Introduction to Quantum Groups

The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich's formalism of `non-commutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.

In many quantum gravity approaches, the cosmological constant is introduced by deforming the gauge group into a quantum group. In three dimensions, the quantization of the Chern-Simons formulation of gravity provided the first example of such a deformation. The Turaev-Viro model, which is an example of a spin-foam model, is also defined in terms of a quantum group. By extension, it is believed that in four dimensions, a quantum group structure could encode the presence of $\mathrm{\ensuremath{\Lambda}}\ensuremath{\ne}0$. In this article, we introduce by hand the quantum group ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$ into the loop quantum gravity (LQG) framework; that is, we deal with ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$. We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the $\mathrm{U}(N)$ formalism in this deformed case, which is given by the quantum group ${\mathcal{U}}_{q}(\mathfrak{u}(N))$. We are then able to build geometrical observables, such as the length, area or angle operators, etc. We show that these operators characterize a quantum discrete hyperbolic geometry in the three-dimensional LQG case. Our results confirm that a quantum group structure in LQG can be a tool to introduce a nonzero cosmological constant into the theory. Our construction is both relevant for three-dimensional Euclidian quantum gravity with a negative cosmological constant and four-dimensional Lorentzian quantum gravity with a positive cosmological constant.

We study the noncommutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric multipole moment, in the presence of an external magnetic field. First, by introducing a shift for the magnetic field we give the Schrodinger equations in the presence of an external magnetic field both on a noncommutative space and a noncommutative phase space, respectively. Then by solving the Schrodinger equations, we obtain quantum phases of the electric multipole moment both on a noncommutative space and a noncommutative phase space. We demonstrate that these phase are geometric and dispersive.

Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory

This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms. The famous Erlanger Programm of Klein says that is about to study the transfor- mation groups of various spaces, or more precisely the properties invariant under the actions of such groups, i.e., the symmetries. Noncommutative geometry(NCG), which originated from Connes' study in operator alge- bras in 1970's, brought the landscape of many new objects and some astonishing phenomena. Back to the early 1990s, Connes and Moscovici pointed out that in noncommutative geome- try(NCG) while noncommutative spaces are represented by the algebras (usually noncommuta- tive C ∗ -algebras) of continuous functions over noncommutative spaces, the local symmetries are reflected in some Hopf algebras. One of the first noncommutative spaces studied in NCG is the C ∗ -algebra of a foliated space. In the case of codimension n foliations, Connes and Moscovici discovered a Hopf algebra Hn, which governs the local symmetry of leaf spaces of foliations of codimension n. The Hopf algebra Hn is universal in the sense that it depends only on the codimension of a foliation. This family of Hopf algebras {Hn} is very useful in the study of index theory, and later was found to have connections with various different areas of mathematics, c.f. (10), (13). In this paper, we review the application of the Hopf algebra H1 in Rankin-Cohen deformations, which was initiated by Connes and Moscovici (14). We start by recalling the general setting of geometry. Let M be a smooth manifold and F be a foliation on M of codimension n. Let X be a complete flat transversal of F, and F + X be the oriented frame bundle of X. The holonomy pseudogroup acts on X and therefore F + X by transforming X parallelly along paths in leaves of F. The transverse geometry is to study the transversal X along with the action by the holonomy pseudogroup . In what follows we focus on the case when n = 1, and define Connes-Moscovici's Hopf algebra H1. Now the complete transversal X is a flat 1-dim manifold; and the oriented frame bundle F + X is diffeomorphic to X × R + , and is a discrete holonomy pseudogroup acting on X as local diffeomorphisms. We introduce coordinates x on X and y on R + . The lifted action of on F + X is

International audience

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.

The conference Conceptual and Technical Challenges for Quantum Gravity at Sapienza University of Rome, from 8 to 12 September 2014, has provided a beautiful opportunity for an encounterbetween different approaches and different perspectives on the quantum-gravity problem. It contributedto a higher level of shared knowledge among the quantum-gravity communities pursuing each specific research program.There were plenary talks on many different approaches, including in particular string theory, loop quantum gravity, spacetime noncommutativity, causal dynamical triangulations, asymptotic safety and causal sets. Contributions from the perspective of philosophy of science were also welcomed.In addition several parallel sessions were organized. The present volume collects contributions from the Noncommutative Geometry and Quantum Gravity parallel session4, with additional invited contributions from specialists in the field.Noncommutative geometry in its many incarnations appears at the crossroad of many researches in theoretical and mathematical physics:• from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and string theory,• from early considerations on UV-divergencies in quantum field theory to recent models of gauge theories on noncommutative spacetime,• from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions.This volume provides an overview of these various topics, interesting for the specialist as well as accessibleto the newcomer. 4partially funded by CNRS PEPS /PTI ‘‘Metric aspect of noncommutative geometry: from Monge to Higgs’’

This collection of invited mathematical papers by a list of distinguished mathematicians is an outgrowth of the scientific activities at the Center for Geometry and Mathematical Physics of Penn State University. The articles in this text present results or discuss perspectives on work that should be of interest to researchers and graduate students working in symplectic geometry and geometric quantization, deformation quantization, non-commutative geometry and index theory, quantum groups, holomorphic algebraic geometry and moduli spaces, quantum cohomology, algebraic groups and invariant theory, and characteristic classes.

Let [Formula: see text] be a groupoid and assume that [Formula: see text] is a family of algebras with identity. First, we introduce the notion of a weak Hopf (groupoid)[Formula: see text]-coalgebra by that if, for each pair [Formula: see text], there is given a unital homomorphism [Formula: see text] satisfying certain properties, generalizing the notion of Hopf group-coalgebras as introduced by Turaev from groups to groupoids and Hopf algebra structures to weak Hopf algebra structures. Then one considers now the direct sum [Formula: see text] of these algebras. It is an algebra, without identity, except when [Formula: see text] is a finite groupoid, but the product is non-degenerate. The maps [Formula: see text] can be used to define a coproduct [Formula: see text] on [Formula: see text] and the conditions imposed on these maps give that [Formula: see text] is a weak multiplier Hopf algebra. It is [Formula: see text]-cograded as explained in this paper. We study these so-called groupoid-cograded weak multiplier Hopf algebras. They are, as explained above, more general than the weak Hopf group-coalgebras (introduced by Van Daele and Wang), generalizing the Turaev’s Hopf group-coalgebras. Moreover, our point of view makes it possible to use results and techniques from the theory of weak multiplier Hopf algebras in the study of weak Hopf groupoid-coalgebras (and generalizations).

1.8 Non-Commutative Geometry and Quantum Groups

Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.