Gröbner/SAGBI bases for non-commutative setting - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:34:05Z http://mathoverflow.net/feeds/question/73279 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73279/grobner-sagbi-bases-for-non-commutative-setting Gröbner/SAGBI bases for non-commutative setting Chris 2011-08-20T14:12:25Z 2011-10-23T01:18:51Z <p>It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good reference for the general theory of SAGBI/Gröbner bases for non-commutative setting. Actually, I am interesting in the following subjects:</p> <p>(1) Where can I find the general construction of these bases for non-commutative setting?</p> <p>(2) Is there some reference for the specific case of the Universal enveloping algebra of a finite-dimensional semi-simple Lie algebra over \$\mathbb{C}\$?</p> <p>(3) Does exist a reference using this bases in representation theory? For instance, in the study of universal objects defined by generators and relations.</p> <p>THANKS!</p> http://mathoverflow.net/questions/73279/grobner-sagbi-bases-for-non-commutative-setting/73284#73284 Answer by Mark Sapir for Gröbner/SAGBI bases for non-commutative setting Mark Sapir 2011-08-20T15:19:57Z 2011-08-21T07:15:51Z <p>In the non-commutative situation, it is called Groebner-Shirshov basis. See, for example, <a href="http://arxiv.org/abs/0804.1344" rel="nofollow"> this </a> survey paper and references there. The case of universal enveloping algebra was considered by Bergman in G. M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29, 178-218(1978). I do not know about representation theory, but certainly this is one of the main tools of studying finitely presented rings and algebras over a field. The problem, of course, is that a finite Groebner-Shirshov basis (unlike Groebner basis in the commutative case) does not always exist. </p> http://mathoverflow.net/questions/73279/grobner-sagbi-bases-for-non-commutative-setting/73343#73343 Answer by Ronnie Brown for Gröbner/SAGBI bases for non-commutative setting Ronnie Brown 2011-08-21T16:53:59Z 2011-08-21T16:53:59Z <p>You should be interested in the discussion in </p> <p>Evans, G.~A. and Wensley, C.~D. {Complete involutive rewriting systems}. {J. Symbolic Comput.} \textbf{42}~(11-12) (2007) 1034--1051.</p> <p>on the notion of noncommutative involutive systems, which in the commutative case are a well known modification of the Grobner basis theory. Evans' thesis is available at math.RA/0602140. </p> <p>I can't help on the proposed applications. </p> <p>Ronnie Brown</p> http://mathoverflow.net/questions/73279/grobner-sagbi-bases-for-non-commutative-setting/78858#78858 Answer by Vladimir Dotsenko for Gröbner/SAGBI bases for non-commutative setting Vladimir Dotsenko 2011-10-23T01:18:51Z 2011-10-23T01:18:51Z <p>There are lots of papers dealing with representation-theoretic questions and universal enveloping algebras using Gr&ouml;bner bases. Some examples are given by these: <a href="http://s-space.snu.ac.kr/bitstream/10371/12179/1/42.pdf" rel="nofollow">1</a>, <a href="http://www.ams.org/journals/spmj/2011-22-04/S1061-0022-2011-01159-8/home.html" rel="nofollow">2</a>, <a href="http://www.springerlink.com/content/bwn7336156456u7u/" rel="nofollow">3</a>, <a href="http://vm-jn.wspc.com.sg/ijac/06/0604/S0218196796000234.html" rel="nofollow">4</a>.</p>