On Shirshov’s Papers for Lie Algebras

Abstract

AbstractShirshov published six papers on Lie algebras in which he found the following results (in order of publication. 1953–1962): • Some years before Witt [84], the “Shirshov-Witt theorem” [1]. • Some years before Lazard [62], the “Lazard-Shirshov elimination process” [1]. This is often called “Lazard elimination”; see for example [79]. • The first example of a Lie ring that is not representable into any associative ring [2]; see also P. Cartier [37] and P.M. Cohn [43]. • In the same year as Chen-Fox-Lyndon [38], the “Lyndon-Shirshov basis” of a free Lie algebra (Lyndon-Shirshov Lie words) [6]. This is often called the “Lyndon basis”; see for example [63], [79] [64]. • Independently of Lyndon [65], the “Lyndon-Shirshov (associative) words” [6] They are often called “Lyndon words”; see for example [63]. In the literature they are also often called “(Shirshov’s) regular words” or “Lyndon-Shirshov words”; see for example [42], [24], [23], [85], [76], [14]. • The algorithmic criterion to recognize Lie polynomials in a free associative algebra over any commutative ring [6]. The algorithm is based on the property that the maximal (in deg-lex ordering) associative word of any Lie polynomial is an associative Lyndon-Shirshov word. The Friedrichs criterion [45] follows from the Shirshov algorithmic criterion (see [6]). • In the same year as Chen-Fox-Lyndon [38], the “central result on Lyndon-Shirshov words”: any word is a unique non-decreasing product of Lyndon-Shirshov words [6]. This is often called the “Lyndon theorem” or the “Chen-Fox-Lyndon theorem”. • The reduction algorithm for Lie polynomials: the elimination of the maximal Lyndon-Shirshov Lie word of a Lie polynomial in a Lyndon-Shirshov Lie word [6]. The algorithm is based on the Special Bracketing Lemma [6, Lemma 4], which in turn depends on the “central result on Lyndon-Shirshov words” above. • The theorem that any Lie algebra of countable dimension is embeddable into two-generated Lie algebra with the same number of defining relations [6]. • Some years before Viennot [82], the “Hall-Shirshov bases” of a free Lie algebra [7]: a series of bases that contains the Hall basis and the Lyndon-Shirshov basis and depends on an ordering of basic Lie words such that [w]=[[u][v]]>[v]. They are often called “Hall sets”; see for example [79]. • Some years before Hironaka [53] and Buchberger [35], [36], the “Gröbner-Shirshov basis theory” for Lie polynomials (Lie algebras) explicitly and for noncommutative polynomials (associative algebras) implicitly [9]. This theory includes the definition of composition (s-polynomial), the reduction algorithm, the algorithm for producing a Gröbner-Shirshov basis (this is an infinite algorithm of Knuth-Bendix type [35]; see also the software implementations in [48], [87], [15], and the “Composition-Diamond Lemma”. Shirshov’s “Composition-Diamond Lemma” for associative algebras was formulated explicitly in [25] and rediscovered by G. Bergman [78] under the name “non-commutative Gröbner basis theory”. The analogous theory for polynomials (commutative algebras) was found by B. Buchberger [35], [36] under the name “Gröbner basis theory”; similar ideas for (commutative) formal series were found by H. Hironaka [53] under the name “standard basis theory”. • The “Freiheitssatz” and the decidability of the word problem for one-relator Lie algebras [9]. • The first linear basis of the free product of Lie algebras [10]. • The first example showing that an analogue of the Kurosh subgroup theorem is not valid for subalgebras of the free product of Lie algebras [10]. KeywordsWord ProblemAssociative AlgebraFree ProductFree Associative AlgebraLyndon WordThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.