arXiv subjects

Janusz Grabowski

Publications and source records attributed to Janusz Grabowski.

At least 55 records · Page 3Linked to original sources

arXiv2016

Duality for graded manifolds

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation of $N$-manifolds of degree 2, announced in the literature.

math.DG↗

arXiv2016

Splitting theorem for $\mathbb{Z}_2^n$-supermanifolds

Smooth $\mathbb{Z}_2^n$-supermanifolds have been introduced and studied recently. The corresponding sign rule is given by the "scalar product" of the involved $\mathbb{Z}_2^n$-degrees. It exhibits interesting changes in comparison with the sign rule using the parity of the total degree. With the new rule, nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. The classical Batchelor-Gawcedzki theorem says that any smooth supermanifold is diffeomorphic to the "superization" $ΠE$ of a vector bundle $E$. It is also known that this result fails in the complex analytic category. Hence, it is natural to ask whether an analogous statement goes through in the category of $\mathbb{Z}_2^n$-supermanifolds with its local model made of formal power series. We give a positive answer to this question.

math.DG↗

arXiv2016

The category of $\mathbb{Z}_2^n$-supermanifolds

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n>1$, appear in various fields. The corresponding sign rule is determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The $\mathbb{Z}_2^n$-Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. In this article we develop the foundations of the theory: we define $\mathbb{Z}_2^n$-supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of $\mathbb{Z}_2^\bullet$-supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any $n$-fold vector bundle has a canonical `superization' to a $\mathbb{Z}_2^n$-supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the $\mathbb{Z}_2^n$-context.

math.DG↗

arXiv2015

Graded Bundles in the Category of Lie Groupoids

We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that $\mathcal{VB}$-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids, objects defined in a previous publication of the authors, which are graded manifolds in the category of Lie algebroids, showing that they are naturally related via differentiation and integration. In this work we also make an initial study of weighted Poisson-Lie groupoids and weighted Lie bi-algebroids, as well as weighted Courant algebroids.

math.DG↗

arXiv2015

Linear duals of graded bundles and higher analogues of (Lie) algebroids

Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle. They are examples of double structures, graded-linear ($\mathcal{GL}$) bundles, including double vector bundles as a particular case. On $\mathcal{GL}$-bundles we define what we shall call weighted algebroids, which are to be understood as algebroids in the category of graded bundles. They can be considered as a geometrical framework for higher order Lagrangian mechanics. Canonical examples are reductions of higher tangent bundles of Lie groupoids. Weighted algebroids represent also a generalisation of $\mathcal{VB}$-algebroids as defined by Gracia-Saz & Mehta and the $\mathcal{LA}$-bundles of Mackenzie. The resulting structures are strikingly similar to Voronov's higher Lie algebroids, however our approach does not require the initial structures to be defined on supermanifolds.

math-ph↗

arXiv2015

An introduction to loopoids

We discuss a concept of loopoid as a non-associative generalization of (Brandt) groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.

math.GR↗

arXiv2015

Tulczyjew triples in the constrained dynamics of strings

We show that there exists a natural Tulczyjew triple in the dynamics of objects for which the standard kinematic configuration space $TM$, i.e. the tangent bundle, is replaced with its $n$-th exterior power, i.e. the bundle of tangent $n$-vectors. In this framework, which is fully covariant, we geometrically derive phase equations, as well as Euler-Lagrange equations, including nonholonomic constraints into the picture. Dynamics of strings and a constrained Plateau problem in statics are particular cases of this framework.

math-ph↗

arXiv2014

Higher order mechanics on graded bundles

In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. As important examples, we geometrically derive the classical higher order Euler-Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie groupoids.

math-ph↗

arXiv2014

$\mathbb{Z}_2^n$-Supergeometry I: Manifolds and Morphisms

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n \geq 2$, do appear quite frequently. The corresponding sign rules are determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The present paper is the first of a series on $\mathbb{Z}_2^n$-Supergeometry. The new theory exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise (the parity is the parity of the total degree). It is based on the hierarchy: ` $\mathbb{Z}_2^0$-Supergeometry (classical differential Geometry) contains the germ of $\mathbb{Z}_2^1$-Supergeometry (standard Supergeometry), which in turn contains the sprout of $\mathbb{Z}_2^2$-Supergeometry, etc.' The $\mathbb{Z}_2^n$-supergeometric viewpoint provides deeper insight and simplified solutions; interesting relations with Quantum Field Theory and Quantum Mechanics are expected. In this article, we define $\mathbb{Z}_2^n$-supermanifolds and provide examples in the atlas, the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the category of $\mathbb{Z}_2^n$-supermanifolds is closed with respect to the tangent and cotangent functors. The fundamental theorem describing supermorphisms in terms of coordinates is extended to the $\mathbb{Z}_2^n$-context.

math.DG↗

arXiv2014

$\mathbb{Z}_2^n$-Supergeometry II: Batchelor-Gawedzki Theorem

Quite a number of $\mathbb{Z}_2^n$-gradings, $n\geq 2$, appear in Physics and in Mathematics. The corresponding sign rules are given by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The new theory exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise (the parity is the parity of the total degree). Formal series are the appropriate substitute for nilpotency; the category of $\mathbb{Z}_2^\bullet$-manifolds is closed with respect to the tangent and cotangent functors. The $\mathbb{Z}_2^n$-supergeometric viewpoint provides deeper insight and simplified solutions; interesting relations with Quantum Field Theory and Quantum Mechanics are expected. In this article, we introduce split $\mathbb{Z}_2^n$-manifolds as intrinsic superizations of $\mathbb{Z}_2^n\setminus\{0\}$-graded vector bundles and prove that, conversely, any $\mathbb{Z}_2^n$-manifold is noncanonically split. We thus provide a complete proof of the $\mathbb{Z}_2^n$-extension of the so-called Batchelor-Gawedzki Theorem.

math.DG↗

arXiv2014

Modular classes revisited

We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry and its generalizations in a new light and simplifies proofs. It gives, in particular, a nice geometric interpretation of modular classes of twisted-Poisson structures on Lie algebroids.

math.DG↗

arXiv2014

Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings

The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of double vector bundles T*TM, T*TM, and TT*M. We show that there exist an analogous picture in the dynamics of objects for which the configuration space is the vector bundle of n-vectors, if we make use of certain graded bundle structures of degree n, i.e. objects generalizing vector bundles (for which n=1). For instance, the role of TT*M is played in our approach by the vector bundle of n-vectors on the bundle of n-covectors, which is canonically a graded bundle of degree n over the bundle of n-vectors. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.

math.DG↗

arXiv2014

Tulczyjew triples: from statics to field theory

We propose a geometric approach to dynamical equations of physics, based on the idea of the Tulczyjew triple. We show the evolution of these concepts, starting with the roots lying in the variational calculus for statics, through Lagrangian and Hamiltonian mechanics, and concluding with Tulczyjew triples for classical field theories, illustrated with a numer of important examples.

math-ph↗

arXiv2013

Convex bodies of states and maps

We give a general solution to the question when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density states. The same approach can be applied to study convex combinations of quantum channels. The importance of both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k-entangled states are also considered as examples of our techniques.

math-ph↗

arXiv2013

The Supergeometry of Loday Algebroids

A new concept of Loday algebroid (and its pure algebraic version - Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are interpreted as homological vector fields on a `supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given.

math.DG↗

arXiv2013

Brackets

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as the supergeometric setting. The review is supplemented by a few new concepts and examples.

math.DG↗

arXiv2013

Graded contact manifolds and contact Courant algebroids

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear (more generally, n-linear) contact structure. Linear contact structures are proven to be exactly the canonical contact structures on first jets of line bundles. They provide linear Kirillov (or Jacobi) brackets and give rise to the concept of a Kirillov algebroid, an analog of a Lie algebroid, for which the corresponding cohomology operator is represented not by a vector field (de Rham derivative) but a first-order differential operator. It is shown that one can view Kirillov or Jacobi brackets as homological Hamiltonians on linear contact manifolds. Contact manifolds of degree 2 are studied, as well as contact analogs of Courant algebroids. We define lifting procedures that provide us with constructions of canonical examples of the structures in question.

math.DG↗

arXiv2012

Mixed superposition rules and the Riccati hierarchy

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.

math.CA↗