arXiv math/99010731999-01math.QAmath.DG

A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups

Eugene Karolinsky

备注：: LaTeX 2e, 9 pages, to be published in Banach Center Publications; minor changes (typos corrected)

英文摘要：

Let $G$ be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous $G$-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous $G$-spaces and Lagrangian subalgebras in the double $D(g)$ (here $g = Lie G$). A geometric interpretation of some of Poisson homogeneous $G$-spaces is also proposed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Symplectic Leaves of Complex Reductive Poisson-Lie Groups

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.

2000-05-04math.QAmath.SG

Leibniz Algebras, Courant Algebroids, and Multiplications on Reductive Homogeneous Spaces

We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this construction to realize the bracket operations on the sections of Courant algebroids and on the ``omni-Lie algebras'' recently introduced by the second author.

2000-07-06math.DGmath.QA

Sur les triples de Manin pour les algèbres de Lie réductives complexes

We study Manin triples for a reductive Lie algebra, $\g$. First, we generalize results of E. Karolinsky, on the classification of Lagrangian subalgebras (cf. KAROLINSKY E., {\em A Classification of Poisson homogeneous spaces of a compact Poisson Lie group}, Dokl. Ak. Nauk, 359 (1998), 13-15). Then we show that, if $\g$ is non commutative, one can attach, to each Manin triple in $\g$, an other one for a strictly smaller reductive complex Lie subalgebra of $\g$. We study also the inverse process.

1999-12-14math.QA

On quasi-Poisson homogeneous spaces of quasi-Poisson Lie groups

We provide a quasi-Poisson version of the Drinfeld's correspondence between Poisson homogeneous spaces and Lagrangian subalgebras.

2003-03-24math.DGmath.QA

Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization

This paper is a continuation of [KS]. We develop the results of [KS] principally in two directions. First, we generalize the main result of [KS], the connection between the solutions of the classical dynamical Yang-Baxter equation and Poisson homogeneous spaces of Poisson Lie groups. We hope that now we present this result in its natural generality. Secondly, we propose a partial quantization of the results of [KS]. [KS] E. Karolinsky and A. Stolin, Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras, Lett. Math. Phys., 60 (2002), p.257-274; e-print math.QA/0…

2004-02-19math.DGmath.QA

Equivariant quantization on quotients of simple Lie groups by reductive subgroups of maximal rank

We consider a class of homogeneous manifolds including all semisimple coadjoint orbits. We describe manifolds of that class admitting deformation q uantizations equivariant under the action of $G$ and the corresponding quantum group. We also classify Poisson brackets relating to such quantizations.

2002-10-13math.QAmath.SG

更多相关论文收起更多论文

Elliptic Sklyanin integrable systems for arbitrary reductive groups

We present the analogue, for an arbitrary complex reductive group G, of the elliptic integrable systems of Sklyanin. The Sklyanin integrable systems were originally constructed on symplectic leaves, of a quadratic Poisson structure, on a loop group of type A. The phase space, of our integrable systems, is a group-like analogue of the Hitchin system over an elliptic curve E. We consider the moduli space of pairs (P,f), where P is a principal G-bundle on E, and f is a meromorphic section of the adjoint group bundle. We show: 1) The moduli space admits an algebraic Poisson structure. It is relate…

2003-12-27math.AGmath.QAnlin.SI

Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces

Poisson-Lie (PL) dynamical r-matrices are generalizations of dynamical r-matrices, where the base is a Poisson-Lie group. We prove analogues of basic results for these r-matrices, namely constructions of (quasi)Poisson groupoids and of Poisson homogeneous spaces. We introduce a class of PL dynamical r-matrices, associated to nondegenerate Lie bialgebras with a splitting; this is a generalization of trigonometric r-matrices with an abelian base. We prove a composition theorem for PL dynamical r-matrices, and construct quantizations of the polarized PL dynamical r-matrices. This way, we obtain q…

2004-06-14math.QA

Quantum planes and quantum cylinders from Poisson homogeneous spaces

Quantum planes and a new quantum cylinder are obtained as quantization of Poisson homogeneous spaces of two different Poisson structures on classical Euclidean group E(2).

1995-04-25math.QAq-alg

Quantization of Poisson algebraic groups and Poisson homogeneous spaces

This paper consists of two parts. In the first part we show that any Poisson algebraic group over a field of characteristic zero and any Poisson Lie group admits a local quantization. This answers positively a question of Drinfeld. In the second part we apply our techniques of quantization to obtain some nontrivial examples of quantization of Poisson homogeneous spaces.

1995-11-08math.QAq-alg

Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras

Jiang-Hua Lu showed that any dynamical r-matrix for the pair $(g,u)$ naturally induces a Poisson homogeneous structure on $G/U$. She also proved that if $g$ is complex simple, $u$ is its Cartan subalgebra and $r$ is quasitriangular, then this correspondence is in fact 1-1. In the present paper we find some general conditions under which the Lu correspondence is 1-1. Then we apply this result to describe all triangular Poisson homogeneous structures on $G/U$ for a simple complex group $G$ and its reductive subgroup $U$ containing a Cartan subgroup.

2002-02-12math.QA

$\Ughr$ invariant quantization on some homogeneous manifolds

We consider a class of homogeneous manifolds over a simple Lie group which appears in the problem of classification of homogeneous manifolds with reductive subgroups of maximal rank as stabilizer of a point. We prove that any manifold of this class possesses a Poisson bracket admitting a quantization invariant (equivariant) with respect to the corresponding quantum group.

2002-11-05math.QAmath.SG

Quasi-bialgebras and dynamical r-matrices

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…

2004-07-22math.QA

A quantum duality principle for coisotropic subgroups and Poisson quotients

We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual. Namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. As an application, we provide an explicit quantization…

2006-03-15math.QA

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomials), and whenever its velocity bi-vector $\dot{P}=Q(P)$, also homogeneous w.r.t. $\vec{V}$ by $L_{\vec{V}}(Q)=n\cdot Q$ whenever $Q(P)= Or(γ)(P^{\otimes^n})$ is obtained using the orientation morphism $Or$ from a graph cocycle $γ$ on $n$ vertices and $2n-2$ edges in each term, t…

2019-12-29math-phmath.DGmath.MP

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 interpret…

2018-12-31math-phmath.MPmath.QA