From: Robin Chapman Subject: Re: Lie Algebra Date: Sat, 22 Apr 2000 19:33:38 GMT Newsgroups: sci.math Summary: Ado-Iwasawa theorem: Lie algebras arise from matrix groups In article <202784de.7cc44db8@usw-ex0108-061.remarq.com>, Ajit Bhand wrote: > We know that a Lie algebra g is a vector space endowed with > a bilinear operation called bracket. > Is there always a Lie group G corresponding to g(such that > the vector space of left invariant vector fields on G is > precisely g) ? Yes, provided that g is finite dimensional over R (or C). By the theorem of Ado and Iwasawa, each finite-dimensional Lie algebra over R or C is isomorphic to a Lie subalegbra of gl_n(R) or gl_n(C). Here gl_n(K) denotes the Lie algebra of n by n matrices over K. For K = R or C, gl_n(K) is the Lie algebra of the Lie group GL_n(K). Given any subalgebra g of gl_n(K) there is a closed subgroup of GL_n(K) having g as its Lie algebra. The Ado-Iwasawa theorem is not easy. For a proof consult Jacobson's _Lie Algebras_ (Dover). -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Robin Chapman Subject: Re: Lie Algebra Date: Sun, 23 Apr 2000 18:23:44 GMT Newsgroups: sci.math In article <0acc0fdf.42e87755@usw-ex0108-061.remarq.com>, Ajit Bhand wrote: > Thanks! Are Maurer-Cartan equations in any way related to > the construction of the group from the algebra? > Does the map exp:g ->G relate an arbitrary Lie algebra to > it's corrsponding Lie group? I'm not sure I understand these questions. One cannot talk about the "corresponding Lie group" of a Lie algebra, as in general there will be nonisomorphic Lie groups sharing the same Lie algebra. Also one doesn't have the M-C equations until one has a Lie group. A little more detail on the construction of *a* (note indefinite article) Lie group with a given Lie algebra g (finite dimensional over R). Represent g as a subalgebra of gl_n(R) for some n (by Ado-Iwasawa). Then consider the subset exp(g) of GL_n(R). Rather surpisingly at first, exp(g) may not be a subgroup of GL_n(R), so consider the subgroup G it generates. One can prove that G is a closed subgroup of GL_n(R) with Lie algebra g (there are various technical details to be checked). -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy.