Welcome to MathOverflow

Question

5

1

Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on

tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential map sends a vector to a point on a curve naturally defined in terms of the vector;
unital Banach algebras?

(NB. I am not conversant with category theory beyond the words "morphism" and "functor". But a categorically flavored answer that takes my limited knowledge base into account would be preferable. An internet search led me to the notion of a "Cartesian closed category", which doesn't seem to be the sort of thing I have in mind.)

flag	asked Jan 23 2010 at 15:08 Steve Huntsman 8,139●1●22●56

	Exponentiation can mean 1. the function $x \rightarrow \exp(x)$ or 2. the function $(a,b) \rightarrow a^b$. Cartesian closed categories are about the latter. – Dan Piponi Jan 23 2010 at 15:25
	That's why it didn't seem like it to me. – Steve Huntsman Jan 23 2010 at 15:54

Hanno Becker · Accepted Answer · 2010-01-23 15:28:40Z UTC

6

I think the exponential function for unital Banach algebras is a special case of the exponential functions for Lie groups modeled on topological vector spaces: the set of invertible elements in an unital Banach algebra naturally is a Banach Lie group, and the exponential function of this Lie group is the "classical" exponential function for the Banach algebra.

answered Jan 23 2010 at 15:28

Hanno Becker
1,886●7●13

	Nice, but if there is a group structure on generic (nice, finite-dimensional) Riemannian manifolds I don't know about it. – Steve Huntsman Jan 26 2010 at 21:50
	@Hanno Becker @Steve Huntsman just a note: the group of invertible elements in a Banach algebra has natural structure of manifold modeled on the underlying Banach space, and I know Serge Lang's Foundamentals of Differential Geometry as the reference for Banach Manifolds. There the exponential map is associated to any spray on a Banach manifold. – Giuseppe Apr 16 2011 at 20:41

What is an exponential?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.