We study polynomial Poisson algebras with some regularity conditions. Linear (Lie-Berezin-Kirillov) structures on dual spaces of semi-simple Lie algebras, quadratic Sklyanin elliptic algebras of \cite{FO1},\cite{FO2} as well as polynomial algebras recently described by Bondal-Dubrovin-Ugaglia (\cite{Bondal},\cite{Ug}) belong to this class. We establish some simple determinantal relations between the brackets and Casimirs in this algebras. These relations imply in particular that for Sklyanin elliptic algebras the sum of Casimir degrees coincides with the dimension of the algebra. We are discussing some interesting examples of these algebras and in particular we show that some of them arise naturally in Hamiltonian integrable systems. Among these examples is a new class of two-body integrable systems admitting an elliptic dependence both on coordinates and momenta.
The Feynman path integral is used to quantize the symplectic leaves of the Poisson-Lie group SU(2)*. In this way we obtain the unitary representations of U_q(su(2)). This is achieved by finding explicit Darboux coordinates and then using a phase space path integral. I discuss the *-structure of SU(2)* and give a detailed description of its leaves using various parametrizations and also compare the results with the path integral quantization of spin.
We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.
A class of Poisson algebras considered as a Poisson version of the multiparameter quantized coordinate rings of symplectic and Euclidean $2n$-spaces is constructed and the prime Poisson ideals and the symplectic ideals of these Poisson algebras are described. As a result, it is shown that the multiparameter quantized symplectic and Euclidean $2n$-spaces are topological quotients of their classical spaces.
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.
We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Poisson manifold. We study various families of examples. In particular, a global symplectic groupoid for a general class of two-dimensional Poisson domains is constructed.
It is known that any Poisson manifold can be embedded into a bigger space which admites a description in terms of the canonical Poisson structure, i.e., Darboux coordinates. Such a procedure is known as a symplectic realization and has a number of important applications like quantization of the original Poisson manifold. In the present paper we extend the above idea to the case of quasi-Poisson structures which should not necessarily satisfy the Jacobi identity. For any given quasi-Poisson structure $Θ$ we provide a recursive procedure of the construction of a symplectic manifold, as well as t…
We show that each triangular Poisson Lie group can be decomposed into Poisson submanifolds each of which is a quotient of a symplectic manifold. The Marsden-Weinstein-Meyer symplectic reduction technique is then used to give a complete description of the symplectic foliation of all triangular Poisson structures on Lie groups. The results are illustrated in detail for the generalized Jordanian Poisson structures on SL(n).
This paper investigates the Poisson geometry of cluster algebras and the corresponding ideal theory of quantum cluster algebras. We then show how our approach can be applied to the ring theory of quantized coordinate rings. We give a new construction for the Dixmier map constructed by Yakimov from the space of symplectic leaves on $\CC[G]$ to the space of primitive ideals on $\CC_q[G]$ and give further evidence that this map is a homeomorphism.
This manuscript is essentially a collection of lecture notes which were given by the first author at the Summer School Wisl-2019, Poland and written down by the second author. As the title suggests, the material covered here includes the Poisson and symplectic structures (Poisson manifolds, Poisson bi-vectors and Poisson brackets), group actions and orbits (infinitesimal action, stabilizers and adjoint representations), moment maps, Poisson and Hamiltonian actions. Finally, the phase space reduction is also discussed. The very last section introduces the Poisson-Lie structures along with some …
We introduce a new kind of groupoid--a pseudo étale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the idea that symplectic and Poisson geometries are the semiclassical limits of the corresponding quantum geometries, we quantize these noncommutative Poisson manifolds in the framework of deformation quantization.
A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on $\mathbb{C}^4 \times \mathbb{C}^4$ with $3\times 3$ Lax matrices is also presented.
We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G i…
We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of almost Poisson structures. In the absence of fluxes the gauge symmetries close a Poisson gauge algebra and their action is governed by a $P_\infty$-algebra which we construct explicitly from the symplectic embedding. In curved backgrounds they close a field dependent gauge algebra governed by an $L_\infty$-algebra which is not a $P_\infty$-algebra. Our techniqu…
Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures. For deformations of these structures we prove the classification theorems and a general a general index theorem.
These are lecture notes for the course "Poisson geometry and deformation quantization" given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes' theorem, de Rham's theorem and Frobenius' theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinstein's tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introd…
Using Fedosov's approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kaehler-Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.
We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities C^{2n-2}/S_n associated to the type A Weyl group S_n and its reflection representation C^{n-1}. We also compute the full structure of the natural D-module, previously defined by the authors, whose solution space over algebraic distributions identifies with the space of Poisson traces. As a consequence, we deduce bounds on the numbers of finite-dimensional ir…
Let $G_¶$ be a compact simple Poisson-Lie group equipped with a Poisson structure $¶$ and $(M, ø)$ be a symplectic manifold. Assume that $M$ carries a Poisson action of $G_¶$ and there is an equivariant moment map in the sense of Lu and Weinstein which acts to the dual Poisson-Lie group $G^*_¶$, $\m: M\rightarrow G^*_¶$. We prove that $M$ always possesses another symplectic form $\to$ so that the $G$-action preserves $\tildeø$ and there is a new moment map $μ= e^{-1} \circ \m: M\rightarrow \g^*$. Here $e$ is a universal (independent of $M$) invertible equivariant map $e: \g^*\rightarrow G^*_¶$…
As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results is also addressed. In particular, the paper includes an extension to space-times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a "differential" version of the classical and quantum master equations. The quantization…
We expose the basics of the Fedosov quantization procedure, placed in the general framework of symplectic ringed spaces. This framework also includes some Poisson manifolds with non regular Poisson structures, presymplectic manifolds, complex analytic symplectic manifolds, etc.
We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid `with separation of variables' over an arbitrary Kaehler-Poisson manifold.
This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are bre…
A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation $A$ of a Poisson algebra $Z$ should correspond bijectively to the symplectic leaves of $\operatorname{Spec}(Z)$. In this article we consider a Poisson order $A$ over a complex affine Poisson algebra $Z$. We stratify the primitive spectrum $\operatorname{Prim}(A)$ into symplectic cores, which should be thought of as families of non-commutative symplectic leaves. We then introduce a category $A$-$\mathcal{P}$-Mod of $A$-modules adapted to the Poisson structure on $Z$, and we show that when $\operato…
The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of tot…
Let $G$ be a connected complex semi-simple Lie group and ${\mathcal{B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${\mathcal{B}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of P…
Dirac submanifolds are a natural generalization in the Poisson category for symplectic submanifolds of a symplectic manifold. In a certain sense they correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable locus of a Poisson involution. In this paper, we provide a general study for these submanifolds including both local and global aspects. In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable locuses. We discuss the Poisson involutions on a s…
We consider a class of \textit{factorizable} Poisson brackets which includes almost all reasonable Poisson structures. A particular case of the factorizable brackets are those associated with symplectic Lie algebroids. The BRST theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case. This can be viewed as an extension of the Fedosov deformation quantization to a wide class of \textit{irregular} Poisson structures. In a more general case, the factorizable Poisson brackets are shown to be closely conn…
Phase space of a characteristic Hamiltonian system is a symplectic leaf of a factorizable Poisson Lie group. Its Hamiltonian is a restriction to the symplectic leaf of a function on the group which is invariant with respect to conjugations. It is shown in this paper that such system is always integrable.
