Related papers: Background independent geometry and Hopf cyclic co…
This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and…
We argue that the demand of background independence in a quantum theory of gravity calls for an extension of standard geometric quantum mechanics. We discuss a possible kinematical and dynamical generalization of the latter by way of a…
We equip the graded polynomial algebra generated by nonplanar rooted binary trees with a Hopf algebra structure by defining a coproduct which disallows cutting both children of any given vertex, refining Connes-Kreimer's notion of…
We introduce Hopf images of coactions of Hopf algebras and develop their role in the geometry of quantum principal bundles. Assuming cosemisimplicity of the structure Hopf algebra, we show that every quantum principal bundle equipped with a…
The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the…
We describe a bigraded cocommutative Hopf algebra structure on the weight zero compactly supported rational cohomology of the moduli space of principally polarized abelian varieties. By relating the primitives for the coproduct to graph…
A recent result of ours [GM] shows that all Hopf algebra liftings of a given diagram in the sense of Andruskiewitsch and Schneider are cocycle deformations of each other. Here we develop a "non-abelian" cohomology theory, which gives a…
This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…
We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum…
In this article, we focus on a very special class of foliations with complex leaves whose diffeomorphism type is fixed. They have a unique compact leaf and the noncompact leaves all accumulate onto it. We show that the complex structure…
We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple $(A,\mathcal{H},M)$ consisting of a Hopf algebra $\mathcal{H}$, an $\mathcal{H}$-comodule algebra $A$, an $\mathcal{H}$-module $M$, and a…
We introduce a general class of combinatorial objects, which we call \emph{multi-complexes}, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of…
We find a self-dual noncommutative and noncocommutative Hopf algebra acting as a universal symmetry on the modules over inner Frobenius algebras of modular categories (as used in two dimensional boundary conformal field theory) similar to…
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general…
For our own education, we reconstruct the Hopf algebra of Connes and Moscovici obtained by the action of vector fields on a crossed product of functions by diffeomorphisms. We extend the realization of that Hopf algebra in terms of rooted…
We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes-Moscovici Hopf algebra $\mathcal{H}_n$. More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of $\mathcal{H}_n$, and we…
We investigate several Hopf algebras of diagrams related to Quantum Field Theory of Partitions and whose product comes from the Hopf algebras WSym or WQSym respectively built on integer set partitions and set compositions. Bases of these…
REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a…
We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure…
We pursue the study of local index theory for operators of Fourier-integral type associated to non-proper and non-isometric actions of Lie groupoids, initiated in a previous work. We introduce the notion of geometric cocycles for Lie…