From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2398 Path: news.gmane.org!not-for-mail From: "John Baez" Newsgroups: gmane.science.mathematics.categories Subject: Higher-Dimensional Algebra VI: Lie 2-Algebras Date: Sat, 19 Jul 2003 21:43:05 -0700 (PDT) Message-ID: <200307200443.h6K4h5f04326@math-cl-n02.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018633 3956 80.91.229.2 (29 Apr 2009 15:23:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:53 +0000 (UTC) To: categories@mta.ca (categories) Original-X-From: rrosebru@mta.ca Mon Jul 21 10:52:46 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 21 Jul 2003 10:52:46 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19eb2a-0005fW-00 for categories-list@mta.ca; Mon, 21 Jul 2003 10:49:25 -0300 X-Mailer: ELM [version 2.5 PL6] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:2398 Archived-At: Here's a new paper that studies categorified Lie algebras: Higher-Dimensional Algebra VI: Lie 2-algebras John C. Baez and Alissa S. Crans The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category Vect having these 2-vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2-morphisms. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a linear natural transformation called the "Jacobiator", which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang-Baxter equation. We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L-infinity algebras in the sense of Stasheff. We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This paper will soon appear on the mathematics arXiv, but the PDF version on my website looks a tiny bit better: http://math.ucr.edu/home/baez/hda6.pdf