Hopf algebras in noncommutative geometry

Journal of Noncommutative Geometry, 2014

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C * -algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C * -algebras M n (R), M n (C) and M n (H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C⊕H⊕M 3 (C) and A ev = H ⊕ H ⊕ M 4 (C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics minimally coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp(n) (quaternionic unitary group).

Arxiv preprint math/0307277, 2003

After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologies on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities and provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are described.

2008

This dissertation is based on research done at the Mathematical Physics sector of the International School for Advanced Studies of Trieste, during the period from October 2003 to June 2007. It is divided into two parts: the first part (Chapters 1-2) is an account of the general theory and a collection of some general notions and results; the second part (Chapters 3-5) contains the original work, carried out under the supervision of Prof. Ludwik D abrowski and Prof. Giovanni Landi. Part of the original material presented here has been published or submitted as a preprint in the following papers:

Journal of Algebra, 1990

The initial part of this paper presents "Physics for Algebraists" in the context of quantum mechanics combined with gravity. Such physical notions as the Yang-Baxter Equations, position observables, momentum space, momentum and position quantization, etc., are described. Many readers may wish to just read this initial part of the paper. The physics leads to the search for self-dual algebraic structures and finally to non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. The entire paper contains numerous examples. The non-commutative and non-cocommutative Hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted H,w Hz. Among the examples is one obtained by modifying the Weyl algebra. We also give the context in which the compatibility requirements on the structure maps reduce to the Classical Yang-Baxter Equations, and an example related to Drinfel'd's double Hopf algebra D(H). e 1990 Academic Press. Inc.

Communications in Mathematical Physics, 1998

We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

Journal of Mathematical Physics, 2000

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of spacetime, in particular, amounts to a postulated new force or physical effect called cogravity.

This article continues the study of concrete algebra-like structures in our polyadic approach , when the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions. In this way, the associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but " quantized " ; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.

Advances in Mathematics, 2006

Letters in Mathematical Physics, 2006

Communications in Mathematical Physics, 2001