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matt grime
Jan7-06, 07:18 AM
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Quote Quote by George Jones
I made the post in response to your comment
I didn't think that I needed to state explicitly that this showed that a Lie algebra of vector fields was not the example for which you had earlier asked.

I didn't believe it was (since it was clearly using the product of two objects) and as such I was puzzled as to why you'd repeatedly post it in response to a request that asked about something else entirely.

Also, a Lie algebra of vector fields is infinite-dimensional. Do you require that the example be a finite-dimensional Liealgebra? In any case, I know of no 'natural' example.
nope, any 'natural' example, if any one knows of one. it is purely a 'for my own knowledge'

I think that any finite-dimensional Lie algebra is isomorphic to a Lie algebra of matrices, but maybe not in a natural way.
Ie is there an faithful representation of any (finite dimensional) lie algebra (any semisimple lie algebra certainly does)? Even if there is (which is eminently reasonablem though off the top of my head I don't know either) that wouldn't preclude it being a natural example of a space without composition, cf my unnatural example of the span of e,f,h = C^3 giving a copy of sl_2. There is no composition of elements e,f,h in C^3 given, though I am cheating by using a 'forgetful' construction.

Of course 'natural' is an undefined term.