Braided bosonization and inhomogeneous quantum groups

Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map τis a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H)=H ⋈ r H *cop in C. In this paper, we first show that D(H) is a quasitriangular Hopf algebra in C. Next, let H be an ordinary triangular finite-dimensional Hopf algebra. Then one can form quasitriangular Hopf algebras B(H,H) and B(H,D(H)) (in a rigid braided monoidal category) by Majid’s method associated to the ordinary Hopf algebra maps H →H and iH H → D(H), where D(H) is the Drin-fePd quantum double. We show that D (B(H,H)) and B(H,D(H)) are isomorphic Hopf algebras in the braided monoidal category.

We introduce the notion of $(G, \Gamma)$-crossed action on a tensor category, where $(G, \Gamma)$ is a matched pair of finite groups. A tensor category is called a $(G, \Gamma)$-crossed tensor category if it is endowed with a $(G, \Gamma)$-crossed action. We show that every $(G,\Gamma)$-crossed tensor category $\mathcal{C}$ gives rise to a tensor category $\mathcal{C}^{(G, \Gamma)}$ that fits into an exact sequence of tensor categories $\operatorname{Rep} G \longrightarrow \mathcal{C}^{(G, \Gamma)} \longrightarrow \mathcal{C}$. We also define the notion of a $(G, \Gamma)$-braiding in a $(G, \Gamma)$-crossed tensor category, which is connected with certain set-theoretical solutions of the QYBE. This extends the notion of $G$-crossed braided tensor category due to Turaev. We show that if $\mathcal{C}$ is a $(G, \Gamma)$-crossed tensor category equipped with a $(G, \Gamma)$-braiding, then the tensor category $\mathcal{C}^{(G, \Gamma)}$ is a braided tensor category in a canonical way.

We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess–Zumino–Novikov–Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet -algebras as an example of the applications of our general construction of the braided tensor category structure.

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.

We define the important concept of a braided tensor category due to Joyal and Street [JS93]. This concept has been introduced to formalize the char-acteristic properties of the tensor categories of modules over braided bial-gebras as well as the idea of crossing in link and tangle diagrams. After defining braided tensor categories, we show that braids form a braided ten-sor category that is universal in some precise sense. We also give the “centre construction” which is the categorical version of Drinfeld’s quantum dou-ble.KeywordsBraid GroupTensor CategoryTensor FunctorChapter XIIIBraid CategoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Drinfeld center of enriched monoidal categories

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavior of these functors in the case when the base category is braided is also considered.

We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën–Vaquié–Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

Coherence theorems via knot theory

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 H be a coalgebra. In this paper, we show that H is a dual quasi-bialgebra if and only if the category [Formula: see text] of comodules is a tensor category; and H is a braided dual quasi-bialgebra if and only if [Formula: see text] is a braided tensor category. If H is a braided dual quasi-Hopf algebra, it is shown that the antipode of H is inner, i.e., s2(h) = ∑ τ (h1)h2τ-1(h3).

Coquasitriangular universal ${\cal R}$ matrices on quantum Lorentz and quantum Poincar\'e groups are classified. The results extend (under certain assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on those objects are described.

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.

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.