From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4821 Path: news.gmane.org!not-for-mail From: Johannes Huebschmann Newsgroups: gmane.science.mathematics.categories Subject: Re: Lie algebras and failure of PBW Date: Thu, 7 May 2009 22:39:37 +0200 (CEST) Message-ID: Reply-To: Johannes Huebschmann NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241798620 29623 80.91.229.12 (8 May 2009 16:03:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 8 May 2009 16:03:40 +0000 (UTC) Original-X-From: categories@mta.ca Fri May 08 18:03:32 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 1M2SXr-00022v-OW for gsmc-categories@m.gmane.org; Fri, 08 May 2009 18:03:31 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1M2Ruq-0001Lc-Pz for categories-list@mta.ca; Fri, 08 May 2009 12:23:12 -0300 X-X-Sender: huebschm@cyprus.labomath.univ-lille1.fr, categories@mta.ca Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4821 Archived-At: Dear Michael Thank you for your message. My message was perhaps a bit cryptic. By statement of the PBW theorem I mean that, essentially, relative to the PBW filtration of the universal algebra UL of the Lie algebra L, the canonical algebra morphism from the symmetric algebra SL to the associated graded object E^0(UL) is an isomorphism. This then implies that the canonical map from L to UL is injective. More precisely: The universal algebra UL and the symmetric algebra SL both acquire filtered cocommutative coalgebra structures, and the canonical morphism SL --> E^0(UL) is one of Hopf algebras. One way to make precise the statement of the PBW theorem is to require the existence of an isomorphism UL --> SL of filtered coalgebras such that the associated graded morphism E^0(UL) --> SL is the inverse to the canonical morphism SL --> E^0(UL). Certainly the freeness of the Lie algebra is enough to guarantee the statement of the PBW theorem. More generally, L projective as a module over the ground ring still suffices I guess. Indeed, the arguments you give in Subsection 5.3 of your 1996 JPAA algebra paper imply this. Best regards Johannes On Wed, 6 May 2009, Michael Barr wrote: > 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 >> >> >> >> >> >