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Why is a Lie group wanted instead of a semigroup, what does the group structure give? References on this would be much appreciated.

I'm currently pondering manifolds and lie groups and their associations with certain computer vision problems. Semigroups or other algebraic objects may be an interesting idea to study in this venture and I wanted to know whether or not a Lie Semigroup is a structure that makes sense to study.

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2 Answers 2

Of course such things exist... The very first google search that should come to your mind will give you references about them!

For example, none other than Robert Langland's PhD thesis is entitled «Semi-groups and representations of Lie groups» and was written in 1960; the subject pre-existed, though. He published it as [Langlands, R. P. On Lie semi-groups. Canad. J. Math. 12 1960 686--693. MR0121667 (22 #12401)] He refers to [Hille, E., Lie theory of semi-groups of linear transformations, Bull. of the AMS, vol. 56, 1950] and to [Hille, E and R.S. Phillips, Functional Analysis and Semi-Groups, AMS Coll. Publ. 31, 1957] as the founding papers.

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This is a good reminder that (Lie) semigroups have been popular for quite a while in functional analysis settings. On the other hand, representation theory is usually more far-reaching for Lie groups and has influenced mathematical physics crucially. It depends on the context of the questions being asked. –  Jim Humphreys Nov 29 '10 at 23:14

Much of the literature on the representation theory of the group $GL_n$ is really about the representation theory of the monoid $M_n$. (One might argue that it's really about the representation theory of the category Vec.)

There's plenty of literature on Lie monoids; e.g. by Lex Renner.

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As I suggested to Mariano, context matters a lot here. Do you really encounter inverses? Both Renner and Putcha have developed algebraic semigroups in many directions, though I am still uncertain about what follows from their work. The comparison of general linear groups and the monoid of endomorphisms seems to be especially natural but also somewhat exceptional among Lie types. At the same time, algebraic monoids are themselves special among Lie monoids or semigroups. –  Jim Humphreys Nov 29 '10 at 23:21

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