A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P ), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an ‘integral’ of the Koszul-Schouten bracket [ , ]P of differential forms in the sense that the exterior d...

We introduce a new class of Poisson structures on a Riemannian manifold. A Poisson structure in this class will be called a Killing-Poisson structure. The class of Killing-Poisson structures contains the class of symplectic structures, the class of Poisson structures studied in (Differential Geometry and its Applications, Vol. 20, Issue 3 (2004), 279–291) and the class of Poisson structures ind...

Using the recently developed groupoidal description of Schwinger’s picture Quantum Mechanics, a new approach to Dirac’s fundamental question on role Lagrangian in Mechanics is provided. It shown that function [Formula: see text] groupoid configurations (or kinematical groupoid) quantum system determines state von Neumann algebra histories system. This function, which we call q-Lagrangian, can b...

Motivated by the notion of reduced enveloping algebra, we introduce central reductions of universal enveloping Dalgebras of restricted Lie algebroids. In the case of a tangent Lie algebroid the central reductions are sheaves of twisted crystalline differential operators. For a nilpotent p-character of a classical semisimple Lie algebra, we discover a connection between the reduced enveloping al...

‎We consider the uniqueness problem of algebroid functions‎ ‎on an angular domain‎. ‎Several theorems are established to extend the uniqueness theory of meromorphic functions to algebroid functions‎.

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sum E⊕C where E is a given Courant algebroid and C is a flat, pseudo-Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describ...

In this paper, we first obtain Xiong inequality of algebroid function on annuli and using this result we prove uniqueness theorem of algebroid functions on annuli concerning to their multiple values and derivative.

Pseudo-Galois extensions are shown to be depth two extensions. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups or involutive Hopf algebras and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observ...

In this note we study exceptional algebroids, focusing on their relation to type IIB superstring theory. We show that a IIB-exact algebroid (corresponding the group $E_{n(n)}\times \mathbb{R}^+$, for $n\le 6$) locally has standard form given by tangent bundle. derive possible twists, flat $\mathfrak{gl}(2,\mathbb{R})$-connection, covariantly closed pair of 3-forms, and 5-form, comment physical ...

We establish some fundamental relations between Dirac subbundles L for the generalized Courant algebroid (A⊕A, φ+W ) over a differentiable manifold M and the associated Dirac subbubndles L̃ for the corresponding Courant algebroid Ã⊕ Ã over M × IR.