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Topic: Poisson algebra

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	ipedia.com: Lie algebra Article (Site not responding. Last check: 2007-10-21)
		A subalgebra of the Lie algebra g is a linear subspace h of g such that [x, y] ∈ h for all x, y ∈ h.
		An ideal of the Lie algebra g is a subspace h of g such that [a, y] ∈ h for all a ∈ g and y ∈ h.
		The ideals are precisely the kernels of homomorphisms, and the fundamental theorem on homomorphisms is valid for Lie algebras.
	www.ipedia.com /lie_algebra_1.html (1126 words)

	Poisson algebra
		A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz' law.
		More precisely, a Poisson algebra is a vector space over a field K equipped with two bilinear products, and [,] such that forms an associative K-algebra and [,], called the Poisson bracket, forms a Lie algebra, and for any three elements x,y and z, [x,yz]=[x,y]z+y[x,z] (i.e.
		If A is a noncommutative associative algebra, then the commutator [x,y]≡xy-yx turns it into a Poisson algebra.
	www.brainyencyclopedia.com /encyclopedia/p/po/poisson_algebra.html (139 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		While a direct quantization of the reduced symplectic manifolds and concordant induced representations of Poisson algebras may be possible in certain examples, a systematic approach intending to mimic the classical reduction/induction procedure in some quantum fashion ought to start from a quantization of the `unconstrained' system.
		Obviously, the operator algebras $\A$ and $\B$ are to be seen as the quantizations of the Poisson algebras $A$ and $\cin(P)$, respectively (barring boundedness considerations).
		Firstly, the quantization of the Lie-Poisson algebra ${\rm pol}_{{\Bbb R}}[\g^]$ is the operator algebra* ${\cal U}(\g)_{\rm sa}$, which consists of the symmetric elements of the universal enveloping algebra of $\g$ (hence the quantization of the complexified Poisson algebra ${\rm pol}_{{\Bbb C}}[\g^*]$ is ${\cal U}(\g)$ itself).
	www.ma.utexas.edu /mp_arc/papers/93-289 (9809 words)

	[No title]
		We quantize the algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive.
		In fact, in the usual Poisson formulation, the Jacobi identity is the infinitesimal form of Poisson theorem which states that the bracket of two integrals of motion is also an integral of motion.
		We denote by $A$ the algebra of $C^\infty$-functions on $M$ and by ${\cal P}(f,g)$ the Poisson bracket of $f,g\in A$.
	www.ma.utexas.edu /mp_arc/papers/96-39 (4095 words)

	Hamiltonian mechanics - Wikipedia, the free encyclopedia
		The Poisson bracket acts on functions on the symplectic manifold, thus giving the space of functions on the manifold the structure of a Lie algebra.
		Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras.
		A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A
	en.wikipedia.org /wiki/Hamiltonian_mechanics (1197 words)

	Colloquium Talks
		Dirac structures, Courant algebroids and generalized complex structures are new chapters of Poisson geometry, which have developed in the last years and are motivated by the appearance of these structures in the dynamics of constrained systems and in supersymmetries in string theory.
		algebra under weaker conditions on the commutative operation $xy$ of the Poisson algebra: associativity is replaced by the
		Allowing to say that Poisson algebra corresponds to the classical mechanics one can say that the notion of Poisson-Jordan algebra include also the quantum mechanics (and possibly some others).
	math.haifa.ac.il /TOUFIK/colloquium/colloquium.html (1636 words)

	6.2 Loop quantum gravity
		To assure that the quantum Heisenberg equations have the correct classical limit, the algebra of the elementary operator has to be isomorphic to the Poisson algebra of the elementary observables.
		For the reasons illustrated in section 5, the algebra of elementary observables we choose for the quantization is the loop algebra, defined in section 6.1.
		Thus, the kinematic of the quantum theory is defined by a unitary representation of the loop algebra.
	relativity.livingreviews.org /Articles/lrr-1998-1/node15.html (582 words)

	Poisson Structures (Site not responding. Last check: 2007-10-21)
		Poisson structures arose in the works of Lagrange, Poisson and Lie in XIX century; their modern formulation is due to Lichnerowicz, Kirillov and Weinstein.
		This is a brief introduction to the algebra of Poisson brackets and constitutes my contribution to 1999/2000 YAS Seminar at Tor Vergata University.
		A booklet mainly devoted to Poisson calculus (Schouten-Nijenhuis brackets and all that).
	www.math.unifi.it /~caressa/math/poisson.html (324 words)

	[No title]
		Furthermore, this last Lie algebra is given by the homotopy groups modul* *o tor- sion of the loop space of the complement of the subspace arrangement.
		This work is motivated by r* ecent results relating the Lie algebras* of (i) and (ii) arising in the context of cla* ssical con- figuration spaces, and resolves a conjecture of the second two authors concerni ng the generalization of these results to spaces arising from certain hyperplane arran *gements.
		Moreover, the Poisson bracket relations are given by the universal infinitesimal Poisson braid relati* *ons: q-1[Bi;j+ Bi;k+ Bj;k; Bm;k]= 0 for m = i or m = j, and q-1[Bi;j; Bk;l]= 0 for {i; j} \ {k; l} = ;.
	hopf.math.purdue.edu /CohenD-CohenF-Xicotencatl/loop.txt (2863 words)

	Citebase - Duality for Lie-Rinehart algebras and the modular class
		Finally, we show that a Poisson algebra having suitable properties determines a certain module for the corrresponding Lie-Rinehart algebra and hence modular class whose square yields the module and characteristic class for its Lie-Rinehart algebra mentioned before.
		Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry dg-ga/9703001.
		A Lie-Rinehart algebra consists of a commutative algebra and a Lie algebra with additional structure which generalizes the mutual structure of interaction between the algebra of functions and the Lie algebra of smooth vector fields on a smooth manifold.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:dg-ga/9702008 (1178 words)

	[No title]
		The present paper is concerned entirely wit* h this first class case and deals with the reduction of a Poisson* algebra via homological methods,* * although there is considerable motivation from topology, particularly via the models cen* *tral to rational homotopy theory.
		The use of to denote the free graded commuta* tive algebra* on a graded vector space means that the only necessary change in our treatment is to* * specify the resolution degree as the one implied by the degree on s with ffi being of resol* *ution degree 1.
		The spectral sequence and the * Weil algebra, Preprint 1989 [Hu5] Graded Lie-Rinehart algebras, graded Poisson* algebras, and BRST-quantiz* *ation.
	hopf.math.purdue.edu /Stasheff/hrcpa-final.txt (3915 words)

	Pohlmeyer charges, DDF states and string-gauge duality \| The String Coffee Table
		The advantage that I see in using classical DDF invariants is that they satisfy a nice Poisson algebra (isomorphic to that of worldsheet oscillators), hence manifestly know about the massive spectrum of the string (except for the ground state mass and degeneracy) and are easily generalized to the superstring.
		If the quantum algebra of observables is constructed as a subalgebra of an auxiliary larger algebra, then it may have representations which do not arise as subrep’ns of some rep’n of the larger algebra.
		An infinitesimal symmetry is a derivation (or a Lie algebra of derivations) of the algebra (i.e.
	golem.ph.utexas.edu /string/archives/000300.html (8455 words)

	Citebase - The general form of the *-commutator on the Grassman algebra
		Authors: Tyutin, I. We study the general form of the -commutator treated as a deformation of the Poisson* bracket on the Grassman algebra.
		We show that, up to a similarity transformation, there are other deformations of the Poisson bracket in addition to the Moyal commutator (one at even and one at odd n, n is the number of the generators of the Grassman algebra) which are not reduced to the Moyal commutator by a similarity transformation.
		Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poiss...
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0101068 (991 words)

	Deformation Quantization
		Deformation quantization is a quantization scheme which is applied mainly for systems of classical mechanics, where one concentrates on the algebra of observables: classically the observables are the smooth complex-valued functions on the phase space which is modeled by a symplectic or Poisson manifold.
		They are an associative and commutative algebra with respect to the pointwise product and from the symplectic or Poisson structure on the manifold they inherit a Poisson bracket which makes the classical observables into a Poisson algebra.
		While the structure of the observable algebra is quite well understood a physically reasonable concept for the states seemed to be missing.
	idefix.physik.uni-freiburg.de /~stefan/quant_eng.html (1076 words)

	Wilde, Lecomte: Existence of star-products on exact symplectic manifolds
		It is shown that if a manifold admits an exact symplectic form, then its Poisson Lie algebra has non trivial formal deformations and the manifold admits star-products.
		The non-formal derivations of the star-products and the deformations of the Poisson Lie algebra of an arbitrary symplectic manifold are studied.
		LECOMTE, Cohomology of the Lie algebra of smooth vector fields of a manifold, associated to the Lie derivative of smooth forms, J. Math.
	www.numdam.org /numdam-bin/item?id=AIF_1985__35_2_117_0 (209 words)

	Super Algebras (Was: Poisson groups/actions)
		I guess the term Super algebra is kind of a choice dependent because I am thinking of a super algebra if the space is broken into even and odd part and only this matters to the structure.
		By the way, > this winds up implying that the Poisson bracket is always degenerate > at the identity of G. I guess it should be clear from the form of the filed P but just by hand waving is it related to the fact that at the identity h=e we have ghg^(-1)=geg^(-1)=e.
		However I hope that some how one can get the corresponding classical space which should be the same as the space corresponding to the classical group but may be trajectory of points will be somehow different, I don't know but if something like this happens then one can use it to visualize the differences.
	www.lns.cornell.edu /spr/1999-03/msg0015399.html (764 words)

	[No title]
		Poisson series algebra in the problem of celestial body rotation around it's mass center
		All the operations of the method mentioned are prepared as simple action over Poisson's series depending on several variables which are based on Andoyer's angles and integrals of the unperturbed problem.
		In the second case these are built through intermediate canonical transformations that transfer the top of some hyperbolid to a point that corresponds to a regular precession of the satellite.
	www.turpion.org /php/paper.phtml?journal_id=rd&paper_id=15 (169 words)

	Huebschmann Abstract (Site not responding. Last check: 2007-10-21)
		The space R contains spaces of representations where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and, on these representation spaces, the Poisson algebra restricts to stratified symplectic Poisson algebras constructed elsewhere earlier.
		When G is the unitary group, these smaller spaces arise as moduli spaces of semistable holomorphic parabolic vector bundles with flags and weights determined by the chosen conjugacy classes.
		In the general case, the Poisson algebra on R gives a description of the variation of the stratified symplectic Poisson structures on the smaller representation spaces as the chosen conjugacy classes move.
	www.maths.warwick.ac.uk /mrc/1997-98/huebschmann-abs.html (133 words)

	on gerbes and boundary states
		In case somebody is interested, which I seriously doubt, the Moyal algebra (and in particular the Hamiltonian algebra), has a central extension, at least on the torus: [T_m, T_n] = sin(mxn) T_m+n + k.m delta_m+n where m, n are momenta in Z^2, mxn is the cross product, and k is a constant vector.
		This distinction between inner and outer derivations of the non-commutatiev algebra and how the inner ones describe 'inner' gauge groups, while the outer ones are related to the space-time symmetry is very neat.
		The algebraic structures involved with the B-field change depending upon whether there is the presence of NS 5-brane charges in which case the B-field is non-torsion or whether the B-field is flat (torsion), the effects of the KK-monopole electric charges associated with the B-field etc., and whether the B-field is e.g.
	www.pych-one.com /new-5655015-4397.html (9971 words)

	some bibliography: K
		Abstract: Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie structure together with the Leibniz law.
		The non-commutative Poisson algebra structures on the infinite-dimensional algebras are studied.
		An analogue of Ado's theorem for Leibniz algebras.
	www.justpasha.org /math/bib/k.html (6561 words)

	GASC Seminar Talk (Site not responding. Last check: 2007-10-21)
		Abstract: We discuss the Poisson and symplectic structures compatible with a cluster algebra structure.
		The main example is the cluster algebra structure of the decorated Teichmüller space.
		The compatible Poisson structure is Weil-Petersson bracket on the decorated Teichmüller space, whereas the compatible symplectic form is the Weil-Petersson symplectic form on the Teichmüller space.
	www.math.neu.edu /gasc/abs/shapiro04.html (71 words)

	AMCA: Lie-Rinehart algebras, descent, and quantization by Johannes Huebschmann (Site not responding. Last check: 2007-10-21)
		AMCA: Lie-Rinehart algebras, descent, and quantization by Johannes Huebschmann
		The quantization problem for a Poisson algebra is related to that of constructing suitable representations of a Lie-Rinehart algebra associated with the Poisson algebra in a natural fashion, as explained in our paper entitled ``Poisson cohomology and quantization'', J. für die reine und angewandte Mathematik 408 (1990), 57-113.
		The old question whether reduction after quantization coincides with quantization after reduction thus appears as a descent problem for (i) suitable Lie-Rinehart algebras defined in terms of Poisson structures and (ii) for the requisite additional data including the notions of prequantum module and polarization.
	at.yorku.ca /c/a/j/f/28.htm (337 words)

	Egilsson: Linear Hamiltonian circle actions that generate minimal Hilbert bases
		The orbit space of a linear Hamiltonian circle action and the reduced orbit space, at zero, are examples of singular Poisson spaces.
		The orbit space inherits the Poisson algebra of functions invariant under the linear circle action and the reduced orbit space inherits the Poisson algebra obtained by restricting the invariant functions to the reduced space.
		A consequence of this relation is also that the number of generators needed to generate the algebra of invariant functions is minimal.
	www.numdam.org /numdam-bin/item?id=AIF_2000__50_1_285_0 (389 words)

	DC MetaData for: Poisson Brackets of Wilson Loops and Derivations of Free Algebras (Site not responding. Last check: 2007-10-21)
		DC MetaData for: Poisson Brackets of Wilson Loops and Derivations of Free Algebras
		Poisson Brackets of Wilson Loops and Derivations of Free Algebras
		Abstract: We describe a finite analogue of the Poisson Algebra of Wilson
	www.esi.ac.at /Preprint-shadows/esi243.html (138 words)

	DC MetaData for: Deformation Quantizations of the Poisson Algebra of Laurent Polynomials (Site not responding. Last check: 2007-10-21)
		DC MetaData for: Deformation Quantizations of the Poisson Algebra of Laurent Polynomials
		Deformation Quantizations of the Poisson Algebra of Laurent Polynomials
		algebra of Laurant polynomials which is not equivalent to the Moyal
	www.esi.ac.at /Preprint-shadows/esi581.html (128 words)

	Re: Quantum groups (was: Re: Poisson groups/actions)
		(Note that the Poisson algebra axioms imply that {H,.} is a derivation of the algebra of functions on M, which is the same thing as a vector field.) If we're lucky this vector field will generate a flow on M - namely, time evolution!
		Well, the first thing to make sure you understand is that the algebra of real-valued functions on a group is already a Hopf algebra.
		Deformation quantization is all about replacing the ordinary pointwise product of functions on a Poisson manifold by a new noncommutative product, where the "amount of noncommutativity" is measured by the deformation parameter hbar.
	www.lns.cornell.edu /spr/1999-03/msg0015293.html (1357 words)

	week220
		A "Poisson algebra" is a commutative associative algebra that has a bracket operation {a,b} making it into a Lie algebra, with the property that
		And, it's the operad whose algebras are graded Poisson algebras with a bracket of degree k-1.
		So: whenever we have a k-fold loop space, its homology is a graded Poisson algebra with a bracket of degree k-1.
	math.ucr.edu /home/baez/week220.html (3572 words)

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