Quantum doubles in monoidal categories

A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional (minimal) triangular Hopf algebras (A,R) introduced by Drinfeld. Only recently Etingof and the author completely solved this problem for semisimple A over algebraically closed fields of characteristics 0 and p>>dim(A) (any p if one assumes that A is also cosemisimple). In this paper we take the first step towards solving this problem for finite-dimensional pointed Hopf algebras over an algebraically closed field k of characteristic 0. We first prove that the fourth power of the antipode of any triangular pointed Hopf algebra A is the identity. We do that by focusing on minimal triangular pointed Hopf algebras (A,R) (every triangular Hopf algebra contains a minimal triangular sub Hopf algebra) and proving that the group algebra of the group of grouplike elements of A (which must be abelian) admits a minimal triangular structure and consequently that A has the structure of a biproduct. We also generalize our result on the order of the antipode to any finite-dimensional quasitriangular Hopf algebra A whose Drinfeld element u acts as a scalar in any irreducible representation of A (e.g. when A^* is pointed). Second, we describe a method of construction of finite-dimensional pointed Hopf algebras which admit a minimal triangular structure, and classify all their minimal triangular structures. We conclude the paper by proving that any minimal triangular Hopf algebra which is generated as an algebra by grouplike elements and skew primitive elements is isomorphic to a minimal triangular Hopf algebra constructed using our method.

Recall that a triangular Hopf algebra A is said to have the Chevalley property if the tensor product of any two simple A-modules is semisimple, or, equivalently, if the radical of A is a Hopf ideal. There are two reasons to study this class of triangular Hopf algebras: First, it contains all known examples of finite-dimensional triangular Hopf algebras; second, it can be, in a sense, completely understood. Namely, it was shown in our previous work with Andruskiewitsch that any finite-dimensional triangular Hopf algebra with the Chevalley property is obtained by twisting of a finite-dimensional triangular Hopf algebra with R-matrix of rank at most 2 which, in turn, is obtained by modifying the group algebra of a finite supergroup. This provides a classification of such Hopf algebras. The goal of this paper is to make this classification more effective and explicit, i.e. to parameterize isomorphism classes of finite-dimensional triangular Hopf algebras with the Chevalley property by group-theoretical objects, similarly to how it was done in our previous work in the semisimple case. This is achieved in Theorem 2.2, where these classes are put in bijection with certain septuples of data. In the semisimple case, the septuples reduce to the quadruples, and we recover the result of our previous paper. In the minimal triangular pointed case, we recover a classification theorem from a previous paper of the second author.

We consider quasitriangular Hopf algebras in braided tensor categories introduced by Majid. It is known that a quasitriangular Hopf algebra H in a braided monoidal category C induces a braiding in a full monoidal subcategory of the category of H-modules in C. Within this subcategory, a braided version of the bosonization theorem with respect to the category C will be proved. An example of braided monoidal categories with quasitriangular structure deviating from the ordinary case of symmetric tensor categories of vector spaces is provided by certain braided supersymmetric tensor categories. Braided inhomogeneous quantum groups like the dilaton free q-Poincare group are explicit applications.

Double Quantum Groups

We study versions of the categories of Yetter-Drinfel'd mod- ules over a Hopf algebra H in a braided monoidal category C. Contrary- wise to Bespalov's approach, all our structures live in C. This forces H to be transparent or equivalently to lie in Muger's center Z2(C) of C. We prove that versions of the categories of Yetter-Drinfel'd modules in C are braided monoidally isomorphic to the categories of (left/right) modules over the Drinfel'd double D(H) > C for H nite. We obtain that these categories polarize into two disjoint groups of mutually iso- morphic braided monoidal categories. We conclude that if H > Z2(C), then D(H)C embeds as a subcategory into the braided center category Z1(HC) of the category HC of left H-modules in C. For C braided, rigid and cocomplete and a quasitriangular Hopf algebra H such that H > Z2(C) we prove that the whole center category of HC is monoidally isomorphic to the category of left modules over Aut(HC) # H - the bosonization of the braided Hopf algebra Aut(HC) which is the coend in HC. A family of examples of a transparent Hopf algebras is discussed.

A smash coproduct in braided monoidal category C is constructed and some conditions making the smash coproduct a Hopf algebra or braided Hopf algebra are given. It is shown that the smash coproductB ×H inHM is equivalent to the transmutation of Hopf algebra. Thus a method for transmutation theory is provided. Let σ be 2-co-cycle andH a commutation Hopf algebra. A Hopf algebraHσ is constructed.Hσ≅Hσ whereHσ is a transmutation ofHσ. The braided groups from some solutions of quantum Yang-Baxter equation are obtained.

Nichols algebras, Hopf algebras in braided categories with distinguished properties, were discovered several times. They appeared for the first time in the thesis of W. Nichols [72], aimed to construct new examples of Hopf algebras. In this same paper, the small quantum group \(u_q(sl_3)\), with q a primitive cubic root of one, was introduced. Independently they arose in the paper [84] by Woronowicz as the invariant part of his non-commutative differential calculus. Later there were two unrelated attempts to characterize abstractly the positive part \(U^+_q(\mathfrak g)\) of the quantized enveloping algebra of a simple finite-dimensional Lie algebra \(\mathfrak g\) at a generic parameter q. First, Lusztig showed in [64] that \(U^+_q(\mathfrak g)\) can be defined through the radical of a suitable invariant bilinear form. Second, Rosso interpreted \(U^+_q(\mathfrak g)\) in [74, 75] via quantum shuffles. These two viewpoints were conciliated later, as alternative definitions of the same notion of Nichols algebra. Other early appearances of Nichols algebras are in [65, 77]. As observed in [17, 18], Nichols algebras are basic invariants of pointed Hopf algebras, their study being crucial in the classification program of Hopf algebras; see also [10]. More recently, they are the subject of an intriguing proposal in Conformal Field Theory [79]. This is an introduction from scratch to the notion of Nichols algebra. I was invited to give a mini-course of two lessons, 90 min each, at the Geometric, Algebraic and Topological Methods for Quantum Field Theory, Villa de Leyva, Colombia, in July 2015. The theme was Nichols algebras that requires several preliminaries and some experience to be appreciated; a selection of the ideas to be presented was necessary. These notes intend to preserve the spirit of the course, discussing some motivational background material in Sect. 4.1, then dealing with braided vector spaces and braided tensor categories in Sect. 4.2, arriving at last to the definition and main calculation tools of Nichols algebras in Sect. 4.3. I hope that the various examples and exercises scattered through the text would serve the reader to absorb the beautiful concept of Nichols algebra and its many facets. Section 4.4 is a survey of the main examples of, and results on, Nichols algebras that I am aware of; here the pace is faster and the precise formulation of some statements is referred to the literature. I apologize in advance for any possible omission. This section has intersection with, and is an update of, the surveys [1, 2, 19], to which I refer for further information.

Hopf (bi-)modules and crossed modules in braided monoidal categories

Weak C-cleft extensions, weak entwining structures and weak Hopf algebras

We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided monoidal category of Yetter–Drinfeld modules over a Hopf algebra. This gives rise to the notion of ribbon Yetter–Drinfeld modules over a Hopf algebra, which form ribbon categories. This gives an invariant of tangles.

The Yang-Baxter equation plays a fundamental role in various areas of\nmathematics. Its solutions, called braidings, are built, among others, from\nYetter-Drinfel'd modules over a Hopf algebra, from self-distributive\nstructures, and from crossed modules of groups. In the present paper these\nthree sources of solutions are unified inside the framework of Yetter-Drinfe' d\nmodules over a braided system. A systematic construction of braiding structures\non such modules is provided. Some general categorical methods of obtaining such\ngeneralized Yetter-Drinfel'd (=GYD) modules are described. Among the braidings\nrecovered using these constructions are the Woronowicz and the Hennings\nbraidings on a Hopf algebra. We also introduce the notions of crossed modules\nof shelves / Leibniz algebras, and interpret them as GYD modules. This yields\nnew sources of braidings. We discuss whether these braidings stem from a\nbraided monoidal category, and discover several non-strict pre-tensor\ncategories with interesting associators.\n

We apply quantum group methods to quantum computing, starting with the notion of interacting Frobenius Hopf algebras for ZX calculus with noncommutative algebra and noncocommutative coalgebra. We introduce the notion of *-structures in ZX calculus at this algebraic level and construct examples based on the quantum group u q (sl 2) at a root of unity. We provide an abstract formulation of the Hadamard gate related to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX calculus to take place in a braided tensor category. In the ribbon case, the Hadamard gate coming from braided self-duality obeys a modular identity. We give the example of b q (sl 2), the self-dual braided version of u q (sl 2).

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad. In Part 4 we define general monoidal categories equipped with an action operad equivariant structure, and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras. Many illustrations and examples are included. Assuming only basic category theory, this monograph is intended for graduate students and researchers. In addition to being a coherent reference for the topics covered, this book is also suitable for a graduate student seminar and a reading course.

Let A and B be multiplier Hopf algebras, and let R ∈ M(B ⊗ A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ⊗ R B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ⊗ R B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group.

The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.