mizar-items: Exploring fine-grained dependencies in the Mizar Mathematical Library

The Mizar Mathematical Library (MML) is a rich database of formalized mathematical proofs (see http://mizar.org). Owing to its large size (it contains more than 1100 "articles" summing to nearly 2.5 million lines of text, expressing more than 50000 theorems and 10000 definitions using more than 7000 symbols), the nature of its contents (the MML is slanted toward pure mathematics), and its classical foundations (first-order logic, set theory, natural deduction), the MML is an especially attractive target for research on foundations of mathematics. We have implemented a system, mizar-items, on which a variety of such foundational experiements can be based. The heart of mizar-items is a method for decomposing the contents of the MML into fine-grained "items" (e.g., theorem, definition, notation, etc.) and computing dependency relations among these items. mizar-items also comes equipped with a website for exploring these dependencies and interacting with them.Comment: Accepted at CICM 2011: Conferences in Intelligent Computer Mathematics, Track C: Systems and Project

Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e. mod(E)=0. Further, relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and that of a Poisson map.Comment: 20 page

Jacobi-Nijenhuis algebroids and their modular classes

Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids

Lie groupoids in information geometry

We demonstrate that the proper general setting for contrast (potential) functions in statistical and information geometry is the one provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. If the two-form is non-degenerate, it defines a `pseudo-Riemannian' metric on the Lie algebroid and a family of Lie algebroid torsion-free connections, including the Levi-Civita connection of the metric. In this framework, the two-point functions are just functions on the pair groupoid M\ti M with the `standard' metric and affine connection on the Lie algebroid \sT M. We study also reductions of such systems and infinite-dimensional examples. In particular, we find a contrast function defining the Fubini-Study metric on the Hilbert projective space.Comment: 19 page

The graded Jacobi algebras and (co)homology

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.Comment: 20 pages, a few typos corrected; final version to be published in J. Phys. A: Math. Ge