Janusz Grabowski

Lie algebroids and Poisson-Nijenhuis structures

Abstract Poisson-Nijenhuis structures for an arbitrary Lie algebroid are defined and studied by means of complete lifts of tensor fields.

Higher vector bundles and multi-graded symplectic manifolds

Abstract A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F , or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain N n -grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known “universal Legendre transformation” T ∗ E ≃ T ∗ E ∗ , identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg’s picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin–Vilkovisky formalisms. This is also a natural framework for studying n -fold Lie algebroids and related structures.

Tangent lifts of Poisson and related structures

The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector fields. These tangent lifts are studied with application to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

GEOMETRICAL MECHANICS ON ALGEBROIDS

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.

Geometry of quantum systems: density states and entanglement

Various problems concerning the geometry of the space of Hermitian operators on a Hilbert space are addressed. In particular, we study the canonical Poisson and Riemann?Jordan tensors and the corresponding foliations into K?hler submanifolds. It is also shown that the space of density states on an n-dimensional Hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space of rank-k states, k = 1, ..., n, is a smooth manifold of (real) dimension 2nk ? k2 ? 1 and this stratification is maximal in the sense that every smooth curve in , viewed as a subset of the dual to the Lie algebra of the unitary group , at every point must be tangent to the strata it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition , an abstract criterion of entanglement is proved.

Variational calculus with constraints on general algebroids

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler–Lagrange equations. Constrained systems are introduced in the variational and geometrical settings. The constrained Euler–Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers the majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler–Lagrange equations in classical mechanics for E = TM.

Jacobi structures revisited

Jacobi algebroids, i.e. graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero. Jacobi bialgebroids are defined in the same manner. A lifting procedure of elements of this Grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding Lie brackets is developed. This gives the possibility of associating canonically a Lie algebroid with any local Lie algebra in the sense of Kirillov.

Superposition rules, lie theorem, and partial differential equations

Abstract A rigorous geometric proof of the Lie theorem on nonlinear superposition rules for solutions of nonautonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of the Lie theorem for the case of systems of partial differential equations.

Algebroids — general differential calculi on vector bundles☆

Abstract A notion of an algebroid— a generalized of a Lie algebroid structure on a vector bundle is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T M can be obtained in the framework of a general algebroid. Also a compatibility condition which leads, in general, to a concept of a bialgebroid.

QUANTUM BI-HAMILTONIAN SYSTEMS

We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.