Poisson Homogeneous Spaces Research Articles

Dirac geometry and integration of Poisson homogeneous spaces

Dirac geometry and integration of Poisson homogeneous spaces

A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincaré

In this contribution we present a general procedure that allows the construction of noncommutative spaces with quantum group invariance as the quantization of their associated coisotropic Poisson homogeneous spaces coming from a coboundary Lie bialgebra structure. The approach is illustrated by obtaining in an explicit form several noncommutative spaces from (3+1)D (A)dS and Poincaré coisotropic Lie bialgebras. In particular, we review the construction of the \kappaκ-Minkowski and \kappaκ-(A)dS spacetimes in terms of the cosmological constant L. Furthermore, we present all noncommutative Minkowski and (A)dS spacetimes that preserve a quantum Lorentz subgroup. Finally, it is also shown that the same setting can be used to construct the three possible 6D \kappaκ-Poincaré spaces of time-like worldlines. Some open problems are also addressed.

Cluster algebra structures on Poisson nilpotent algebras

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.

Poisson orders on large quantum groups

We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras [9] belonging to a one-parameter family; we call these algebras large quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of [13]. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras Uq⊃Uq⩾⊃Uq+ are also described. Besides all (multiparameter) big quantum groups of De Concini–Kac–Procesi and big quantum super groups at roots of unity, our framework also contains the quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeiler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. The previous approaches to the above problems relied on reductions to rank two cases and direct calculations of Poisson brackets, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on perfect pairings between restricted and non-restricted integral forms.

All noncommutative spaces of κ-Poincaré geodesics

Noncommutative spaces of geodesics provide an alternative way of introducing noncommutative relativistic kinematics endowed with quantum group symmetry. In this paper we present explicitly the seven noncommutative spaces of time-, space- and light-like geodesics that can be constructed from the time-, space- and light-versions of the κ-Poincaré quantum symmetry in (3 + 1) dimensions. Remarkably enough, only for the light-like (or null-plane) κ-Poincaré deformation the three types of noncommutative spaces of geodesics can be constructed, while for the time-like and space-like deformations both the quantum time-like and space-like geodesics can be defined, but not the light-like one. This obstruction comes from the constraint imposed by the coisotropy condition for the corresponding deformation with respect to the isotropy subalgebra associated to the given space of geodesics, since all these quantum spaces are constructed as quantizations of the corresponding classical coisotropic Poisson homogeneous spaces. The known quantum space of geodesics on the light cone is given by a five-dimensional homogeneous quadratic algebra, and the six nocommutative spaces of time-like and space-like geodesics are explicitly obtained as six-dimensional nonlinear algebras. Five out of these six spaces are here presented for the first time, and Darboux generators for all of them are found, thus showing that the quantum deformation parameter κ −1 plays exactly the same algebraic role on quantum geodesics as the Planck constant ℏ plays in the usual phase space description of quantum mechanics.

The noncommutative space of light-like worldlines

The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) κ-deformation of the (3+1) Poincaré group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg–Weyl algebras whose noncommutative parameter is just the Planck scale parameter κ−1. Moreover, it is shown that the usual time-like κ-deformation of the Poincaré group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like κ-deformation.

Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

The complete classification of classical r-matrices generating quantum deformations of the (3 + 1)-dimensional (A)dS and Poincaré groups such that their Lorentz sector is a quantum subgroup is presented. It is found that there exists three classes of such r-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case are naturally expressed in terms of quantum null-plane coordinates, and they are always defined by homogeneous quadratic algebras. Finally, non-relativistic and ultra-relativistic limits giving rise to novel Newtonian and Carrollian noncommutative spacetimes are also presented.

Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups M = G/H equipped with an additional Poisson structure π which is compatible with a Poisson–Lie structure Π on G. Since the infinitesimal version of Π defines a unique Lie bialgebra structure δ on the Lie algebra , we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space M ⊥ = G*/H ⊥ of a given homogeneous space M with respect to a coisotropic Lie bialgebra. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to M ⊥ thus showing that an even richer duality framework between M and M ⊥ arises from them. In order to analyze physical implications of these notions, the case of M being a Minkowski or (anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding complementary dual reductive and symmetric spaces M ⊥ are explicitly constructed in the case of the well-known κ-deformation, where the cosmological constant Λ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that M ⊥ is a reductive space is shown to provide a natural condition for the representation theory of the quantum analogue of M that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally, despite these dual spaces M ⊥ are not endowed in general with a G*-invariant metric, we show that their geometry can be described by making use of K-structures.

Quantum Orbit Method in the Presence of Symmetries

We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

Integrability of quotients in Poisson and Dirac geometry

We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We also give explicit constructions of Lie groupoids integrating two interesting families of geometric structures: (i) a special class of Poisson homogeneous spaces of symplectic groupoids integrating Poisson groups and (ii) Dirac homogeneous spaces.

The κ-(A)dS noncommutative spacetime

The (3+1)-dimensional κ-(A)dS noncommutative spacetime is explicitly constructed by quantizing its semiclassical counterpart, which is the κ-(A)dS Poisson homogeneous space. This turns out to be the only possible generalization of the well-known κ-Minkowski spacetime to the case of non-vanishing cosmological constant, under the condition that the time translation generator of the corresponding quantum (A)dS algebra is primitive. Moreover, the κ-(A)dS noncommutative spacetime is shown to have a quadratic subalgebra of local spatial coordinates whose first-order brackets in terms of the cosmological constant parameter define a quantum sphere, while the commutators between time and space coordinates preserve the same structure of the κ-Minkowski spacetime. When expressed in ambient coordinates, the quantum κ-(A)dS spacetime is shown to be defined as a noncommutative pseudosphere.

Drinfel’d double structures for Poincaré and Euclidean groups

All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel’d double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel’d double structures of the Lie algebra is presented.

Noncommutative spaces of worldlines

The space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincaré group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincaré group, the quantization of this Poisson bracket gives rise to a noncommutative space of worldlines with quantum group invariance. As an oustanding example, the Poisson homogeneous space of worldlines coming from the κ-Poincaré deformation is explicitly constructed, and shown to define a symplectic structure on the space of worldlines. Therefore, the quantum space of κ-Poincaré worldlines is just the direct product of three Heisenberg-Weyl algebras in which the parameter κ−1 plays the very same role as the Planck constant ħ in quantum mechanics. In this way, noncommutative spaces of worldlines are shown to provide a new suitable and fully explicit arena for the description of quantum observers with quantum group symmetry.

Cayley--Klein Poisson Homogeneous Spaces

The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson-Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley-Klein geometries. The kinematical interpretation for the semi-Riemannian and pseudo-Riemannian Cayley-Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.

The Poincaré group as a Drinfel’d double

The eight nonisomorphic Drinfel’d double (DD) structures for the Poincaré Lie group in (2 + 1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1 + 1) Poincaré group are also identified and constructed, while in (3 + 1) dimensions no Poincaré DD structure does exist. Each of the DD structures here presented has an associated canonical quasitriangular Poincaré -matrix whose properties are analysed. Some of these -matrices give rise to coisotropic Poisson homogeneous spaces with respect to the Lorentz subgroup, and their associated Poisson Minkowski spacetimes are constructed. Two of these (2 + 1) noncommutative DD Minkowski spacetimes turn out to be quotients by a Lorentz Poisson subgroup: the first one corresponds to the double of with trivial Lie bialgebra structure, and the second one gives rise to a quadratic noncommutative Poisson Minkowski spacetime. With these results, the explicit construction of DD structures for all Lorentzian kinematical groups in (1 + 1) and (2 + 1) dimensions is completed, and the connection between (anti-)de Sitter and Poincaré -matrices through the vanishing cosmological constant limit is also analysed.

Generalized Kähler geometry and current algebras in classical N=2 superconformal WZW model

I examine the Generalized Kähler (GK) geometry of classical [Formula: see text] superconformal WZW model on a compact group and relate the right-moving and left-moving Kac–Moody superalgebra currents to the GK geometry data using biholomorphic gerbe formulation and Hamiltonian formalism. It is shown that the canonical Poisson homogeneous space structure induced by the GK geometry of the group manifold is crucial to provide [Formula: see text] superconformal [Formula: see text]-model with the Kac–Moody superalgebra symmetries. Then, the biholomorphic gerbe geometry is used to prove that Kac–Moody superalgebra currents are globally defined.

On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action

Let be the space of bilinear forms on with defining matrices endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action of the Poisson–Lie group on . A classification is given of all possible quadratic brackets on preserving the Poisson property of the action, thus endowing with the structure of a Poisson homogeneous space. Besides the product Poisson structure on , there are two other (mutually dual) structures, which (unlike the product Poisson structure) admit reductions by the Dirac procedure to a space of bilinear forms with block upper triangular defining matrices. Further generalisations of this construction are considered, to triples with the Poisson action , and it is shown that then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are investigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid. Bibliography: 30 titles.

AdS Poisson homogeneous spaces and Drinfel’d doubles

The correspondence between Poisson homogeneous spaces over a Poisson–Lie group G and Lagrangian Lie subalgebras of the classical double is revisited and explored in detail for the case in which is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group , namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for , while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on that arise from two Drinfel’d double structures on . The first one realises as a quotient of by the Poisson-subgroup , while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises as a coisotropic Poisson homogeneous space.

Homogeneous spaces of Dirac groupoids

A Poisson structure on a homogeneous space of a Poisson groupoid is homogeneous if the action of the Lie groupoid on the homogeneous space is compatible with the Poisson structures. According to a result of Liu, Weinstein and Xu, Poisson homogeneous spaces of a Poisson groupoid are in correspondence with suitable Dirac structures in the Courant algebroid defined by the Lie bialgebroid of the Poisson groupoid. We show that this correspondence result fits into a more natural context: the one of Dirac groupoids , which are objects generalizing Poisson groupoids and multiplicative closed 2-forms on groupoids.

Kazama-Suzuki models of N = 2 superconformal field theory and Manin triples

Kazama-Suzuki coset models is an interesting class of N=2 supersymmetric models of conformal field theory which are used to build realistic models of superstring in 4 dimensions. We formulate Kazama-Suzuki construction of N=2 superconformal coset models using more general language of Manin triples and represent the corresponding N=2 Virasoro superalgebra currents in explicit form. A correspondence between the Kazama-Suzuki models and Poisson homogeneous spaces is established also.