American Institute of Mathematical Sciences

Roughly speaking, \begin{document}$ {\mathbb Z}_2^n $\end{document}-manifolds are 'manifolds' equipped with \begin{document}$ {\mathbb Z}_2^n $\end{document}-graded commutative coordinates with the sign rule being determined by the scalar product of their \begin{document}$ {\mathbb Z}_2^n $\end{document}-degrees. We examine the notion of a symplectic \begin{document}$ {\mathbb Z}_2^n $\end{document}-manifold, i.e., a \begin{document}$ {\mathbb Z}_2^n $\end{document}-manifold equipped with a symplectic two-form that may carry non-zero \begin{document}$ {\mathbb Z}_2^n $\end{document}-degree. We show that the basic notions and results of symplectic geometry generalise to the 'higher graded' setting, including a generalisation of Darboux's theorem.

In this paper we look at the question of integrability, or not, of the two natural almost complex structures \begin{document}$ J^{\pm}_\nabla $\end{document} defined on the twistor space \begin{document}$ J(M, g) $\end{document} of an even-dimensional manifold \begin{document}$ M $\end{document} with additional structures \begin{document}$ g $\end{document} and \begin{document}$ \nabla $\end{document} a \begin{document}$ g $\end{document}-connection. We measure their non-integrability by the dimension of the span of the values of \begin{document}$ N^{J^\pm_\nabla} $\end{document}. We also look at the question of the compatibility of \begin{document}$ J^{\pm}_\nabla $\end{document} with a natural closed \begin{document}$ 2 $\end{document}-form \begin{document}$ \omega^{J(M, g, \nabla)} $\end{document} defined on \begin{document}$ J(M, g) $\end{document}. For \begin{document}$ (M, g) $\end{document} we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection \begin{document}$ \nabla $\end{document}. In all cases \begin{document}$ J(M, g) $\end{document} is a bundle of complex structures on the tangent spaces of \begin{document}$ M $\end{document} compatible with \begin{document}$ g $\end{document}. In the case \begin{document}$ M $\end{document} is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.

Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced. It is shown that, given a groupoid \begin{document}$ G\rightrightarrows \Omega $\end{document} associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global". The microscopic point of view leads to the introduction of an additional layer over the grupoid \begin{document}$ G $\end{document}, giving rise to a suitable algebraic structure of 2-groupoid. On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid. The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.

We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it.

This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmüller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra.

The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.

The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.

Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.

In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frölicher–Nijenhuis, then in the work of Gerstenhaber and Nijenhuis–Richardson in cohomology theory.

A theory of local convexity for a second order differential equation (${\text{sode}}$) on a Lie algebroid is developed. The particular case when the ${\text{sode}}$ is homogeneous quadratic is extensively discussed.

Brackets by another name - Whitehead or Samelson products - have a history parallel to that in Kosmann-Schwarzbach's "From Schouten to Mackenzie: notes on brackets". Here I sketch the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics.

In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns; unless particularized, bracket will be the generic term including product and brace. The path leads beyond binary to multi-linear \begin{document}$ n $\end{document}-ary operations, either for a single \begin{document}$ n $\end{document} or for whole coherent congeries of such assembled into what is known now as an \begin{document}$ \infty $\end{document}-algebra, such as in homotopy Gerstenhaber algebras. It also leads to more subtle invariants. Along the way, attention will be called to interaction with 'physics'; indeed, it has been a two-way street.

Given a foliation, there is a well-known notion of holonomy, which can be understood as an action that differentiates to the Bott connection on the normal bundle. We present an analogous notion for Lie subalgebroids, consisting of an effective action of the minimal integration of the Lie subalgebroid, and provide an explicit description in terms of conjugation by bisections. The construction is done in such a way that it easily extends to singular subalgebroids, which provide our main motivation.