Poisson Subalgebra Research Articles

On the Lichnerowicz-Poisson cohomology complex and quasi-Poisson algebras

This article explores F-commutative algebra (A,·) with Poisson structure defined by a non-Lie algebra bivector Q. We analyse the Lichnerowicz-Poisson coboundary δ*Q and its implications as a quasi-Poisson structure, where the Schouten bracket does not vanish, complicating cohomology computations. The examined open question is whether the deviating term is a coboundary of Poisson cohomology. Our main objectives are to construct a Poisson subalgebra on which the twisted term δ2Q(Q) vanishes, and to study the existence of Poisson structures P on a Poisson ideal satisfying the condition δ2P(·) = δ2Q(Q).

The classical limit of mean-field quantum spin systems

The theory of strict deformation quantization of the two-sphere S2⊂R3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers A1/N=MN+1(C) and A0 = C(S2) form a continuous bundle of C*-algebras over the base space I={0}∪1/N*⊂[0,1], one can define a strict deformation quantization of A0 where quantization is specified by certain quantization maps Q1/N:Ã0→A1/N, with Ã0 being a dense Poisson subalgebra of A0. Given now a sequence of such HN, we show that under some assumptions, a sequence of eigenvectors ψN of HN has a classical limit in the sense that ω0(f) ≔ limN→∞⟨ψN, Q1/N(f)ψN⟩ exists as a state on A0 given by ω0(f)=1n∑i=1nf(Ωi), where n is some natural number. We give an application regarding spontaneous symmetry breaking, and moreover, we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere S2.

Covariant variational evolution and Jacobi brackets: Particles

The formulation of covariant brackets on the space of solutions to a variational problem is analyzed in the framework of contact geometry. It is argued that the Poisson algebra on the space of functionals on fields should be read as a Poisson subalgebra within an algebra of functions equipped with a Jacobi bracket on a suitable contact manifold.

Strict deformation quantization of the state space of Mk(ℂ) with applications to the Curie–Weiss model

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

On cobrackets on the Wilson loops associated with flat $\mathrm{GL}(1, \mathbf{R})$-bundles over surfaces

Let $S$ be a closed connected oriented surface of genus $g \gt 0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(\pi_1(S), \mathrm{GL}(1, \mathbf{R}))/\mathrm{GL}(1, \mathbf{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbf{R})$-bundles over $S$. There is a surjective Lie algebra homomorphism from the Goldman Lie algebra onto $W_1(g)$. We classify all cobrackets on $W_1(g)$ up to coboundary, that is, we compute $H^1(W_1(g), W_1(g) \wedge W_1(g)) \cong \mathrm{Hom}(\mathbf{Z}^{2g}, \mathbf{R})$. As a result, there is no cohomology class corresponding to the Turaev cobracket on $W_1(g)$.

On covariant Poisson brackets in classical field theory

How to give a natural geometric definition of a covariant Poisson bracket in classical field theory has for a long time been an open problem—as testified by the extensive literature on “multisymplectic Poisson brackets,” together with the fact that all these proposals suffer from serious defects. On the other hand, the functional approach does provide a good candidate which has come to be known as the Peierls–De Witt bracket and whose construction in a geometrical setting is now well understood. Here, we show how the basic “multisymplectic Poisson bracket” already proposed in the 1970s can be derived from the Peierls–De Witt bracket, applied to a special class of functionals. This relation allows to trace back most (if not all) of the problems encountered in the past to ambiguities (the relation between differential forms on multiphase space and the functionals they define is not one-to-one) and also to the fact that this class of functionals does not form a Poisson subalgebra.

Hamiltonian structure of reduced fluid models for plasmas obtained from a kinetic description

We consider the Hamiltonian structure of reduced fluid models obtained from a kinetic description of collisionless plasmas by Vlasov–Maxwell equations. We investigate the possibility of finding Poisson subalgebras associated with fluid models starting from the Vlasov–Maxwell Poisson algebra. In this way, we show that the only possible Poisson subalgebra involves the moments of zeroth and first order of the Vlasov distribution, meaning the fluid density and the fluid velocity. We find that the bracket derived in [B.A. Shadwick, G.M. Tarkenton, E.H. Esarey, Phys. Rev. Lett. 93 (2004) 175002] which involves moments of order 2 is not a Poisson bracket since it does not satisfy the Jacobi identity.

An introduction to motivic Hall algebras

We give an introduction to Joyceʼs construction of the motivic Hall algebra of coherent sheaves on a variety M. When M is a Calabi–Yau threefold we define a semi-classical integration map from a Poisson subalgebra of this Hall algebra to the ring of functions on a symplectic torus. This material will be used in Bridgeland (2011) [3] to prove some basic properties of Donaldson–Thomas curve-counting invariants on Calabi–Yau threefolds.

Finite-Dimensional Simple Poisson Modules

We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.

FINITE-DIMENSIONAL NON-COMMUTATIVE POISSON ALGEBRAS. II

Here is a structure theorem of a finite-dimensional non-commutative Poisson algebra A. A nice element ε of A will be found, so that the Lie module action of an element of a large Poisson subalgebra of A on A is described in terms of ε and the ordinary associative commutator. Consequently, we can figure out a structure of A when the Jacobson radical rad A satisfies (rad A)2 = 0. This structure theorem leads us to a classification of the finite-dimensional simple Poisson A-modules.

A Groenewold-Van Hove Theorem for $S^2$

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S^2 which is irreducible on the subalgebra generated by the components {S_1,S_2,S_3} of the spin vector. We also show that there does not exist such a quantization of the Poisson subalgebra P consisting of polynomials in {S_1,S_2,S_3}. Furthermore, we show that the maximal Poisson subalgebra of P containing {1,S_1,S_2,S_3} that can be so quantized is just that generated by {1,S_1,S_2,S_3}.

The graded modules for the graded contact cartan algebras

In this paper we investigate the graded modules for the graded contact Cartan algebras K(n, m) and K(n). For a canonical basis of uPTG module, we derive a commutator formula and then realize Shen's mixed product module in uPTG module ν(n, m) for H(n, m). Considering the Poisson subalgebra K as 1-dimensional central extension of H(n,m), we describe the irreducible PTG modules for K(n,m) and K(n) respectively. In particular, for arbitrary K(n,m), we recover Holmes' work for K(n,1)

On the geometric quantisation of constrained systems

Geometric quantisation is applied to first-class constrained systems. Certain inconsistencies associated with the prequantisation of open gauge algebras and observables in, e.g., the prequantum operator constraint method may be cured at the quantum level by the introduction of suitable polarisations. The limitations of this possibility are discussed. Furthermore it is shown how a large Poisson subalgebra of the ideal of constraint functions can be consistently implemented at the quantum level.

Constraints and polarizations

Geometric quantization is applied to first-class constrained systems. It is shown how certain inconsistencies associated with the prequantization of open gauge algebras and observables can be cured at the quantum level by the introduction of suitable polarizations. Furthermore it is shown a large Poisson subalgebra of the ideal of constrait functions can be consistently implemented at the quantum level and how reparametrization invariance of the constraint surface relates to a partially flat local Kähler structure on phase space.

On deformations of a subalgebra of the poisson lie algebra C ? ((?2n )

We prove that there exists an infinite-dimensional Poisson subalgebra in a C∞(ℝ2n) invariant with respect to the Moyal bracket but not rigid, i.e., admitting nontrivial deformations.