N. Ciccoli, A. J. -L. Sheu
The purpose of this paper is to study covariant Poisson structures on the complex Grassmannian obtained as quotients by coisotropic subgroups of the standard Poisson--Lie SU(n). Properties of Poisson quotients allow to describe Poisson embeddings generalizing those obtained in math.SG/9802082.
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We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian $G_k(n)$ and show that any such bracket endows $G_k(n)$ with a structure of a Poisson homogeneous space with respect to the natural action of $SL_n$ equipped with an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin-Drinfeld classification. Moreover, every compatible Poisson structure can be obtained this way.
We show that the standard SU(n)-covariant Poisson sphere $S^{2n-1}$ is embedded in the nonstandard $SU(n+1)$-covariant Poisson complex projective spaces $CP^{n}$.
We prove that the standard Poisson structure on the Grassmannian Gr(k, n) is invariant under the action of the Coxeter element c =(1 2 ... n). In particular, its symplectic foliation is invariant under c. As a corollary, we obtain a second, Poisson geometric proof of the result of Knutson, Lam, and Speyer that the Coxeter element c interchanges the Lusztig strata of Gr(k, n). We also relate the main result to known anti-invariance properties of the standard Poisson structures on cominuscule flag varieties.
We compute the Poisson cohomology of the one-parameter family of SU(2)-covariant Poisson structures on the homogeneous space S^{2}=CP^{1}=SU(2)/U(1), where SU(2) is endowed with its standard Poisson--Lie group structure,thus extending the result of Ginzburg \cite{Gin1} on the Bruhat--Poisson structure which is a member of this family. In particular, we compute several invariants of these structures, such as the modular class and the Liouville class. As a corollary of our computation, we deduce that these structures are nontrivial deformations of each other in the direction of the standard rota…
We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra structure. As an example, we find a pre-Lie algebra structure underlying the standard 3D differential structure on $\C_q[SU_2]$. At the noncommutative geometry level we show that the enveloping algebra $U(\cm)$ of a Lie algebra $\cm$, viewed as quantisation of $\cm^*$, admits a connected differential exterior algebra of classical dimension if and only if $\cm$…
The Poisson gauge algebra is a semi-classical limit of complete non-commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity $Θ^{ab}(x)$, especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the cov…
We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis…
The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrodinger equation (MNLS). We construct via gauge transformations an isomorphism from C(R,M) to the phase space of the MNLS equation so that the GNLS flow corresponds to the MNLS flow. The existence of global solutions to the Cauchy problem for GNLS and the hierarchy of commuting flows …
A geometric prequantization formula for the Poisson-Gerstenhaber bracket of forms found within the DeDonder-Weyl Hamiltonian formalism earlier is presented. The related aspects of covariant geometric quantization of field theories are sketched. In particular, the importance of the framework of Clifford and spinor bundles and superconnections in this context is underlined.
The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten- Nijenhuis bracket of covariant symmetric tensor fields defined by the co- tangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded restriction to the fiberwise polynomial algebra; they must be p-related (p is the projection of TM onto M) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of TM which produce graded brackets on the fibe…
The prequantization map for a Poisson-Gerstenhaber algebra of dynamical variables represented by differential forms within the polysymplectic formulation of the De Donder--Weyl covariant Hamiltonian field theory is presented and the corresponding prequantum Schroedinger equation for a non-homogeneous form valued wave function is derived. This is the first step toward understanding the procedures of covariant precanonical field quantization from the point of view of geometric quantization.
We study a holomorphic Poisson structure defined on the linear space $S(n,d):= {\rm Mat}_{n\times d}(\mathbb{C}) \times {\rm Mat}_{d\times n}(\mathbb{C})$ that is covariant under the natural left actions of the standard ${\rm GL}(n,\mathbb{C})$ and ${\rm GL}(d,\mathbb{C})$ Poisson-Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the $d=1$ special case gives a Poisson structure on $S(n,1)$, and we construct a local Poisson map from the Cartesian product of $d$ independent copies of $S…
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