Quantization of lie Bialgebras via the Formality of the operad of Little Disks

We give a proof of the Etingof–Kazhdan theorem on quantization of Lie bialgebras based on the formality of the chain operad of little disks and show that the Grothendieck–Teichmüller group acts non-trivially on the corresponding quantization functors. Partially supported by an NSF Grant and A. Sloan Research Fellowship Received: November 2004 Revision: April 2006 Accepted: May 2006     

Loop spaces, and coherence for monoidal and braided monoidal bicategories

We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpreted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.     

Categorification and Group Extensions

We review several known categorification procedures, and introduce a functorial categorification of group extensions (Section 4.1) with applications to non-Abelian group cohomology (Section 4.2). The obstruction to the existence of group extensions (Section 4.2.4, Equation (9)) is interpreted as a coboundary condition (Proposition 4.5).     

Iterated monoidal categories

We develop a notion of an n-fold monoidal category and show that it corresponds in a precise way to the notion of an n-fold loop space. Specifically, the group completion of the nerve of such a category is an n-fold loop space, and free n-fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher H-space structure of an n-fold loop space. We also show directly that this operad has the same homotopy type as the n-th Smith filtration of the Barratt-Eccles operad and the n-th filtration of Berger's complete graph operad. Moreover, this operad contains an equivalent preoperad which gives rise to Milgram's small model for when n=2 and is very closely related to Milgram's model of for n>2.     

On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there isa notion of a centralizer C_C K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z₂C C_CC (not tobe confused with the center Z₁ of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the X_i being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenC_CC_CK = K and dim K·dim C_CK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=C_CK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z₂K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z₂K. (3) For every finite group G there is a braided tensor*-category C such that Z₂CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10.     

A(∞)TH上的辫子Monoidal范畴

Let A and H be Hopf algebra,T-smash product A (∞)T H generalizes twisted smash product A*H.This paper shows a necessary and sufficient condition for T-smash product module category A(∞)T H M to be braided monoidal category.     

Smash Products for Secondary Homotopy Groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. The second author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract.     

余拟三角弱Hopf群代数与辫子Monoidal范畴

In this paper, we first give the definitions of a crossed left π-H-comodules over a crossed weak Hopf π-algebra H, and show that the category of crossed left π-H-comodules is a monoidal category. Finally, we show that a family σ = {σα,β: Hα Hβ→ k}α,β∈πof k-linear maps is a coquasitriangular structure of a crossed weak Hopf π-algebra H if and only if the category of crossed left π-H-comodules over H is a braided monoidal category with braiding defined by σ.     

Fusion Operators and Cocycloids in Monoidal Categories

The Yang–Baxter equation has been studied extensively in the context of monoidal categories. The fusion equation, which appears to be the Yang–Baxter equation with a term missing, has been studied mainly in the context of Hilbert spaces. This paper endeavours to place the fusion equation in an appropriate categorical setting. Tricocycloids are defined; they are new mathematical structures closely related to Hopf algebras.     

On categories associated to a Quasi-Hopf algebra

A quasi-Hopf algebra H can be seen as a commutative algebra A in the center 𝒵(H-Mod) of H-Mod. We show that the category of A-modules in 𝒵(H-Mod) is equivalent (as a monoidal category) to H-Mod. This can be regarded as a generalization of the structure theorem of Hopf bimodules of a Hopf algebra to the quasi-Hopf setting.