Related papers: The Orbit Method for Poisson Orders

Take S to be a 4-dimensional Sklyanin (elliptic) algebra that is module-finite over its center Z; thus, S is PI. Our first result is the construction of a Poisson Z-order structure on S such that the induced Poisson bracket on Z is…

For any differential graded (DG for short) Poisson algebra $A$ given by generators and relations, we give a "formula" for computing the universal enveloping algebra $A^e$ of $A$. Moreover, we prove that $A^e$ has a Poincar\'e-Birkhoff-Witt…

We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the…

Poisson transversals are those submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In a previous note we proved a normal form theorem around such submanifolds. In this communication, we…

In Kapranov, M. {\it Noncommutative geometry based on commutator expansions,} J. reine angew. Math {\bf 505} (1998) 73-118, a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the…

Kirillov's orbit theory provides a powerful tool for the investigation of irreducible unitary representations of many classes of Lie groups. In a previous paper we used a modification hereof, called monomial linearisation, to construct a…

We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g.…

The quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polynomial algebra $\K[X_1,..., X_n]$ with Poisson bracket…

Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the…

We study the algebraic structure of the Poisson algebra P(O) of polynomials on a coadjoint orbit O of a semisimple Lie algebra. We prove that P(O) splits into a direct sum of its center and its derived ideal. We also show that P(O) is…

To a tree of semi-simple algebras we associate a qurve (or formally smooth algebra) S. We introduce a Zariski- and etale quiver describing the finite dimensional representations of S. In particular, we show that all quotient varieties of…

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do…

Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the…

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide.…

There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${{\boldsymbol{\mathsf U}}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations…

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental…

A Poisson analog of the Dixmier-Moeglin equivalence is established for any affine Poisson algebra $R$ on which an algebraic torus $H$ acts rationally, by Poisson automorphisms, such that $R$ has only finitely many prime Poisson $H$-stable…

Let $\Sigma $ be a compact connected and oriented surface with nonempty boundary and let $G$ be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal $G$-bundles over $\Sigma$ which are…

We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this…

Poisson algebras are, just like Lie algebras, particular cases of Lie-Rinehart algebras. The latter were introduced by Rinehart in his seminal 1963 paper, where he also introduces the notion of an enveloping algebra and proves --- under…