From: foltinek@math.utexas.edu (Kevin Foltinek)
Subject: Re: Definition of "Lie"
Date: 17 Apr 2000 12:58:30 -0500
Newsgroups: sci.math
Summary: [missing]

In article <20000415192926.05038.00000367@nso-da.aol.com>
kramsay@aol.commangled (Keith Ramsay) writes:

> The tangent space at the identity of a Lie group is a Lie algebra
> (the "infinitesimal" elements of the group is how I think some
> physicists put it) which tells you a fair bit about the group.

My opinion: this approach leaves one wondering "but where does the
bracket come from?" (because the tangent space is merely a vector
space).  Of course one can answer this question using various
constructions, but I much prefer defining the Lie algebra of a Lie
group to be the set of left-invariant vector fields on the group;
easily seen to be isomorphic (as a vector space) to the tangent space
at the identity, but also comes with a bracket for free.  And the
constructions I mentioned above are absorbed into the definition.

Kevin.
==============================================================================

From: Dev Sinha <dps@math.brown.edu>
Subject: Re: combinatorics of free Lie algebras: (some answers)
Date: 29 Jun 2000 09:30:01 -0500
Newsgroups: sci.math.research

Thank you to all who replied to my questions.  There were two main
references which people gave me: the book on Lie Groups and Lie Algebras
by Bourbaki (in particular chapter 2) and a book on Free Lie Algebras by
Reutenauer.

Corollary 4.14 of Reutenauer's book states:
The dimension of the space of Lie polynomials of degree n on q letters is
equal to:

		1/n  \sum_(d|n) \mu(d) q^{n/d}

The proof proceeds by first constructing a basis for this module -
Reutenauer uses bases due to Hall.  These bases are fairly
straightforward - one could gather what they must be from trying to find
some canonical reduction for brackets using anti-commutativity and the
Jacobi identity.  What was more impressive to me was how using the
embedding of the free Lie algebra into its enveloping algebra - namely
the free associative algebra, and the fact that the Hall bases also give a
type of basis for this algebra, and the fact that we already know a basis
for this algebra, and using the Mobius inversion formula, one can prove
the Corollary above.

Thanks again to those who replied to my first message.

Best,
Dev

> I have a basic question for which I do not know a reference with which to
> get started.
> Does anyone have some good insight/ references for the combinatorics of
> free Lie algebras?  In particular, if I take the free Lie algebra on k
> generators over some (good) ring R (over the rational numbers is the case 
> I am most interested in), what can you say about the rank of the R-module 
> generated by all possible brackets involving n of these generators?
> Are the asymptotics of this function known?
> What if I take the free graded Lie algebra on generators which commute (as
> opposed to anti-commute)?