On Shirshov’s Papers for Lie Algebras

Abstract

Shirshov 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].