From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4822 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Lie algebras and failure of PBW Date: Wed, 6 May 2009 21:44:14 -0400 (EDT) Message-ID: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241798631 29657 80.91.229.12 (8 May 2009 16:03:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 8 May 2009 16:03:51 +0000 (UTC) To: Johannes Huebschmann , categories@mta.ca Original-X-From: categories@mta.ca Fri May 08 18:03:43 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1M2SY1-00028R-By for gsmc-categories@m.gmane.org; Fri, 08 May 2009 18:03:41 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1M2RqO-00010t-OD for categories-list@mta.ca; Fri, 08 May 2009 12:18:36 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4822 Archived-At: It is not entirely clear what the PBW theorem is supposed to say over an arbitrary ring. Cartan-Eilenberg prove that if g is a K-free Lie algebra (K is an arbitrary ring with 1), then the enveloping algebra is K-free and on the same sort of basis as when K is a field (assume the basis is ordered, then you can take the set of increasing sequences as the basis of g^e). Although they don't, it is simple to show that if g is K-projective, so is g^e, although the idea of a basis is no longer meaningful. If g is an arbitrary K-Lie algebra, then I have no idea what a PBW theorem could say. Michael On Wed, 6 May 2009, Johannes Huebschmann wrote: > Dear Friends and Colleagues > > On p. 331 of > > Magnus-Karras-Solitar, Combinatorial group theory > > there is a hint at an unpublished > manuscript of R. Lyndon [1955] containing an example of a Lie > algebra over an integral domain > for which the statement of the PBW theorem is not true. > I did not find this example in the literature > not did I find any other hint at it. > Does anybody know anything about it? > > > > Many thanks in advance > > Johannes > > > > >