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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:

(1) Where can I find the general construction of these bases for non-commutative setting?

(2) Is there some reference for the specific case of the Universal enveloping algebra of a finite-dimensional semi-simple Lie algebra over $\mathbb{C}$?

(3) Does exist a reference using this bases in representation theory? For instance, in the study of universal objects defined by generators and relations.


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up vote 6 down vote accepted

In the non-commutative situation, it is called Groebner-Shirshov basis. See, for example, this 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.

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There are lots of papers dealing with representation-theoretic questions and universal enveloping algebras using Gröbner bases. Some examples are given by these: 1, 2, 3, 4.

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You should be interested in the discussion in

Evans, G.~A. and Wensley, C.~D. {Complete involutive rewriting systems}. {J. Symbolic Comput.} \textbf{42}~(11-12) (2007) 1034--1051.

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.

I can't help on the proposed applications.

Ronnie Brown

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