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1
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Isomorphisms of Poisson and Jacobi brackets

100%
Grabowski J.
Banach Center Publications
|
2000
|
tom 51
|
nr 1
79-85
EN
We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as, for example, those given by Poisson or contact structures. We admit degenerate structures as well, which seems to be new in the literature.
2
Content available remote

Poisson Lie groups and their relations to quantum groups

100%
Grabowski J.
Banach Center Publications
|
1995
|
tom 34
|
nr 1
55-64
EN
The notion of Poisson Lie group (sometimes called Poisson Drinfel'd group) was first introduced by Drinfel'd [1] and studied by Semenov-Tian-Shansky [7] to understand the Hamiltonian structure of the group of dressing transformations of a completely integrable system. The Poisson Lie groups play an important role in the mathematical theories of quantization and in nonlinear integrable equations. The aim of our lecture is to point out the naturality of this notion and to present basic facts about Poisson Lie groups together with some relations to the recent work on quantum groups.
3
Content available remote

On quantum and classical Poisson algebras

64%
Grabowski J., Poncin N.
Banach Center Publications
|
2007
|
tom 76
|
nr 1
313-324
EN
Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular a somewhat unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, whether or not the line bundles themselves are isomorphic.
4
Content available remote

Binary operations in classical and quantum mechanics

64%
Grabowski J., Marmo G.
Banach Center Publications
|
2003
|
tom 59
|
nr 1
163-172
EN
Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply properties which are usually additionally required.
5
Content available remote

Lie algebraic characterization of manifolds

64%
Grabowski J., Poncin N.
Open Mathematics
|
2004
|
tom 2
|
nr 5
811-825
EN
Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.
6
Content available remote

On characterization of Poisson and Jacobi structures

64%
Grabowski J., Urbański P.
Open Mathematics
|
2003
|
tom 1
|
nr 1
123-140
EN
We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.
7
Content available remote

Closed subgroups in Banach spaces

51%
D. Ancel F., Dobrowolski T., Grabowski J.
Studia Mathematica
|
1994
|
tom 109
|
nr 3
277-290
EN
We show that zero-dimensional nondiscrete closed subgroups do exist in Banach spaces E. This happens exactly when E contains an isomorphic copy of $c_0$. Other results on subgroups of linear spaces are obtained.
8
Content available remote

Topological type of weakly closed subgroups in Banach spaces

51%
Dobrowolski T., Grabowski J., Kawamura K.
Studia Mathematica
|
1996
|
tom 118
|
nr 1
49-62
EN
The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in $ℓ^1$ which are interesting from the Banach space theory point of view are discussed.
9
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Quotient groups of linear topological spaces

44%
Grabowski J., Wojtyński W.
Colloquium Mathematicum
|
1990
|
tom 59
|
nr 1
35-51
10
Content available remote

Free subgroups of diffeomorphism groups

38%
Grabowski J.
Fundamenta Mathematicae
|
1988
|
tom 131
|
nr 2
103-121
11
Content available remote

The Lie structure of C* and Poisson algebras

32%
Grabowski J.
Studia Mathematica
|
1985
|
tom 81
|
nr 3
259-270
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