Related papers: Fundamentals of direct limit Lie theory
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie…
We show that countable direct limits of finite-dimensional Lie groups do not have small subgroups. The same conclusion is obtained for suitable direct limits of infinite-dimensional Lie groups.
Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions,…
We introduce the theory of local minimal models for Kan simplicial manifolds, which provide the appropriate generalization of minimal Kan simplicial sets to geometric contexts. We use this to obtain the first proof of Lie's third theorem…
We prove that direct limits of finite dimensional Lie algebroids and their prolongations can be endowed with structures of convenient spaces.
We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test…
Let G be a regular Lie group which is a directed union of regular Lie groups G_i (all modelled on possibly infinite-dimensional, locally convex spaces). We show that G is the direct limit of the G_i as a regular Lie group whenever G admits…
Let G be a Lie group which is the union of an ascending sequence of Lie groups G_n (all of which may be infinite-dimensional). We study the question when G is the direct limit of the G_n's in the category of Lie groups, topological groups,…
This survey is about the fundamentals of the theory of finite dimensional Lie groups over the field of real numbers. The notion of the tangent space of a manifold at a point is considered to be defined via the well known chart and vector…
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of…
We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices.
Weil prolongations of a Lie group are naturally Lie groups. It is not known in the theory of infinite-dimensional Lie groups how to construct a Lie group with a given Lie algebra as its Lie algebra or whether there exists such a Lie group…
We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative…
We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.
We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…
We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such…
We classify all real three dimensional Lie bialgebras. In each case, their automorphism group as Lie bialgebras is also given.
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of \'etale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to…
We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras $\mathcal{L}$…
We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular…