Publications (21)4.32 Total impact
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Article: Mixed superposition rules and the Riccati hierarchy
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ABSTRACT: Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, characterizing systems admitting a mixed superposition rule. This somehow unexpected result says that such systems are exactly Lie systems, i.e., they admit a standard superposition rule. This provides a new and powerful tool for finding Lie systems, which is applied here to studying the Riccati hierarchy and to retrieving some known results in a more efficient and simpler way.03/2012; -
Article: Segre maps and entanglement for multipartite systems of indistinguishable particles
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ABSTRACT: We elaborate the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. The entanglement is characterized in terms of generalized Segre maps, supplementing thus an algebraic approach to the problem by a more geometric point of view.11/2011; -
Article: Entanglement for multipartite systems of indistinguishable particles
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ABSTRACT: We analyze the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. We use the representation theory of symmetry groups to formulate a unified approach to this problem in terms of simple tensors with appropriate symmetry. For an arbitrary parastatistics, we define the S-rank generalizing the notion of the Schmidt rank. The S-rank, defined for all types of tensors, serves for distinguishing entanglement of pure states. In addition, for Bose and Fermi statistics, we construct an analog of the Jamiolkowski isomorphism.12/2010; -
Article: Lie families: theory and applications
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ABSTRACT: We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study relations of these families, called Lie families, with the theory of Lie and quasi-Lie systems and apply our theory to provide common time-dependent superposition rules for certain Lie families.03/2010; -
Article: Wigner's Theorem and geometry of extreme positive maps
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ABSTRACT: We consider transformation maps on the space of states which are symmetries in the sense of Wigner. Due to the convex nature of the space of states, the set of these maps has a convex structure. We investigate the possibility of a complete characterization of extreme maps of this convex body, to be able to contribute to the classification of positive maps. Our study provides a variant of Wigner's theorem originally proved for ray transformations in Hilbert spaces.04/2009; -
Article: Quasi-Lie schemes: theory and applications
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ABSTRACT: A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to solve some of these equations, but also gives a geometrical explanations for some, already known, ad hoc methods of dealing with such problems.10/2008; -
Article: On the Relation between States and Maps in Infinite Dimensions
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ABSTRACT: Relations between states and maps, which are known for quantum systems in finitedimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators L2(H2 , H1 ){\mathcal{L}}_2({\mathcal{H}}_2 , {\mathcal{H}}_1 ) and the corresponding tensor products H1ÄH2*{\mathcal{H}}_1\otimes{\mathcal{H}}_2^* of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map C : L1(L2(H2, H1)) ® L¥(L(H2), L1(H1)){\mathcal{C}} : {\mathcal{L}}_1({\mathcal{L}}_2({\mathcal{H}}_2, {\mathcal{H}}_1)) \to {\mathcal{L}}_\infty({\mathcal{L}}({\mathcal{H}}_2), {\mathcal{L}}_1({\mathcal{H}}_1)) from trace-class operators on L2 (H2, H1){\mathcal{L}}_2 ({\mathcal{H}}_2, {\mathcal{H}}_1) (with the nuclear norm) into compact operators mapping the space of all bounded operators on H2{\mathcal{H}}_2 into trace class operators on H1{\mathcal{H}}_1 (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.Open Systems & Information Dynamics 11/2007; 14(4):355-370. · 1.17 Impact Factor -
Article: Symmetries, Group Actions, and Entanglement
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ABSTRACT: We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of entanglement measures.Open Systems & Information Dynamics 11/2006; 13(4):343-362. · 1.17 Impact Factor -
Article: Superposition rules, lie theorem, and partial differential equations
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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.Reports on Mathematical Physics. 11/2006; -
Article: Quantum Bi-Hamiltonian Systems
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ABSTRACT: 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. Comment: 14 pages; the paper is posted for archival purposes10/2006; -
Article: Geometry of quantum systems: density states and entanglement
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ABSTRACT: Various problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into K\"ahler submanifolds. It is also shown that the space $\cD(\cH)$ of density states on an $n$-dimensional Hilbert space $\cH$ is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space $\cD^k(\cH)$ of rank-$k$ states, $k=1,...,n$, is a smooth manifold of (real) dimension $2nk-k^2-1$ and this stratification is maximal in the sense that every smooth curve in $\cD(\cH)$, viewed as a subset of the dual $u^*(\cH)$ to the Lie algebra of the unitary group $U(\cH)$, at every point must be tangent to the strata $\cD^k(\cH)$ it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition $\cH=\cH^1\ot\cH^2$, an abstract criterion of entanglement is proved.08/2005; -
Article: Courant algebroid and Lie bialgebroid contractions
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ABSTRACT: Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, Poisson-Nijenhuis structures, and triangular Lie bialgebroids as particular examples.03/2004; -
Article: Homology and modular classes of Lie algebroids
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ABSTRACT: For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evans, J.-H. Lu, and A. Weinstein.11/2003; -
Article: The graded Jacobi algebras and (co)homology
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ABSTRACT: Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten's gauging of exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This, in turn, gives canonical generating operators for triangular Jacobi algebroids. One gets, in particular, the Lichnerowicz-Jacobi homology operators associated with classical Jacobi structures. Courant-Jacobi brackets are obtained in a similar way and use to define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi structure. All this offers a new flavour in understanding the Batalin-Vilkovisky formalism.08/2002; -
Article: Binary operations in classical and quantum mechanics
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ABSTRACT: 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 the properties which usually are additionally required. In particular, it is proved that locality of a Loday bracket on sections of a one-dimensional vector bundle forces skew-symmetry, i.e. a local Lie algebra structure in the sense of A.A.Kirillov.02/2002; -
Article: Jacobi structures revisited
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ABSTRACT: Jacobi algebroids, that is 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 and can be viewed also as odd Jacobi brackets on the supermanifolds associated with the vector bundles. 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 brackets is developed. This gives the possibility of associating canonically a Lie algebroid with any local Lie algebra in the sense of Kirillov. Comment: 20 pages11/2001; -
Article: Non-antisymmetric versions of Nambu-Poisson and algebroid brackets
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ABSTRACT: We show that we can skip the skew-symmetry assumption in the definition of Nambu-Poisson brackets. In other words, a n-ary bracket on the algebra of smooth functions which satisfies the Leibniz rule and a n-ary version of the Jacobi identity must be skew-symmetric. A similar result holds for a non-antisymmetric version of Lie algebroids.05/2001; -
Article: Reduction of Time-Dependent Systems Admitting a Superposition Principle
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ABSTRACT: The problem of differential equation systems admitting a nonlinear superposition principle is analyzed from a geometric perspective. We show how it is possible to reduce the problem of finding the general solution of such a differential equation system defined by a Lie group G to a pair of simpler problems, one in a subgroup H and the other on a homogeneous space. The theory is illustrated with several examples and applications.Acta Applicandae Mathematicae 01/2001; 66(1):67-87. · 0.90 Impact Factor -
Article: On Filippov algebroids and multiplicative Nambu-Poisson structures
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ABSTRACT: We discuss relations between linear Nambu-Poisson structures and Filippov algebras and define Filippov algebroids which are n-ary generalizations of Lie algebroids. We also prove results describing multiplicative Nambu- Poisson structures on Lie groups. In particular, we show that simple Lie groups do not admit multiplicative Nambu-Poisson structures of order n>2. Comment: Latex, 22 pages, to appear in Diff. Geom. Appl02/1999; -
Article: Generalized n-Poisson brackets on a symplectic manifold
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ABSTRACT: On a symplectic manifold a family of generalized Poisson brackets associated with powers of the symplectic form is studied. The extreme cases are related to the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can be obtained in a similar way. Comment: Latex, 10 pages, to appear in Mod. Phys. Lett. AModern Physics Letters A 02/1999; · 1.08 Impact Factor
Top co-authors
Top Journals
Institutions
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2008–2010
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Universidad de Zaragoza
- Departamento de Física Teórica
Zaragoza, Aragon, Spain
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2001
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Policlinico Federico II di Napoli
Napoli, Campania, Italy
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1999
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University of Warsaw
Warsaw, Masovian Voivodeship, Poland
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