fusion category $\mathcal{C}$. Therefore the generalized Kitaev models provide realizations of the target space of the Turaev-Viro TQFT based on $\mathcal{C}$.
We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show that it coincides with the triangulation of a sphere cut out by a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic interpretation of many hyperplane arrangements that are not…
We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study t…
A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. We define the term "quantum category". The definition of antipode for a bialgebroid is less resolved in the literature. Our suggestion is that the kind of dualization occurring in Barr's star-autonomous categories is more suitable than autonomy (= compactness = rigidity). This leads to our definition of quantum groupoid intended as a "Hopf algebra with several objects".
Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains the list of all crystallographic arrangements, a large subclass of the class of simplicial hyperplane arrangements. Supposing that the rank is at least three, the classification yields Cartan schemes of type $A$ and $B$, an infinite family of series involving the types $C$ and $D$, and $74$ sporadic examples.
A q-analogue of Erdelyi's formula for the Hankel transform of the product of Laguerre polynomials is derived using the quantum linking groupoid between the quantum SU(2) and E(2) groups. The kernel of the q-Hankel transform is given by the 1\varphi1-q-Bessel function, and then the transform of a product of two Wall polynomials times a q-exponential is calculated as a product of two Wall polynomials times a q-exponential.
We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{iπ/\ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({\mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({\mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociati…
We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi) via vacant double groupoids as explained in \http://arxiv.org/abs/math.QA/0308228. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.
This is an introduction to double algebras which is the structure modelled by the properties of the convolution product in Hopf algebras, weak Hopf algebras and in Hopf algebroids. We show that Hopf algebroids with a Frobenius integral can be characterized as the double algebras in which the two multiplications satisfy distributivity. We discuss questions of duality, antipode, Maschke theorem and examples.
From smooth dynamical twists to twistors of quantum groupoids
该研究通过结合Gutt星积，将光滑动力学扭转化为量子共轭体的扭，为构造等价星积提供了新框架。
We relate quantum 6J symbols of various types (quantum versions of Wigner and Racah symbols) to Ocneanu cells associated with AN Dynkin diagrams. We check explicitly the algebraic structure of the associated quantum groupoids and analyze several examples (A3, A4). Some features relative to cells associated with more general ADE diagrams are also discussed.
Given a double cover $π: \mathcal{G} \rightarrow \hat{\mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $τ_π$ and $τ_π^{ref}$, thereby associating to a Jandl $n$-gerbe $\hatλ$ on $\hat{\mathcal{G}}$ a Jandl $(n-1)$-gerbe $τ_π(\hatλ)$ on the quotient loop groupoid of $\mathcal{G}$ and an ordinary $(n-1)$-gerbe $τ^{ref}_π(\hatλ)$ on the unoriented quotient loop groupoid of $\mathcal{G}$. For $n =1,2$, we interpret the character theory (resp. centre) of the category of Real $\hatλ$-twisted $n$-vector bundles over $\hat{\mathcal{G}}$ in terms of flat secti…
Hamiltonian Cycles for Finite Weyl Groupoids
该研究证明有限Weyl广群的Cayley图均存在哈密顿环，绘制了低秩不可约情形的哈密顿环，还给出秩3不可约情形下邻接矩阵的第二大特征值并判定其是否为二部Ramanujan图。
Every monoidal functor G: C --> M has a canonical factorization through the category of bimodules over some monoid R in M such that the factor U: C -->_R M_R is strongly unital. Using this result and the characterization of the forgetful functors M_A -->_R M_R of bialgebroids A over R given by Schauenburg together with their bimonad description given by the author recently here we characterize the "long" forgetful functors M_A -->_R M_R --> M of both bialgebroids and weak bialgebras.
Computing finite Weyl groupoids
该研究提出算法计算对角型Nichols代数与余伴随李超代数的广义根系，进而得到计算对应Kharchenko PBW基中Lyndon字及超字的算法，为获得有限根系对角型Nichols代数的极小表示提供关键数据。
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretat…
In this paper, we show that $\C{G}$-Frobenius algebras (for $\C{G}$ a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of $D(k[\C{G}])$, where $D(k[\C{G}])$ is the Drinfeld double of the quantum groupoid $k[\C{G}]$.
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