Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

We systematically study noncommutative and nonassociative algebras $A$ and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of $A$-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras $A$ the full subcategory of symmetric $A$-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.     

Synthetic Braided Geometry. I

In a previous paper we dealt with supergeometryfrom a synthetic standpoint, showing that the totalityof vector fields on a superized version of microlinearspace is a Lie superalgebra. The main purpose of this paper is to generalize the methods tosymmetric braided geometry. Nonsymmetric braidedgeometry will be discussed in a sequel to thispaper.     

Synthetic Vector Analysis

Classical vector analysis is rife with geometric and physical ideas, but appears precarious from a modern viewpoint of pure mathematics. Modern vector analysis with differential forms is surely up to the contemporary standard of mathematical rigor, but geometric ideas are completely lost in the bulk of dull calculations. The main goal in this paper is to show that synthetic differential geometry, which has replenished differential geometry with nilpotent infinitesimals, can mathematically sanitize classical vector analysis by eradicating its total confusion between approximate calculations and infinitesimal calculations, thereby helping it retrieve mathematical rigor.     

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.     

Braided Identities, Quantum Groups, and Clifford Algebras

A Lie algebra in a braided category is constructed within the algebra structure of the positive part of the Drinfeld—Jimbo quantum group of type An such that its universal enveloping algebra is a braided Hopf algebra. Similarities with Clifford algebras are discussed.     

Infinitesimal Calculus of Variations

It is desirable that physical laws should beformulated infinitesimally, while it is well known thatthe calculus of variations, which has long beenconcerned with local or global horizons, gives aunifying viewpoint of various arenas of modern physics.The principal objective of this paper is toinfinitesimalize the calculus of variations by makinguse of the vanguard of modern differential geometry,namely, synthetic differential geometry, in whichnilpotent infinitesimals of various orders areabundantly and coherently available. Our treatment iscompletely coordinate-free, the decomposition of a stateinto its position and velocity components beingreplaced by the vertical-horizontal decompositionassociated with an appropriate connection. Within ournewly established infinitesimal calculus of variations, generalized conservation laws of momentum andenergy are demonstrated.     

Quantum poincaré group for isotropic quantum space-time

Possibilities of isotropic deformation of space-time are studied. The result is the two-parameter deformation. A differential calculus on the quantum space-time is constructed and the quantum differential geometry is formulated. A group of rigid motion of quantum space-time is investigated. This group is an example of a quantized braided group.     

Quantum and braided group Riemannian geometry

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 example, including quantum spheres and quantum planes.     

Braided Covariance of the Braided Differential Bialgebras Under Quantized Braided Groups

The braided differential bialgebras on braided matrix algebras (with bothmultiplicative and additive coproducts) and on quantum hyperplanes (withadditive coproduct) are proven to be covariant under the braided coactions ofthe quantized braided groups, which contain the usual quantum group-covarianceas a special case. This means that the above braided differential bialgebras havemore and richer symmetries. It is also shown that the braided matrix algebraitself and the related braided differential algebra constitute two braided rings withthe two above-mentioned coproducts.     

A metric approach to Fréchet geometry

The aim of this article is to present the category of bounded Fréchet manifolds in respect to which we will review the geometry of Fréchet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some examples from Riemannian geometry are given.     

Differential calculi on noncommutative bundles

We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a connection with respect to a differential calculus and consider questions of existence and uniqueness. At the end these constructions are applied to basic examples of noncommutative bundles over a coquasitriangular Hopf algebra.     

A Spin Decomposition of the Verlinde Formulas for Type A Modular Categories

A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU(N). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context.     

Theory of microcubes

Kock and Lavendhomme have begun to couch the standard theory of iterated tangents within the due framework of synthetic differential geometry. Generalizing their theory of microsquares, we give a general theory of microcubes, its three-dimensional generalization, in which an unexpected generalization of the Jacobi identity of vector fields with respect to Lie brackets and a synthetic treatment of Bianchi's first identity are discussed.     

Synthetic Hamiltonian mechanics

This paper deals with some infinitesimal aspects of Hamiltonian mechanics from the standpoint of synthetic differential geometry. Fundamental results concerning Hamiltonian vector fields, Poisson brackets, and momentum mappings are discussed. The significance of the Lie derivative in the synthetic context is also consistently stressed. In particular, the notion of an infinitesimally Euclidean space is introduced, and the Jacobi identity of vector fields with respect to Lie brackets is established naturally for microlinear, infinitesimally Euclidean spaces by using Lie derivatives instead of a highly combinatorial device such as P. Hall's 42-letter identity.     

A Rigidity Result for Extensions of Braided Tensor <Emphasis Type="Italic">C</Emphasis>*–Categories Derived from Compact Matrix Quantum Groups

Let G be a classical compact Lie group and G _μ the associated compact matrix quantum group deformed by a positive parameter μ (or \({\mu\in{\mathbb R}\setminus\{0\}}\) in the type A case). It is well known that the category of unitary representations of G _μ is a braided tensor C*–category. We show that any braided tensor *–functor \({\rho: \text{Rep}(G_\mu)\to\mathcal{M}}\) to another braided tensor C*–category with irreducible tensor unit is full if |μ| ≠ 1. In particular, the functor of restriction RepG _μ → Rep(K) to a proper compact quantum subgroup K cannot be made into a braided functor. Our result also shows that the Temperley–Lieb category \({\mathcal{T}_{\pm d}}\) for d > 2 can not be embedded properly into a larger category with the same objects as a braided tensor C*–subcategory.     

Non-trivially associated tensor categories and left coset representatives

In this talk I will motivate and describe a generalisation of the group doublecross product construction, involving a set of coset representatives for the left action of a subgroup on a group. From this data a non-trivially associated tensor category can be made. I shall briefly mention the corresponding double construction, which gives a non-trivially associated braided tensor category, containing a braided Hopf algebra. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

From Quantum Quasi-Shuffle Algebras to Braided Rota–Baxter Algebras

In this letter, we use quantum quasi-shuffle algebras to construct Rota–Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota–Baxter algebras, the relevant object of Rota–Baxter algebras in a braided tensor category. Examples of such new algebras are provided using quantum multi-brace algebras in a category of Yetter–Drinfeld modules.     

Some comments on bosonisation and biproducts

We collect here some less well-known results and formulae about the bosonisation construction which turns braided groups into quantum groups. We clarify the relation with biproduct Hopf algebras (the constructions are not the same), the response to twisting of braided groups and the abstract characterisation via automorphisms of the forgetful functor for the category of (co)modules of a braided group.     

Braided Coadditive Differential Complexes on Quantized Braided Groups

Braided coadditions of differential complexes on some further generalized quantized braided matrix algebras are constructed. With these coadditions the generalized algebraic systems form a kind of braided (additive) differential Hopf algebras. This is a generalization and unification of some existing results. The coadditions of differential complexes on the usual braided matrices and quantum matrices, etc., can be obtained as special cases.