We develop a general theory of `quantum' diffeomorphism groups based on the universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its various quotients. Explicit formulae are introduced for this construction, as well as dual quasitriangular and braided R-matrix versions. Among the examples, we construct the $q$-diffeomorphisms of the quantum plane $yx=qxy$, and recover the quantum matrices $M_q(2)$ as those respecting its braided group addition law.
The cross coproduct braided group $Aut(C) \rcocross B$ is obtained by Tannaka-Krein reconstruction from $C^B\to C$ for a braided group $B$ in braided category $C$. We apply this construction to obtain partial solutions to two problems in braided group theory, namely the tensor problem and the braided double. We obtain $Aut(C)\rcocross Aut(C)\isom Aut(C)\lcross Aut(C)$ and higher braided group `spin chains'. The example of the braided group $B(R)\lcross B(R)\lcross...\lcross B(R)$ is described explicitly by R-matrix relations. We also obtain $Aut(C)\rcocross Aut(C)^*$ as a dual quasitriangular …
We classify braided generalized near-group fusion categories whose global dimension is not an integer; there are exactly two up to Grothendieck equivalence and taking products with braided pointed fusion categories.
We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups.
We show that the core of a weakly group-theoretical braided fusion category $\C$ is equivalent as a braided fusion category to a tensor product $\B \boxtimes \D$, where $\D$ is a pointed weakly anisotropic braided fusion category, and $\B \cong \vect$ or $\B$ is an Ising braided category. In particular, if $\C$ is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Fro…
We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theoretical or solvable in terms of the factorization of its Frobenius-Perron dimension and the Frobenius-Perron dimensions of its simple objects. As an application, we prove that every non-degenerate braid…
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generali…
Let $(H, R)$ be a finite dimensional quasitriangular Hopf algebra over a field $k$, and $_H\mathcal{M}$ the representation category of $H$. In this paper, we study the braided autoequivalences of the Drinfeld center $^H_H\mathcal{YD}$ trivializable on $_H\mathcal{M}$. We establish a group isomorphism between the group of those autoequivalences and the group of quantum commutative bi-Galois objects of the transmutation braided Hopf algebra $_RH$. We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group $\mathrm{BM}(k, H,R)$ in […
We prove that any fusion category over $\mathbb{C}$ with exactly one non-invertible simple object is spherical. Furthermore, we classify all such categories that come equipped with a braiding.
We describe Lagrangian algebras in twisted Drinfeld centres for finite groups. Using the full centre construction, we establish a 1-1 correspondence between Lagrangian algebras and module categories over pointed fusion categories.
We investigate braid group representations associated with unitary braided vector spaces, focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order. We verify this conjecture for the two infinite families of Gaussian and group-type braided vector spaces, as well as the generalization to quasi-braided vector spaces of group-type.
For an abelian group $ A $, we study a close connection between braided crossed $ A $-categories with a trivialization of the $ A $-action and $ A $-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a monoidal category $\mathcal{C}$ is given by an element $O(T)\in H^2(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T)=0$, the set of obstructions form a torsor over $\operatorname{Hom}(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$, where $\o…
We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.
This is a survey of general aspects of the theory of braided Hopf algebras with emphasis on a special class of braided graded Hopf algebras named tobas. The interest on tobas arises from problems of classification of pointed Hopf algebras. We discuss tobas from different points of view following ideas of Lusztig, Nichols and Schauenburg. We then concentrate on braided Hopf algebras in the Yetter-Drinfeld category over H, where H is the group algebra of a non abelian finite group. We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider.
Let k be a field. Let also (F, G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (σ, τ) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra k^G{}^τ#_σ kF combine into a braided Hopf algebra. We also discuss diagonal realizations of such braided Hopf algebras in the category of Yetter-Drinfeld modules over a finite group.
A generalized Hopf algebra structure for the positive (negative) part of the Drinfeld-Jimbo quantum group of type A_n is established without make any use of the usual deformation of the abelian part of sl_{n+1}.
The paper deals with braided Clifford algebras, understood as Chevalley-Kahler deformations of braided exterior algebras. It is shown that Clifford algebras based on involutive braids can be naturally endowed with a braided quantum group structure. Basic group entities are constructed explicitly.
We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $\cal B$ (consisting of invertible central $\cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categorie…
For a finite group $G$, Turaev introduced the notion of a braided $G$-crossed fusion category. The classification of braided $G$-crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a $G$-crossed Frobenius $\star$-algebra and give a classification of (strict) $G$-crossed extensions of a commutative Frobenius $\star$-algebra $R$ equipped with a given action of $G$, in terms of the second group cohomology $H^2(G,R^\times)$. Now suppose that $\mathcal{B}$ is a non-degene…
A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations of the 3-cocycles and quasi-bicharacters of finite abelian groups which are direct product of two cyclic groups.
We show that a weakly integral braided fusion category C such that every simple object of C has Frobenius-Perron dimension at most 2 is solvable. In addition, we prove that such a fusion category is group-theoretical in the extreme case where the universal grading group of C is trivial.
We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's cha…
Motivated by the recently introduced concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PS_n, as the quotient of the braid group B_n by the relations σ_iσ_{i+1}^{-1}σ_i=σ_{i+1}σ_i^{-1}σ_{i+1}, with 1\leq i\leq n-2. It turns out that PS_n is isomorphic to the quotient of B_n by the commutator subgroup [P_n, P_n] of the pure braid group P_n (which amounts to saying that [P_n, P_n] coincides with the normal subgroup of B_n generated by the elements [σ_i^2, σ_{i+1}^2], with 1\leq i\leq n-2), and that PS_n is a linear group.
This is the second part of the paper. Results of the first part about crossed modules are applied here to study of quantum groups in braided categories. Correct cross product in the class of quantum braided groups is built. Criterion when quantum braided group is cross product is otained.
We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring $k[x]$ of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of $S^1$ are also obtained by the same method.
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We study the bosonisation of any braided group provides as a trivial principal bundle in three ways.
Let $H$ be a Hopf algebra in braided category $\cal C$. Crossed modules over $H$ are objects with both module and comodule structures satisfying some comatibility condition. Category ${\cal C}^H_H$ of crossed modules is braided and is concrete realization of general categorical construction. For quantum braided group $(H,{\cal R})$ corresponding braided category ${\cal C}^{\cal R}_H$ of modules is identifyed with full subcategory in ${\cal C}_H^H$. Connection with crossproducts is discussed. Correct cross product in the class of quantum braided groups is built. Radford's--Majid's theorem gives…
A key invariant of a braided categorical group is its quadratic form, introduced by Joyal and Street. We show that the categorical group is braided equivalent to a simultaneously skeletal and strictly associative one if and only if the polarization of this quadratic form is the symmetrization of a bilinear form. This generalizes the result of Johnson-Osorno that all Picard groupoids can simultaneously be strictified and skeletalized, except that in the braided case there is a genuine obstruction.
Let $(H,R)$ be a quasitriangular weak Hopf algebra over a field $k$. We show that there is a braided monoidal equivalence between the Yetter-Drinfeld module category $^H_H\mathscr{YD}$ over $H$ and the category of comodules over some braided Hopf algebra ${}_RH$ in the category $_H\mathscr{M}$. Based on this equivalence, we prove that every braided bi-Galois object $A$ over the braided Hopf algebra ${}_RH$ defines a braided autoequivalence of the category $^H_H\mathscr{YD}$ if and only if $A$ is quantum commutative. In case $H$ is semisimple over an algebraically closed field, i.e. the fusion …
Double-bosonisation associates to a braided group in the category of modules of a quantum group, a new quantum group. We announce the semiclassical version of this inductive construction.
A braided generalization of the concept of Hopf algebra (quantum group) is presented. The generalization overcomes an inherent geometrical inhomogeneity of quantum groups, in the sense of allowing completely pointless objects. All braid-type equations appear as a consequence of initial axioms. Braided counterparts of basic algebraic relations between fundamental entities of the standard theory are found.
We prove a coherence theorem for actions of groups on monoidal categories. As an application we prove coherence for arbitrary braided $G$-crossed categories.
Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct $\undΔL=L\und\tens L$ where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category…
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their associativity and commutativity morphisms.
Quantum Lorentz groups H admitting quantum Minkowski space V are selected. Natural structure of a quantum space G = V x H is introduced, defining a quantum group structure on G only for triangular H (q=1). We show that it defines a braided quantum group structure on G for |q|=1.
We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type $A,$ we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided difference operators. The relations satisfied by the mutually commuting elements called Dunkl elements in the deformed Fomin-Kirillov algebra are determined. The Dunkl elements correspond to the truncated…
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we study the braided group $G=G(X,r)$ of an involutive square-free solution $(X,r)$ of finite order $n$ and cyclic index $p=p(X,r)$ and the group algebra $\textbf{k} [G]$ over a field $\textbf{k}$. We show that $G$ contains a $G$-invariant normal subgroup $\mathcal{F}_p$ of finite index $p^n$, $\mathcal{F}_p$ is isomorphic to th…
We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the s…
With the motivation of giving a more precise estimation of the quantum Brauer group of a Hopf algebra $H$ over a field $k$ we construct an exact sequence containing the quantum Brauer group of a Hopf algebra in a certain braided monoidal category. Let $B$ be a Hopf algebra in $\C=_H ^H\YD$, the category of Yetter-Drinfel'd modules over $H$. We consider the quantum Brauer group $\BQ(\C; B)$ of $B$ in $\C$, which is isomorphic to the usual quantum Brauer group $\BQ(k; B\rtimes H)$ of the Radford biproduct Hopf algebra $B\rtimes H$. We show that under certain symmetricity condition on the braidin…
We analyze the structure of the Witt group W of braided fusion categories introduced in the previous paper arXiv:1009.2117v2. We define a "super" version of the categorical Witt group, namely, the group sW of slightly degenerate braided fusion categories. We prove that sW is a direct sum of the classical part, an elementary Abelian 2-group, and a free Abelian group. Furthermore, we show that the kernel of the canonical homomorphism S: W --> sW is generated by Ising categories and is isomorphic to Z/16Z. Finally, we give a complete description of etale algebras in tensor products of braided fus…
We define the braided differential algebras which can be interpreted as quantization of the differential operator algebra defined on some algebraic varieties supplied with the action of the group GL(m). The algebra is generated by right invariant or coajoint vector fields. Our main example is gl^*(m) and coadjoint orbits in it. The Heisenberg double on the quantum group Fun_q(GL(m)is a particular case of the suggested construction.
The well known incompatibility between inhomogeneous quantum groups and the standard q-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantum ISO(p,N-p) containing SO_q(p,N-p) with |q|=1 are constructed for N=2p, 2p+1, 2p+2. Their Poisson analogues (obtained first) are presented as an introduction to the quantum case.
The connection between braided Hopf algebra structure and the quantum group covariance of deformed oscillators is constructed explicitly. In this context we provide deformations of the Hopf algebra of functions on SU(1,1). Quantum subgroups and their representations are also discussed.
$*$-structures on quantum and braided spaces of the type defined via an R-matrix are studied. These include $q$-Minkowski and $q$-Euclidean spaces as additive braided groups. The duality between the $*$-braided groups of vectors and covectors is proved and some first applications to braided geometry are made.
A differential calculus of the first order over multi-braided quantum groups is developed. In analogy with the standard theory, left/right-covariant and bicovariant differential structures are introduced and investigated. Furthermore, antipodally covariant calculi are studied. The concept of the *-structure on a multi-braided quantum group is formulated, and in particular the structure of left-covariant *-covariant calculi is analyzed. A special attention is given to differential calculi covariant with respect to the action of the associated braid system. In particular it is shown that the lef…
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.
For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided autoequivalences of the Drinfeld center of the braided tensor category. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.
We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different automorphism groups arise as automorphism groups of two generated free braided associative algebras. This list covers all three automorphism groups that arise in the case of quantum planes \cite{AC}.
We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson Lie groups are nilpotent radicals of parabolic subgroups. We also provide examples of moment maps in this new context coming from moduli spaces of flat connections on surfaces.
We give a characterization of Drinfeld centers of fusion categories as non-degenerate braided fusion categories containing a Lagrangian algebra. Further we study the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers and show that its formal properties are similar to those of the classical Witt group.
We consider quantum group theory on the Hilbert space level. We find all unitary representations of three braided quantum groups related to the quantum ``ax+b'' group. First we introduce an auxiliary braided quantum group, which is apparently not related to the quantum ``ax+b'' group, but easy to work with. We find all unitary representations of this quantum group. Then we use this result to find all unitary representations of another braided quantum group, whose C*-algebra is generated by two (out of three) generators generating (in the sense of Woronowicz) C*-algebra of the quantum ``ax+b''-…
Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group $(A,\overline A,{\cal R})$ the corresponding braided category of modules ${\cal C}_{\cO{A,\overline A}}$ is identified with a full subcategory in $\DY{\cal C}_A^A$. The connection with cross products is …
This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied,the notion of `braided -commutative' or `braided-cocommutative' Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group aut(C) is given. The theory has important implications for the theory of quasitrian…
We clarify the relation between the approach to $q$-Minkowski space of Carow-Watamura et al. with an approach based on the idea of $2\times 2$ braided Hermitean matrices. The latter are objects like super-matrices but with Bose-Fermi statistics replaced by braid statistics. We also obtain new R-matrix formulae for the $q$-Poincare group in this framework.
Let G be a classical compact Lie group and G_μthe associated compact matrix quantum group deformed by a positive parameter μ(or a nonzero and real μin the type A case). It is well known that the category Rep(G_μ) of unitary f.d. representations of G_μis a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_μ) to another braided tensor C*-category with irreducible tensor unit is full if |μ|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows…
Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$, while for the braided monoidal structures we will compute the explicit form of the abelian 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided…
We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual pair of canonical forms. The role of Levi-Civita connection is naturally generalised to connections with vanishing torsion and cotorsion, which we introduce. We then provide the corresponding quantum group and braided group formulations with the universal quantum differential calculus. We also give general constructions for examples, including quantum spher…
