
TL;DR
This paper establishes conditions under which quotient groups of diffeological groups form Lie algebras, extending the Lie functor to infinite-dimensional and elastic diffeological spaces, and applies this to integrate certain Banach-Lie algebras.
Contribution
It introduces a framework for Lie algebras of quotient groups in diffeological spaces, including infinite-dimensional cases, and characterizes convenience via the diffeological tangent functor.
Findings
Conditions for Lie algebra structures on quotient groups are provided.
Convenient infinite-dimensional manifolds are characterized as elastic diffeological spaces.
Some Banach-Lie algebras are integrated into diffeological 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.…
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Taxonomy
TopicsAdvanced Topics in Algebra
