On the Heisenberg Invariance and the Elliptic Poisson Tensors

We study different algebraic and geometric properties of Heisenberg invariant Poisson polynomial quadratic algebras. We show that these algebras are unimodular. The elliptic Sklyanin?COdesskii?CFeigin Poisson algebras ${q_{n,k}(\mathcal E)}$ are the main important example. We classify all quadratic H-invariant Poisson tensors on ${{\mathbb C}^n}$ with n ?? 6 and show that for n ?? 5 they coincide with the elliptic Sklyanin?COdesskii?CFeigin Poisson algebras or with their certain degenerations.     

Nearly associative deformation quantization

We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket.     

Double quantization of CP type orbits by generalized Verma modules

It is known that symmetric orbits in g^* for any simple Lie algebra g are equipped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the “canonical” R-matrix. We realize quantization of the Poisson pencil CPⁿ type orbits (i.e. orbits in sl(n + 1)^* whose real compact form is CPⁿ) by means of q-deformed Verma modules.     

New realizations of observables in dynamical systems with second class constraints

In the Dirac bracket approach to dynamical systems with second class constraints observables are represented by elements of a quotient Dirac bracket algebra. We describe families of new realizations of this algebra through quotients of the original Poisson bracket algebra. Explicit expressions for generators and brackets of the algebras under consideration are found.     

A generalized Hamiltonian formalism unifying classical and quantum mechanics

A Poisson bracket structure is defined on associative algebras which allows for a generalized Hamiltonian dynamics. Both classical and quantum mechanics are shown to be special cases of the general formalism.     

Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of Poisson structures with background'.     

Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of ‘Poisson structures with background’.     

Canonical maps between semidirect products with applications to elasticity and superfluids

It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.     

Deformation theory and quantization. I. Deformations of symplectic structures

We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C^∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C^∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.     

Canonical transformations of local functionals and sh-Lie structures

In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector bundle. We consider those automorphisms of the vector bundle which induce mappings on the space of functionals preserving the Poisson bracket and refer to such automorphisms as canonical automorphisms.We determine how such automorphisms relate to the corresponding sh-Lie structure. If a Lie group acts on the bundle via canonical automorphisms, there are induced actions on the space of local functionals and consequently on the corresponding sh-Lie algebra. We determine conditions under which the sh-Lie structure induces an sh-Lie structure on a corresponding reduced space where the reduction is determined by the action of the group. These results are not directly a consequence of the corresponding theorems on Poisson manifolds as none of the algebraic structures are Poisson algebras.     

Time-dependent mechanical symmetries and extended Hamiltonian systems

Time-dependent mechanical symmetries are discussed in the framework of an extended Hamiltonian system. The Lie-algebraic structure of the time-dependent symmetry is made clear by introducing an extended Poisson bracket. Moreover, the relationship between the symmetry algebras of the classical and the quantum system is established.     

Continual Lie algebra bicomplexes and integrable models

We introduce bicomplex structures associated with Saveliev-Vershik continual Lie algebras, and derive non-linear dynamical systems resulting from the bicomplex conditions. Examples related to classes of continual Lie algebras, including contact Lie, Poisson bracket, and Hilbert-Cartan ones are discussed. Using the bicomplex linearization problem, we derive corresponding conservation laws. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Derived Bracket Construction and Manin Products

We will extend the classical derived bracket construction to any algebra over a binary quadratic operad. We will show that the derived product construction is a functor given by the Manin white product with the operad of permutation algebras. As an application, we will show that the operad of prePoisson algebras is isomorphic to the Manin black product of the Poisson operad with the preLie operad. We will show that differential operators and Rota–Baxter operators are, in a sense, Koszul-dual to each other.     

Generalized Poisson algebras and Hamiltonian dynamics

Hamiltonian dynamics can be formulated entirely in terms of a Poisson manifold, that is, one for which the algebra of smooth functions is a Poisson algebra. The latter is a commutative associative algebraA together with a skew-symmetric bracket which is a derivation onA. It is shown that a Poisson algebra can be generalized by replacingA by algebras which do not necessarily commute. These allow for algebraic generalizations of Hamiltonian dynamics in both classical and quantum forms. Particular examples are models of classical and quantum electrons.     

On the variational noncommutative poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.     

A Class of Integrable Spin Calogero-Moser Systems

 We introduce a class of spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of A _n -type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids. Received: 19 October 2001 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research partially supported by NSF grant DMS00-72171.     

ClassicalN=1W-superalgebras from Hamiltonian reduction

A combinatorial proof is presented of the fact that the space of supersymmetric Lax operators admits a Poisson structure analogous to the second Gel'fand-Dickey bracket of the generalized KdV hierarchies. This allows us to prove that the space of Lax operators of odd order has a symplectic submanifold-defined by (anty)symmetric operators-which inherits a Poisson structure defining classicalW-superalgebras extending theN=1 supervirasoro algebra. This construction thus yields an infinite series of extended superconformal algebras.Address after October 1991: Physikalisches Institut der Universität Bonn, FRG     

The algebra of Weyl symmetrised polynomials and its quantum extension

The Algebra of Weyl symmetrised polynomials in powers of Hamiltonian operatorsP andQ which satisfy canonical commutation relations is constructed. This algebra is shown to encompass all recent infinite dimensional algebras acting on two-dimensional phase space. In particular the Moyal bracket algebra and the Poisson bracket algebra, of which the Moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra. We propose the introduction of a second deformation, by the replacement of the Heisenberg algebra forP, Q with aq-deformed commutator, and construct algebras ofq-symmetrised Polynomials.Research supported in part by the Department of Energy under Grant DE/FG02/88/ER25065, and by a grant from the Alfred P. Sloan Foundation and the Fulbright Commission     

Generalized Drinfel'd-Sokolov hierarchies

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct theW _n ^(l) algebras, first discussed for the casen=3 andl=2 by Polyakov and Bershadsky.