In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.
An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommut…
We introduce the filtered *-bialgebra which is a noncommutative analog of the *-bialgebra of multivariate polynomials with the canonical coproduct and counit. We study the associated filtered convolutions, random walks and random variables. The GNS representations of the limit states lead to filtered fundamental operators which are the CCR fundamental operators on the multiple symmetric Fock space, multiplied by appropriate projections. The importance of filtered random variables and fundamental operators stems from the fact that by addition and strong limits one obtains from them the main typ…
We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev-Reshetikhin-Takhtajan RTT prescription with non-triangular R-matrices which are denoted $R_{03}$ and $R_{14}$ in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1/1). We give the spectral decomposition which involves two, resp., three, projectors. These projectors are then used to provide the Baxterisation procedure with one, resp., two, parameters. Further, the projectors are used to construct the noncommutative p…
Let $G$ be a Poisson Lie group and $\g$ its Lie bialgebra. Suppose that $\g$ is a group Lie bialgebra. This means that there is an action of a discrete group $Γ$ on $G$ deforming the Poisson structure into coboundary equivalent ones. Starting from this we construct a non-trivial stack of Hopf-Poisson algebras and prove the existence of associated deformation quantizations. This non-trivial stack is a stack of functions on the formal Poisson group, dual of the starting $Γ$ Poisson-Lie group. To quantize this non-trivial stack we use quantization of a $Γ$ Lie bialgebra which is the infinitesimal…
All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universa…
All bialgebra structures for centrally extended Galilei algebra are classified. The corresponding Lie-Poisson structures on centrally extended Galilei group are found.
The notions of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras are introduced. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. The notion of Manin triple of transposed Hom-Poisson algebras is introduced, and its equivalence to the transposed Hom-Poisson bialgebras is investigated. The notion of $\mathcal{O}$-operator is exploited to illustrate the relations existing between transposed Hom-Poisson and Hom-pre-Lie Poisson algebras.
We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermut…
In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G* are complex Lie groups. We also prove that a Hermitian structure on the Lie algebra $\mathfrak{g}$ with ad-invariant metric induces a structure of the same type on the double Lie algebra ${\mathcal D}\mathfrak{g}= \mathfrak{g}\oplus\mathfrak{g}^*$, with respect to the canonical ad-invariant metric of neutral signature on ${\m…
We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras naturally compatible with a reductive decomposition, we extend the description of the moduli space of classical dynamical r-matrices of Etingof and Schiffmann. We construct, in each gauge orbit, an explicit analytic representative l. We translate the notion of duality for dynamical Poisson groupoids into a duality for Lie quasi-bialgebras. It is shown that duality maps the dynamical Poisson groupoid for l and a Lie quasi-bialgebra to the dynamical Poisson groupoid…
We quantize the Poisson-Lie group SL(2,R)^* as a bialgebra using the product of Kontsevich. The coproduct is a deformation of the coproduct that comes from the group structure. The resulting bialgebra structure is isomorphic to the quantum universal enveloping algebra U_hsl(2,R).
We present a quasi-Lie bialgebra (QLBA) quantization problem which comes from an algebraic reformulation of the Nambu-Goto string theory and invariant charges by Pohlmeyer and Rehren. This QLBA structure depends on a symmetric bivector (coming from a Minkowski metric) and is built on the free Lie algebra on a finite dimensional vector space. We solve this problem when the bivector has rank 1 or 2.
We introduce the analogue of Manin's universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras $P$ and $U$, where $U$ is finite dimensional, we construct a Poisson algebra $\mathcal{B}(P,\, U)$ together with a Poisson algebra homomorphism $ψ_{\mathcal{B}(P,\,U)} \colon P \to U \otimes \mathcal{B}(P,\, U)$ satisfying a suitable universal property. $\mathcal{B}(P,\, U)$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $P=U$ is a finite dimensional Poisson alg…
Motivated by the universal obstruction to the deformation quantization of Poisson structures in infinite dimensions we introduce the notion of quantizable odd Lie bialgebra. The main result of the paper is a construction of a highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so called quantizable Poisson structures.
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized as a ``cocycle perturbation'' of the linear Poisson bracket. We construct a certain twisted group C*-algebra A, which is shown to be a strict deformation quantization of G. Motivated by the data at the Poisson (classical) level, we then construct on A its locally compact quantum group structures: comultiplication, counit, antipode and Haar weight, as well as …
Let G be a connected, simply connected Poisson-Lie group with quasitriangular Lie bialgebra g. An explicit description of the double D(g) is given, together with the embeddings of g and g^*. This description is then used to provide a construction of the double D(G). The aim of this work is to describe D(G) in sufficient detail to be able to apply the procedures of Semenov-Tian-Shansky and Drinfeld for the classification of symplectic leaves and Poisson homogeneous spaces for Poisson-Lie groups.
We introduce the notion of Gamma-Lie bialgebra, where Gamma is a group. These objects give rise to cocommutative co-Poisson algebras, for which we construct quantization functors. This enlarges the class of co-Poisson algebras for which a quantization is known. Our result relies on our earlier work, where we showed that twists of Lie bialgebras can be quantized; we complement this work by studying the behavior of this quantization under compositions of twists.
We give relations between dynamical Poisson groupoids, classical dynamical Yang--Baxter equations and Lie quasi-bialgebras. We show that there is a correspondance between the class of bidynamical Lie quasi-bialgebras and the class of bidynamical Poisson groupoids. We give an explicit, analytical and canonical equivariant solution of the classical dynamical Yang--Baxter equation (classical dynamical $\ell$-matrices) when there exists a reductive decomposition $\g=ł\oplus\m$, and show that any other equivariant solution is formally gauge equivalent to the canonical one. We also describe the dual…
We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the "Moyal product" which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the …
In this paper we prove formality of the exterior algebra on V+V* endowed with the big bracket considered as a graded Poisson algebra. We also discuss connection of this result to bialgebra deformations of the symmetric algebra of V considered as bialgebra.
We introduce a new 2-parameter family of sigma models exhibiting Poisson-Lie T-duality on a quasitriangular Poisson-Lie group $G$. The models contain previously known models as well as a new 1-parameter line of models having the novel feature that the Lagrangian takes the simple form $L=E(u^{-1}u_+,u^{-1}u_-)$ where the generalised metric $E$ is constant (not dependent on the field $u$ as in previous models). We characterise these models in terms of a global conserved $G$-invariance. The models on $G=SU_2$ and its dual $G^\star$ are computed explicitly. The general theory of Poisson-Lie T-dual…
We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumve…
In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and describe some constructions of them. Then we study the formal distribution (noncommutative) Poisson algebras and coefficient (noncommutative) Poisson algebras. Next, we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.
We complete the study of the Poisson-Sigma model over Poisson-Lie groups. Firstly, we solve the models with targets $G$ and $G^*$ (the dual group of the Poisson-Lie group $G$) corresponding to a triangular $r$-matrix and show that the model over $G^*$ is always equivalent to BF-theory. Then, given an arbitrary $r$-matrix, we address the problem of finding D-branes preserving the duality between the models. We identify a broad class of dual branes which are subgroups of $G$ and $G^*$, but not necessarily Poisson-Lie subgroups. In particular, they are not coisotropic submanifolds in the general …
Given a Lie algebra, there uniquely exists a Poisson algebra which is called a Lie-Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday algebra. The noncommutative Poisson algebras are called the Loday-Poisson algebras. In the super/graded cases, the Loday-Poisson bracket is regarded as a noncommutative version of classical (linear) Schouten-Nijenhuis bracket. It will be shown that the Loday-Poisson algebras form a special subclass of Aguiar's dual-prePoisson algebras. We also study a proble…
We introduce and study infinitesimal BiHom-bialgebras, BiHom-Novikov algebras, BiHom-Novikov-Poisson algebras, and find some relations among these concepts. Our main result is to show how to obtain a left BiHom-pre-Lie algebra from an infinitesimal BiHom-bialgebra
We develop a theory of quasi-Lie bialgebroids using a homological approach. This notion is a generalization of quasi-Lie bialgebras, as well as twisted Poisson structures with a 3-form background which have recently appeared in the context of string theory, and were studied by Ševera and Weinstein using a different method.
Let $\g$ be a finite dimensional complex Lie algebra and $ł\subset \g$ a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group $\Exp ł$ with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra $\Dł$. Let $\Nc_ł(0)\subset ł$ be an open domain parameterizing a neighborhood of the identity in $\Exp ł$ by the exponential map. We present dynamical $r$-matrices with values in $\g\wedge \g$ over the Poisson Lie base manifold $\Nc_ł(0)$.
In this paper, first we introduce the notion of quadratic Rota-Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight, and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Lie algebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota-Baxter Lie algebras pl…
In this paper we propose a procedure for a noncommutative derived Poisson reduction, in the spirit of the Kontsevich-Rosenberg principle: "a noncommutative structure of some kind on $A$ should give an analogous commutative structure on all schemes ${\mathrm{Rep}}_n(A)$". We use double Poisson structures as noncommutative Poisson structures and noncommutative Hamiltonian spaces -- as first introduced by M. Van den Bergh -- to define (derived) zero loci of Hamiltonian actions and a noncommutative Chevalley-Eilenberg and BRST constructions, showing how we recover the corresponding commutative con…
It is shown by Connes, Douglas and Schwarz that gauge theory on noncommutative torus describes compactifications of M-theory to tori with constant background three-form field. This indicates that noncommutative gauge theories on more general manifolds also can be useful in string theory. We discuss a framework to noncommutative quantum gauge theory on Poisson manifolds by using the deformation quantization. The Kontsevich formula for the star product was given originally in terms of the perturbation expansion and it leads to a non-renormalizable quantum field theory. We discuss the nonperturba…
We introduce a new type of noncommutative Poisson structure on associative algebras. It induces Poisson structures on the moduli spaces classifying semisimple modules. Path algebras of doubled quivers and preprojective algebras have noncommutative Poisson structures given by the necklace Lie algebra.
We introduce a noncommutative Poisson random measure on a von Neumann algebra. This is a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Poissonization is a functor from the category of von Neumann algebras with normal semifinite faithful weights to the category of von Neumann algebras with normal faithful states. Poissonization is a natural adaptation of the second quantization to the context of von Neumann algebras. The construction is compatible with normal (weight-preserving) homomorphisms and unital normal completely positiv…
Recantly, William Crawley-Boevey proposed the definition of a Poisson structure on a noncommutative algebra $A$ based on the Kontsevich principle. His idea was to find the {\it weakest} possible structure on $A$ that induces standard (commutative) Poisson structures on all representation spaces $ \Rep_V(A) $. It turns out that such a weak Poisson structure on $A$ is a Lie algebra bracket on the 0-th cyclic homology $ \HC_0(A) $ satisfying some extra conditions; it was thus called in an {\it $ H_0$-Poisson structure}. This paper studies a higher homological extension of this construction. In ou…
In this paper we introduce a kind of "noncommutative neighbourhood" of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a "distortion" of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue o…
All coboundary Lie bialgebras and their corresponding Poisson--Lie structures are constructed for the oscillator algebra generated by $\{å,\ap,\am,\bb\}$. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal $R$--matrices.
A.A. Kirillov introduced the family algebras in 2000. In this paper we study the noncommutative Poisson bracket P on the classical family algebra. We show that P is the first-order deformation from the classical family algebra to the quantum family algebra. We will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary therefore the deformation is infinitesimally trivial. In the last part of this paper we also talk about Mackey's analogue and the quantization problem of the family algebras.
It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on $\mathbb{P}^1$ by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver $Γ_n$ on two vertices and $n$ equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modeled on the quiver $Γ_n$. We p…
A thorough analysis of Lie super-bialgebra structures on Lie super-algebras osp(1|2) and super-e(2) is presented. Combined technique of computer algebraic computations and a subsequent identification of equivalent structures is applied. In all the cases Poisson-Lie brackets on supergroups are found. Possibility of quantizing them in order to obtain quantum groups is discussed. It turns out to be straightforward for all but one structures for super-E(2) group.
We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It is illustrated with the specializations of the algebras of quantum matrices at roots of unity and more generally all quantum Schubert cell algebras.
All factorizable Lie bialgebra structures on complex reductive Lie algebras were described by Belavin and Drinfeld. We classify the symplectic leaves of the full class of corresponding connected Poisson-Lie groups. A formula for their dimensions is also proved.
Let G be the group of all formal power series starting with x with coefficients in a field k of zero characteristic (with the composition product), and let F[G] be its function algebra. C. Brouder and A. Frabetti introduced a non-commutative, non-cocommutative graded Hopf algebra H, via a direct process of ``disabelianisation'' of F[G], i.e. taking the like presentation of the latter as an algebra but dropping the commutativity constraint. In this paper we apply a general method to provide four one-parameters deformations of H, which are quantum groups whose semiclassical limits are Poisson ge…
