# Shirshov basis

*Širšov basis*

A particular basis for free Lie algebras introduced in [a1]. It is identical, up to symmetries, to the Lyndon basis (cf Lyndon word; Lie algebra, free).

A word is a sequence of letters , that is, elements chosen from a set called an alphabet. A word is usually written as , or abbreviated by a single symbol: . The length of is equal to the number of letters in , i.e. . One may concatenate words , and this operation is concisely written as . The set of all words over is denoted by .

Shirshov's original description, as given in [a2], is as follows. Let be a set totally ordered by a relation (cf. Totally ordered set). Extend the order to all words by setting and for all and such that .

Let be the set of words strictly greater, with respect to , than any of their circular shifts (). Shirshov's lemma [a1] shows that any word is a non-decreasing product of words in : with and . As for Lyndon words (cf. Lyndon word), words in lead to a basis of the free Lie algebra (over ; cf. Lie algebra, free). Indeed, only a bracketing of words in is needed. This is done inductively as follows. Set for . Otherwise, a may be written as with , and . Then one defines

The set is the Shirshov basis for the free Lie algebra over .

#### References

[a1] | A.I. Shirshov, "On bases for free Lie algebras" Algebra i Logika Sém. , 1 (1962) pp. 14–19 (In Russian) |

[a2] | X. Viennot, "Algèbres de Lie libres et monoïdes libres" , Lecture Notes in Mathematics , 691 , Springer (1978) |

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Shirshov basis.

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