Related papers: Lie algebroid structures on a class of affine bund…
This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy…
Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this…
The intention of this article is to make an attempt of classification of transitive Lie algebroids and on this basis to construct a classifying space. The realization of the intention allows to describe characteristic classes of transitive…
In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for…
A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in…
The purpose of this work is to study Lie superalgebroid structures on the space of superdifferential $1$-forms over the supermanifolds whose superfunctions are the differential forms on its underlying manifold. These superalgbroids are…
We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real…
We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal…
Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Lie bialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to be triangular coboundary. This…
A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the…
In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated.…
Almost Lie algebroids are generalizations of Lie algebroids, when the Jacobiator is not necessary null. A simple example is given, for which a Lie algebroid bracket or a Courant bundle is not possible for the given anchor, but a natural…
We associate a real distribution to any complex Lie algebroid that we call distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain…
In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations…
These notes provide a self-contained introduction to Lie algebroids, Lie-Rinehart algebras and their universal envelopes. This review is motivated by the speculation that higher-spin gauge symmetries should admit a natural formulation as…
Motivated by families of formal moduli problems, in this note we generalize the notion of L-infinity space by allowing sheaves of L-infinity algebras over any (reasonable) nilpotent dg manifold. We discuss various examples including those…
It is shown that the correct mathematical implementation of symmetry in the geometric formulation of classical field theory leads naturally beyond the concept of Lie groups and their actions on manifolds, out into the realm of Lie group…
A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of…
We show that every Lie algebroid $A$ over a manifold $P$ has a natural representation on the line bundle $Q_A = \wedge^{top}A \otimes \wedge^{top} T^*P$. The line bundle $Q_A$ may be viewed as the Lie algebroid analog of the orientation…
Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups…