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Lie algebras of quotient groups

David Miyamoto

Abstract

We give conditions on a diffeological group $G$ and a normal subgroup $H$ under which the quotient group $G/H$ differentiates to a Lie algebra for which $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie functor is instantiated by the tangent structure on elastic diffeological spaces introduced by Blohmann. The requisite conditions on $G$ and $H$ hold, for example, when $G$ is a convenient infinite-dimensional Lie group and $H$ is countable, or when $G$ is finite-dimensional and $H$ is arbitrary. To recognize that convenient infinite-dimensional manifolds are elastic diffeological spaces, we give a characterization of convenience in terms of the diffeological tangent functor: a separated and bornological locally convex topological vector space $E$ is convenient if and only if the natural map $E \times E \to TE$ is an isomorphism of diffeological spaces. As an application, we integrate some classically non-integrable Banach-Lie algebras to diffeological groups.

Lie algebras of quotient groups

Abstract

We give conditions on a diffeological group and a normal subgroup under which the quotient group differentiates to a Lie algebra for which . Our Lie functor is instantiated by the tangent structure on elastic diffeological spaces introduced by Blohmann. The requisite conditions on and hold, for example, when is a convenient infinite-dimensional Lie group and is countable, or when is finite-dimensional and is arbitrary. To recognize that convenient infinite-dimensional manifolds are elastic diffeological spaces, we give a characterization of convenience in terms of the diffeological tangent functor: a separated and bornological locally convex topological vector space is convenient if and only if the natural map is an isomorphism of diffeological spaces. As an application, we integrate some classically non-integrable Banach-Lie algebras to diffeological groups.

Paper Structure

This paper contains 20 sections, 49 theorems, 124 equations, 1 table.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Remarks on the literature
  4. Structure of the paper
  5. A tangent structure
  6. Elastic diffeological spaces
  7. Elastic diffeological vector spaces
  8. Diffeological groupoids
  9. Tangent bundles of orbit spaces
  10. Elasticity of certain quotients
  11. Elasticity of quotient groups
  12. The convenient setting and diffeology
  13. Convenient diffeological vector spaces
  14. From convenient manifolds to diffeology
  15. Integrating central extensions
  16. ...and 5 more sections

Key Result

Theorem 1

For every finite-dimensional Lie algebra $\mathfrak{g}$, there is some Lie group whose Lie algebra is $\mathfrak{g}$.

Theorems & Definitions (148)

  • Theorem : Lie-Cartan
  • Theorem I
  • Theorem II
  • Theorem III
  • Definition 2.1
  • Definition 2.3
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8: Bloh24
  • ...and 138 more