Gröbner-Shirshov bases and their calculation

Abstract

In this survey we give an exposition of the theory of Gröbner-Shirshov bases for associative algebras and Lie algebras. We mention some new Composition-Diamond lemmas and applications.

Poincare-Birkhoff-Witt Theorems

“…Diamond lemmas often originate from Newman's Lemma [72] for graph theory. Shirshov (see [86] and [87]) gave a general version for Lie polynomials in 1962 which Bokut' (see [11] and [12]) extended to associative algebras in 1976, using the term "Composition Lemma." Around the same time (Bokut' cites a preprint by Bergman), Bergman [9] developed a similar result which he instead called the Diamond Lemma.…”

“…Of course the Composition-Diamond Lemma and Gröbner-Shirshov bases have been used to explore many different kinds of algebras (and in particular to find PBW-like bases) that we will not discuss here. See Bokut' and Kukin [13] and Bokut' and Chen [12] for many such examples.…”

Groebner-Shirshov basis of the universal enveloping Rota-Baxter algebra of a Lie algebra

“…In the proof of CD-Lemma we use terminology of [8] and some combinatorial results of [14]. For more detailed exposition of the latter results, see [3].…”

“…g 2 may be presented as(3).2) The edge [w] → g 1 is of positive level d, [w] → g 2 is of level 0. In this case, w = a 1 .…”

Gröbner–Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras

“…For a more detailed discussion on Gröbner-Shirshov bases, see [5] 2. Gröbner-Shirshov bases for the dendriform operad Dendriform algebras are defined by the identities (1).…”