Related papers: Lie groups in tangent join restriction categories
Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles…
A Lie group is a group that is also a differentiable manifold, such that the group operation is continuous respect to the topological structure. To every Lie group we can associate its tangent space in the identity point as a vector space,…
Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these…
We define formal exponential maps for any graded manifold as maps from the formal tangent bundle (that we also define) into the graded manifold. We show that each such map uniquely determines and is determined by its associated Grothendieck…
We consider the problem of existence of representations of topological groupoids on a principal bundle and the classification of such representations up to gauge transformation. Such representations naturally occur in various contexts such…
Reductive G-structures on a principal bundle Q are considered. It is shown that these structures, i.e. reductive G-subbundles P of Q, admit a canonical decomposition of the pull-back vector bundle $i_P^*(TQ) = P \times_Q TQ$ over P. For…
For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation…
We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a…
A toric principal $G$-bundle is a principal $G$-bundle over a toric variety together with a torus action commuting with the $G$-action. In a recent paper, extending the Klyachko classification of toric vector bundles, Chris Manon and the…
Previous work has shown that reverse differential categories give an abstract setting for gradient-based learning of functions between Euclidean spaces. However, reverse differential categories are not suited to handle gradient-based…
Partial connections are (singular) differential systems generalizing classical connections on principal bundles, yielding analogous decompositions for manifolds with nonfree group actions. Connection forms are interpreted as maps…
We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central…
For a weak 2-group, we construct a bicategory of flat 2-group bundles over differentiable stacks as a localization of a functor bicategory. This description is amenable to explicit geometric constructions. For example, we show that flat…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
We develop a new perspective on principal bundles with connection as morphisms from the tangent bundle of the underlying manifold to a classifying dg-Lie groupoid. This groupoid can be identified with a lift of the inner homomorphisms…
Let $X$ be a differentiable manifold endowed with a transitive action $\alpha:A\times X\longrightarrow X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms…
We define locally trivial quantum vector bundles (QVB) and QVB associated to locally trivial quantum principal fibre bundles. There exists a differential structure on the associated vector bundle coming from the differential structure on…
A Poisson structure on a manifold is characterized by the Schouten bracket. The graded algebra of the tangent bundle with the Schouten bracket is a prototype of Lie superalgebra. The Poisson condition means that a cycle in the 2-chain…
We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated tangent bundle of an ordinary manifold, and…
A noncommutative-geometric formalism of framed principal bundles is sketched, in a special case of quantum bundles (over quantum spaces) possessing classical structure groups. Quantum counterparts of torsion operators and Levi-Civita type…