Advances in Geometry

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.

Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of Lie groups and algebras quantum groups, algebras and their duality non-commutative spaces and quantum groups invariant differential calculi elements of quantum groups representations theory tensor products of representations q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator different forms of commutation relations representations q real and roots of unity cases algebraic maps from non-deformed to deformed oscillators path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: relativistic space-time symmetries Poincare group as a classical deformation of Galilei group its representations multiparametric q-deformation of linear groups, twisted groups and algebras q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields unified models of Higgs fields in the frame of non-commutative geometry and others.

We give an introduction to the basic notions of noncommutative geometry including the calculus of infinitesimals with operators, cyclic cohomology and the local index formula. We also explain in details how the infinitesimal calculus based on operators gives a natural home for the infinitesimal line element ds of geometry and leads one to the basic notion of spectral triple, which is the basic paradigm of noncommutative geometry. In order to illustrate these general concepts we then analyse the noncommutative space underlying the quantum group SU q (2) from this spectral point of view, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic cocycle giving the index formula. This specific example allows to illustrate the general notion of locality in noncommutative geometry. The formulas computing the residue are ”local”. Locality by stripping all the expressions from irrelevant details makes them easily computable. The original Chern character is non-local and the cochain whose coboundary is the difference between the original Chern character and the local one is much harder to compute than the local cochains. It is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. The key feature of this spectral triple is its equivariance, i.e. the SU q (2)-symmetry. We explain how this leads naturally to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries and relate this notion to previous work.KeywordsDirac OperatorQuantum GroupPseudodifferential OperatorNoncommutative GeometryChern CharacterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The aim of this contribution is to highlight the relevance of the so-called light-like (or null-plane) κ-Poincaré quantum deformation in relation with the construction of noncommutative spaces in the context of quantum gravity phenomenology. The role of this deformation is enhanced when, in addition to the construction of quantum group covariant noncommutative spacetimes, noncommutative spaces of geodesics preserving the same quantum group symmetry are also considered.

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.

Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract. One success here comes from “universal deformation formulas” (UDFs), expressions built from a Lie algebra which deform any algebra on which the Lie algebra operates as derivations. The most famous of these is the Moyal product, a special case of a class in which the Lie algebra is Abelian. Another comes from recognition that the Belavin–Drinfel’d solutions to the MCYBE are, in fact, infinitesimal deformations for which, in the case of the special linear groups, it is possible to give explicit formulas for the corresponding quantum Yang–Baxter equations. This review paper discusses, necessarily in brief, these and related topics, including “twisting” as a form of UDF and finding formulas for “preferred deformations” of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.

In this paper we study the products of cosymplectic manifolds and a unit circle, which have natural symplectic structures. We have some relations on moduli spaces, Gromov–Witten invariants, and quantum cohomologies of cosymplectic manifolds and the products. As an example we examine the cosymplectic manifold \(M=S^2\times T\times S^1\) and the product \(M\times S^1\), and we also calculate their moduli spaces, Gromov–Witten invariants, and quantum cohomologies.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We intend to produce a theory which generalises topological spaces; we call it non-commutative topology. In non-commutative differential geometry the basic homology theory is the periodic cyclic homology, based on the bi-complex (b, B). In non-commutative topology this structure will be replaced by the bi-complex \((\tilde {b}, d)\); the boundary \(\tilde {b}\) is called topological Hochschild boundary. These ideas combine Connes’ work (Noncommutative geometry, Academic Press, 1994) with ideas of the articles by Teleman and Teleman (St Cerc Math Tom 18(5):753–762, Bucarest, 1966; Rev Roumaine Math Pures Applic 12:725–731, 1967; Central European journal of mathematics. C*-algebras and elliptic theory II. Trends in mathematics, 251–265, Birkhauser, 2008; Local3 index theorem. arXiv: 1109.6095v1, math.KT, 28 Sep. 2011; textitLocal Hochschild homology of Hilbert-Schmidt operators on simplicial spaces. arXiv hal-00707040, Version 1, 11 June 2012; Local algebraic K-theory, 26 Aug 2013 - arXiv.org > math > arXiv:1307.7014, 2013; The local index theorem, HAL-00825083, arXiv 1305.5329, 22 May 2013; \(K_ {i}^{loc}(\ mathbb {C}) \), i = 0, 1 10 set 2013 - arXiv:1309.2421v1 [math.KT] 10 Sep 2013).

The Heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum-mechanical particle. This can be rephrased as follows: the smallest subsets of classical phase space in which the presence of a quantum-mechanical particle can be detected are its Lagrangian submanifolds. For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true "points"; see Weinstein [17]. Now let G be a compact Lie group and G x X ~ X a Hamiltonian action of G on X (see w for definitions). It is well-known that the fixed points of this action form a symplectic submanifold of X. (See for instance Guillemin and Sternberg [5].) However, what can one say about the fixed "points" of G? We will show that they are also the "points'" of a symplectic manifold, Xc. This manifold is the Marsden-Weinstein reduction of X with respect to the zero orbit in g*, and will be described in Sect. 2. (It was introduced in a completely different context fl'om ours by Marsden and Weinstein [12].) Problems in classical mechanics can often be reduced to the study of Hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on Hilbert space. This fact has inspired a number of efforts to "quantize" symplectic geometry by devising schemes for associating Hilbert space to symplectic manifolds. The "no-go" theorems of Groenwald and Van Hove impose some embarrassing limitations on all such schemes; however, it seems to be a useful idea heuristically to think of every symplectic manifold, X~a~ic, 1, as being symbiotically associated with a Hilbert space, Xq,a, t ..... in such a way that the classical observables on the first space correspond to quantum observables on the second space. The heuristics further suggests that if G is a group of symmetries of X~la~,i~, j, it should also be a group of symmetries of Xq,a,,tum. In this heuristic spirit, we will state the main conjecture of this paper:

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

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit $$\hbar \rightarrow 0$$ , and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models.

We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147–182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371–383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248].Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang–Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan–Robbins polytope.

Various types of Lagrange and Finsler geometries, Einstein gravity, and modifications, can be modelled by nonholonomic distributions on tangent bundles/manifolds when the fundamental geometric objects are adapted to nonlinear connection structures. We can convert such geometries and physical theories into almost Kähler/Poisson structures on (co)tangent bundles. This allows us to apply deformation quantization formalism to almost symplectic connections induced by Lagrange–Finsler and/or Einstein fundamental geometric objects. There are constructed respective nonholonomic versions of the trace density maps for the zeroth Hochschild homology of deformation quantization of distinguished algebras (in this work, adapted to nonlinear connection structure). Our main result consists in an algebraic index theorem for Lagrange–Finsler and Einstein spaces. Finally, we show how the Einstein field equations for gravity theories and geometric mechanics models can be imbedded into the formalism of deformation quantization and index theorem.

In deformation quantization, one can associate five characteristic functions to (stable) formality morphisms on cochains and chains and to “two-brane” formality morphisms. We show that these characteristic functions agree.