Abstract
In this survey we give an exposition of the theory of Gröbner–Shirshov bases for associative algebras, Lie algebras, groups, semigroups, -algebras, operads, etc. We mention some new Composition-Diamond lemmas and applications.
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1 Introduction
In this survey we review the method of Gröbner–ShirshovFootnote 1 (GS for short) bases for different classes of linear universal algebras, together with an overview of calculation of these bases in a variety of specific cases.
Shirshov (also spelled Širšov) in his pioneering work ([207], 1962) posed the following fundamental question:
How to find a linear basis of a Lie algebra defined by generators and relations?
He gave an infinite algorithm to solve this problem using a new notion of the composition (later the ‘-polynomial’ in Buchberger’s terminology [65, 66]) of two Lie polynomials and a new notion of completion of a set of Lie polynomials (adding nontrivial compositions; the critical pair/completion (cpc-) algorithm in the later terminology of Knuth and Bendix [138] and Buchberger [67, 68]).
Shirshov’s algorithm goes as follows. Consider a set of Lie polynomials in the free algebra on over a field (the algebra of non-commutative polynomials on over ). Denote by the superset of obtained by adding all non-trivial Lie compositions (‘Lie -polynomials’) of the elements of . The problem of triviality of a Lie polynomial modulo a finite (or recursive) set can be solved algorithmically using Shirshov’s Lie reduction algorithm from his previous paper [203], 1958. In general, an infinite sequence
of Lie multi-compositions arises. The union of this sequence has the property that every Lie composition of elements of is trivial modulo . This is what is now called a Lie GS basis.
Then a new ‘Composition-Diamond lemmaFootnote 2 for Lie algebras’ (Lemma 3 in [207]) implies that the set of all -irreducible (or -reduced) basic Lie monomials in is a linear basis of the Lie algebra generated by with defining relations . Here a basic Lie monomial means a Lie monomial in a special linear basis of the free Lie algebra , known as the Lyndon–Shirshov (LS for short) basis (Shirshov [207] and Chen–Fox–Lyndon [72], see below). An LS monomial is called -irreducible (or -reduced) whenever , the associative support of , avoids the word for all , where is the maximal word of as an associative polynomial (in the deg-lex ordering). To be more precise, Shirshov used his reduction algorithm at each step , , . Then we have a direct system and is what is now called a minimal GS basis (a minimal GS basis is not unique, but a reduced GS basis is, see below). As a result, Shirshov’s algorithm gives a solution to the above problem for Lie algebras.
Shirshov’s algorithm, dealing with the word problem, is an infinite algorithm like the Knuth–Bendix algorithm [138], 1970 dealing with the identity problem for every variety of universal algebras.Footnote 3 The initial data for the Knuth–Bendix algorithm is the defining identities of a variety. The output of the algorithm, if any, is a ‘Knuth–Bendix basis’ of identities of the variety in the class of all universal algebras of a given signature (not a GS basis of defining relations, say, of a Lie algebra).
Shirshov’s algorithm gives linear bases and algorithmic decidability of the word problem for one-relation Lie algebras [207], (recursive) linear bases for Lie algebras with (finite) homogeneous defining relations [207], and linear bases for free products of Lie algebras with known linear bases [208]. He also proved the Freiheitssatz (freeness theorem) for Lie algebras [207] (for every one-relation Lie algebra , the subalgebra , where appears in , is a free Lie algebra). The Shirshov problem [207] of the decidability of the word problem for Lie algebras was solved negatively in [21]. More generally, it was proved [21] that some recursively presented Lie algebras with undecidable word problem can be embedded into finitely presented Lie algebras (with undecidable word problem). It is a weak analogue of the Higman embedding theorem for groups [115]. The problem [21] whether an analogue of the Higman embedding theorem is valid for Lie algebras is still open. For associative algebras a similar problem [21] was solved positively by Belyaev [10]. A simple example of a Lie algebra with undecidable word problem was given by Kukin [142].
Actually, a similar algorithm for associative algebras is implicit in Shirshov’s paper [207]. The reason is that he treats as the subspace of Lie polynomials in the free associative algebra . Then to define a Lie composition of two Lie polynomials relative to an associative word , he defines firstly the associative composition (non-commutative ‘-polynomial’) , with . Then he inserts some brackets by applying his special bracketing lemma of [203]. We can obtain for every in the same way as for Lie polynomials and in the same way as for Lie algebras (‘CD-lemma for associative algebras’) to infer that is a linear basis of the associative algebra generated by with defining relations . All proofs are similar to those in [207] but much easier.
Moreover, the cases of semigroups and groups presented by generators and defining relations are just special cases of associative algebras via semigroup and group algebras. To summarize, Shirshov’s algorithm gives linear bases and normal forms of elements of every Lie algebra, associative algebra, semigroup or group presented by generators and defining relations! The algorithm works in many cases (see below).
The theory of Gröbner bases and Buchberger’s algorithm were initiated by Buchberger (Thesis [65] 1965, paper [66] 1970) for commutative associative algebras. Buchberger’s algorithm is a finite algorithm for finitely generated commutative algebras. It is one of the most useful and famous algorithms in modern computer science.
Shirshov’s paper [207] was in the spirit of the program of Kurosh (1908–1972) to study non-associative (relatively) free algebras and free products of algebras, initiated in Kurosh’s paper [143], 1947. In that paper he proved non-associative analogs of the Nielsen–Schreier and Kurosh theorems for groups. It took quite a few years to clarify the situation for Lie algebras in Shirshov’s papers [200], 1953 and [207], 1962 closely related to his theory of GS bases. It is important to note that Kurosh’s program quite unexpectedly led to Shirshov’s theory of GS bases for Lie and associative algebras [207].
A step in Kurosh’s program was made by his student Zhukov in his Ph.D. Thesis [226], 1950. He algorithmically solved the word problem for non-associative algebras. In a sense, it was the beginning of the theory of GS bases for non-associative algebras. The main difference with the future approach of Shirshov is that Zhukov did not use a linear ordering of non-associative monomials. Instead he chose an arbitrary monomial of maximal degree as the ‘leading’ monomial of a polynomial. Also, for non-associative algebras there is no ‘composition of intersection’ (‘-polynomial’). In this sense it cannot be a model for Lie and associative algebras.Footnote 4
Shirshov, also a student of Kurosh’s, defended his Candidate of Sciences Thesis [199] at Moscow State University in 1953. His thesis together with the paper that followed [203], 1958 may be viewed as a background for his later method of GS bases. In the thesis, he proved the free subalgebra theorem for free Lie algebras (now known as Shirshov–Witt theorem, see also Witt [218], 1956) using the elimination process rediscovered by Lazard [149], 1960. He used the elimination process later [203], 1958 as a general method to prove the properties of regular (LS) words, including an algorithm of (special) bracketing of an LS word (with a fixed LS subword). The latter algorithm is of some importance in his theory of GS bases for Lie algebras (particularly in the definition of the composition of two Lie polynomials). Shirshov also proved the free subalgebra theorem for (anti-) commutative non-associative algebras [202], 1954. He used that later in [206], 1962 for the theory of GS bases of (commutative, anti-commutative) non-associative algebras. Shirshov (Thesis [199], 1953) found the (‘Hall–Shirshov’) series of bases of a free Lie algebra (see also [205] 1962, the first issue of Malcev’s Algebra and Logic).Footnote 5
The LS basis is a particular case of the Shirshov or Hall–Shirshov series of bases (cf. Reutenauer [190], where this series is called the ‘Hall series’). In the definition of his series, Shirshov used Hall’s inductive procedure (see Ph. Hall [114], 1933, Hall [113], 1950): a non-associative monomial is a basic monomial whenever
-
(1)
are basic;
-
(2)
;
-
(3)
if then .
However, instead of ordering by the degree function (Hall words), he used an arbitrary linear ordering of non-associative monomials satisfying
For example, in his Thesis [199], 1953 he used the ordering by the content of monomials (the content of, say, the monomial is the vector ). Actually, the content of may be viewed as a commutative associative word that equals in the free commutative semigroup. Two contents are compared lexicographically (a proper prefix of a content is greater than the content).
If we use the lexicographic ordering, if lexicographically (with the condition ), then we obtain the LS basis.Footnote 6 For example, for the alphabet , with we obtain basic Lyndon–Shirshov monomials by induction:
and so on. They are exactly all Shirshov regular (LS) Lie monomials and their associative supports are exactly all Shirshov regular words with a one-to-one correspondence between two sets given by the Shirshov elimination (bracketing) algorithm for (associative) words.
Let us recall that an elementary step of Shirshov’s elimination algorithm is to join the minimal letter of a word to previous ones by bracketing and to continue this process with the lexicographic ordering of the new alphabet. For example, suppose that . Then we have the succession of bracketings
By the way, the second series of partial bracketings illustrates Shirshov’s factorization theorem [203] of 1958 that every word is a non-decreasing product of LS words (it is often mistakenly called Lyndon’s theorem, see [12]).
The Shirshov special bracketing [203] goes as follows. Let us give as an example the special bracketing of the LS word with the LS subword . The Shirshov standard bracketing is
The Shirshov special bracketing is
In general, if then the Shirshov standard bracketing gives , where . Now, , each is an LS-word, and in the lex ordering (Shirshov’s factorization theorem). Then we must change the bracketing of :
The main property of is that is a monic associative polynomial with the maximal monomial ; hence, .
Actually, Shirshov [207], 1962 needed a ‘double’ relative bracketing of a regular word with two disjoint LS subwords. Then he implicitly used the following property: every LS subword of as above is a subword of some for .
Shirshov defined regular (LS) monomials [203], 1958, as follows: is a regular monomial iff:
-
(1)
is a regular word;
-
(2)
and are regular monomials (then automatically in the lex ordering);
-
(3)
if then .
Once again, if we formally omit all Lie brackets in Shirshov’s paper [207] then essentially the same algorithm and essentially the same CD-lemma (with the same but much simpler proof) yield a linear basis for associative algebra presented by generators and defining relations. The differences are the following:
-
no need to use LS monomials and LS words, since the set is a linear basis of the free associative algebra ;
-
the definition of associative composition for monic polynomials and ,
or
are much simpler than the definition of Lie composition for monic Lie polynomials and ,
or
where , , and are the Shirshov special bracketings of the LS words with fixed LS subwords and respectively.
-
The definition of elimination of the leading word of an associative monic polynomial is straightforward: whenever and . However, for Lie polynomials, it is much more involved and uses the Shirshov special bracketing: whenever .
We can formulate the main idea of Shirshov’s proof as follows. Consider a complete set of monic Lie polynomials (all compositions are trivial). If , where , , and is an LS word, while , then the Lie monomials and are equal modulo the smaller Lie monomials in :
where and . Actually, Shirshov proved a more general result: if and with then
where and . Below we call a Lie polynomial a Lie normal -word provided that .
This is precisely where he used the notion of composition and other notions and properties mentioned above.
It is much easier to prove an analogue of this property for associative algebras (as well as commutative associative algebras): given a complete monic set in (), for with and we have
where and .
Summarizing, we can say with confidence that the work (Shirshov [207]) implicitly contains the CD-lemma for associative algebras as a simple exercise that requires no new ideas. The first author, Bokut, can confirm that Shirshov clearly understood this and told him that “the case of associative algebras is the same”. The lemma was formulated explicitly in Bokut [22], 1976 (with a reference to Shirshov’s paper [207]), Bergman [11], 1978, and Mora [171], 1986.
Let us emphasize once again that the CD-Lemma for associative algebras applies to every semigroup , and in particular to every group, by way of the semigroup algebra over a field . The latter algebra has the same generators and defining relations as , or . Every composition of the binomials and is a binomial . As a result, applying Shirshov’s algorithm to a set of semigroup relations gives rise to a complete set of semigroup relations . The -irreducible words in constitute the set of normal forms of the elements of .
Before we go any further, let us give some well-known examples of algebra, group, and semigroup presentations by generators and defining relations together with linear bases, normal forms, and GS bases for them (if known). Consider a field and a commutative ring or commutative -algebra .
-
The Grassman algebra over is
The set of defining relations is a GS basis with respect to the deg-lex ordering. A -basis is
-
The Clifford algebra over is
where is an symmetric matrix over . The set of defining relations is a GS basis with respect to the deg-lex ordering. A -basis is
-
The universal enveloping algebra of a Lie algebra is
If is a free -module with a well-ordered -basis
then the set of defining relations is a GS basis of . The PBW theorem follows: is a free -module with a -basis,
-
Kandri-Rody and Weispfenning [122] invented an important class of (noncommutative polynomial) ‘algebras of solvable type’, which includes universal enveloping algebras. An algebra of solvable type is
and the compositions modulo , where with . Here is a noncommutative polynomial with all terms less than . They created a theory of GS bases for every algebra of this class; thus, they found a linear basis of every quotient of .
-
A general presentation of a universal enveloping algebra over a field , where with and is as a set of associative polynomials. PBW theorem in a Shirshov’s form. The following conditions are equivalent:
-
(i)
the set is a Lie GS basis;
-
(ii)
the set is a GS basis for ;
-
(iii)
a linear basis for consists of words , where are -irreducible LS words with (in the lex-ordering), see [56, 57];
-
(iv)
a linear basis for consists of the -irreducible LS Lie monomials in ;
-
(v)
a linear basis for consists of the polynomials , where in the lex ordering, , and each is an -irreducible non-associative LS word in .
-
(i)
-
Free Lie algebras over . Hall, Shirshov, and Lyndon provided different linear -bases for a free Lie algebra (the Hall–Shirshov series of bases, in particular, the Hall basis, the Lyndon–Shirshov basis, the basis compatible with the free solvable (polynilpotent) Lie algebra) [194], see also [15]. Two anticommutative GS bases of were found in [34, 37], which yields the Hall and Lyndon–Shirshov linear bases respectively.
-
The Lie -algebras presented by Chevalley generators and defining relations of types , , , , , , , , and . Serre’s theorem provides linear bases and multiplication tables for these algebras (they are finite dimensional simple Lie algebras over ). Lie GS bases for these algebras are found in [49–51].
-
The Coxeter group
for a given Coxeter matrix . Tits [210] (see also [14]) algorithmically solved the word problem for Coxeter groups. Finite Coxeter groups are presented by ‘finite’ Coxeter matrices , , , , , , , , , and . Coxeter’s theorem provides normal forms and Cayley tables (these are finite groups generated by reflections). GS bases for finite Coxeter groups are found in [58].
-
The Iwahory–Hecke (Hecke) algebras over differ from the group algebras of Coxeter groups in that instead of there are relations or , where are units of . Two -bases for are known; one is natural, and the other is the Kazhdan–Lusztig canonical basis [155]. The GS bases for the Iwahory–Hecke algebras are known for the finite Coxeter matrices. A deep connection of the Iwahory–Hecke algebras of type and braid groups (as well as link invariants) was found by Jones [116].
-
Affine Kac–Moody algebras [117]. The Kac–Gabber theorem provides linear bases for these algebras under the symmetrizability condition on the Cartan matrix. Using this result, Poroshenko found the GS bases of these algebras [178–180].
-
Borcherds–Kac–Moody algebras [61–63, 117]. GS bases are not known.
-
Quantum enveloping algebras (Drinfeld, Jimbo). Lusztig’s theorem [154] provides linear canonical bases of these algebras. Different approaches were developed by Ringel [191, 192], Green [110], and Kharchenko [131–135]. GS bases of quantum enveloping algebras are unknown except for the case , see [55, 86, 195, 220].
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Koszul algebras. The quadratic algebras with a basis of standard monomials, called PBW-algebras, are always Koszul (Priddy [184]), but not conversely. In different terminology, PBW-algebras are algebras with quadratic GS bases. See [177].
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Elliptic algebras (Feigin, Odesskii) These are associative algebras presented by generators and homogeneous quadratic relations for which the dimensions of the graded components are the same as for the polynomial algebra in variables. The first example of this type was Sklyanin algebra (1982) generated by , , and with the defining relations , , and . See [175]. GS bases are not known.
-
Leavitt path algebras. GS bases for these algebras are found in Alahmedi et al. [2] and applied by the same authors to determine the structure of the Leavitt path algebras of polynomial growth in [3].
-
Artin braid group . The Markov–Artin theorem provides the normal form and semi-direct structure of the group in the Burau generators. Other normal forms of were obtained by Garside, Birman–Ko–Lee, and Adjan–Thurston. GS bases for in the Artin–Burau, Artin–Garside, Birman–Ko–Lee, and Adjan–Thurston generators were found in [23–25, 89] respectively.
-
Artin–Tits groups. They differ from Coxeter groups in the absence of the relations . Normal forms are known in the spherical case, see Brieskorn, Saito [64]. GS bases are not known except for braid groups (the Artin–Tits groups of type ).
-
The groups of Novikov–Boon type (Novikov [173], Boon [60], Collins [97], Kalorkoti [118–121]) with unsolvable word or conjugacy problem. They are groups with standard bases (standard normal forms or standard GS bases), see [16–18, 77].
-
Adjan’s [1] and Rabin’s [187] constructions of groups with unsolvable isomorphism problem and Markov properties. A GS basis is known for Adjan’s construction [26].
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Markov’s [161] and Post’s [183] semigroups with unsolvable word problem. The GS basis of Post’s semigroup is found in [223].
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Markov’s construction of semigroups with unsolvable isomorphism problem and Markov properties. The GS basis for the construction is not known.
-
Plactic monoids. A theorem due to Richardson, Schensted, and Knuth provides a normal form of the elements of these monoids (see Lothaire [151]). New approaches to plactic monoids via GS bases in the alphabets of row and column generators are found in [29].
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The groups of quotients of the multiplicative semigroups of power series rings with topological quadratic relations of the type embeddable (without the zero element) into groups but in general not embeddable into division algebras (settling a problem of Malcev). The relative standard normal forms of these groups found in [19, 20] are the reduced words for what was later called a relative GS basis [59].
To date, the method of GS bases has been adapted, in particular, to the following classes of linear universal algebras, as well as for operads, categories, and semirings. Unless stated otherwise, we consider all linear algebras over a field . Following the terminology of Higgins and Kurosh, we mean by a ((differential) associative) -algebra a linear space ((differential) associative algebra) with a set of multi-linear operations :
-
Associative algebras, Shirshov [207], Bokut [22], Bergman [11];
-
Associative algebras over a commutative algebra, Mikhalev and Zolotykh [170];
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Associative -algebras, where is a group, Bokut and Shum [59];
-
Lie algebras, Shirshov [207];
-
Lie algebras over a commutative algebra, Bokut et al. [31];
-
Lie p-algebras over with , Mikhalev [166];
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Metabelian Lie algebras, Chen and Chen [75];
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Quiver (path) algebras, Farkas et al. [101];
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Tensor products of associative algebras, Bokut et al. [30];
-
Associative differential algebras, Chen et al. [76];
-
Associative conformal algebras over with , Bokut et al. [45], Bokut et al. [43];
-
Dialgebras, Bokut et al. [38];
-
Pre-Lie (Vinberg–Koszul–Gerstenhaber, right (left) symmetric) algebras, Bokut et al. [35],
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Associative Rota–Baxter algebras over with , Bokut et al. [32];
-
-algebras, Bokut et al. [33];
-
Associative -algebras, Bokut et al. [41];
-
Associative differential -algebras, Qiu and Chen [185];
-
-algebras, Bokut et al. [33];
-
Differential Rota–Baxter commutative associative algebras, Guo et al. [111];
-
Semirings, Bokut et al. [40];
-
Modules over an associative algebra, Golod [108], Green [109], Kang and Lee [123, 124], Chibrikov [90];
-
Small categories, Bokut et al. [36];
-
Non-associative algebras, Shirshov [206];
-
Non-associative algebras over a commutative algebra, Chen et al. [81];
-
Commutative non-associative algebras, Shirshov [206];
-
Anti-commutative non-associative algebras, Shirshov [206];
-
Symmetric operads, Dotsenko and Khoroshkin [98].
At the heart of the GS method for a class of linear algebras lies a CD-lemma for a free object of the class. For the cases above, the free objects are the free associative algebra , the doubly free associative -algebra , the free Lie algebra , and the doubly free Lie -algebra . For the tensor product of two associative algebras we need to use the tensor product of two free algebras, . We can view every semiring as a double semigroup with two associative products and . So, the CD-lemma for semirings is the CD-lemma for the semiring algebra of the free semiring . The CD-lemma for modules is the CD-lemma for the doubly free module , a free module over a free associative algebra. The CD-lemma for small categories is the CD-lemma for the ‘free partial -algebra’ generated by an oriented graph (a sequence , where , is a partial word in iff it is a path; a partial polynomial is a linear combination of partial words with the same source and target).
All CD-lemmas have essentially the same statement. Consider a class of linear universal algebras, a free algebra in , and a well-ordered -basis of terms of . A subset is called a GS basis if every composition of the elements of is trivial (vanishes upon the elimination of the leading terms for ). Then the following conditions are equivalent:
-
(i)
is a GS basis.
-
(ii)
If then the leading term contains the subterm for some .
-
(iii)
The set of -irreducible terms is a linear basis for the -algebra generated by with defining relations .
In some cases ( conformal algebras, dialgebras), conditions (i) and (ii) are not equivalent. To be more precise, in those cases we have .
Typical compositions are compositions of intersection and inclusion. Shirshov [206, 207] avoided inclusion composition. He suggested instead that a GS basis must be minimal (the leading words do not contain each other as subwords). In some cases, new compositions must be defined, for example, the composition of left (right) multiplication. Also, sometimes we need to combine all these compositions. We present here a new approach to the definition of a composition, based on the concept of the least common multiple of two terms and .
In some cases (Lie algebras, (-) conformal algebras) the ‘leading’ term of a polynomial lies outside . For Lie algebras, we have , for (-) conformal algebras belongs to an ‘-semigroup’.
Almost all CD-lemmas require the new notion of a ‘normal -term’. A term in , where , with only one occurrence of is called a normal -term whenever . Given , every -word (that is, an -term) is a normal -word. Given , every Lie -monomial (Lie -term) is a linear combination of normal Lie -terms (Shirshov [207]).
One of the two key lemmas asserts that if is complete under compositions of multiplication then every element of the ideal generated by is a linear combination of normal -terms. Another key lemma says that if is a GS basis and the leading words of two normal -terms are the same then these terms are the same modulo lower normal -terms. As we mentioned above, Shirshov proved these results [207] for (there are no compositions of multiplication for Lie and associative algebras).
This survey continues our surveys with Kolesnikov, Fong, Ke, and Shum [27, 28, 42, 46, 52, 53], Ufnarovski’s survey [213], and the book of the first named author and Kukin [54].
The paper is organized as follows. Section 2 is for associative algebras, Sect. 3 is for semigroups and groups, Sect. 4 is for Lie algebras, and the short Sect. 5 is for -algebras and operads.Footnote 7
To conclude this introduction, we give some information about the work of Shirshov; for more on this, see the book [209]. Shirshov (1921–1981) was a famous Russian mathematician. His name is associated with notions and results on the Gröbner–Shirshov bases, the Composition-Diamond lemma, the Shirshov–Witt theorem, the Lazard–Shirshov elimination, the Shirshov height theorem, Lyndon–Shirshov words, Lyndon–Shirshov basis (in a free Lie algebra), the Hall–Shirshov series of bases, the Cohn–Shirshov theorem for Jordan algebras, Shirshov’s theorem on the Kurosh problem, and the Shirshov factorization theorem. Shirshov’s ideas were used by his students Efim Zelmanov to solve the restricted Burnside problem and Aleksander Kemer to solve the Specht problem.
1.1 Digression on the history of Lyndon–Shirshov bases and Lyndon–Shirshov words
Lyndon [156], 1954, defined standard words, which are the same as Shirshov’s regular words [203], 1958. Unfortunately, the papers (Lyndon [156]) and (Chen et al. [72], 1958) were practically unknown before 1983. As a result, at that time almost all authors (except four who used the names Shirshov and Chen–Fox–Lyndon, see below) refer to the basis and words as Shirshov regular basis and words, cf. for instance [8, 9, 96, 188, 212, 224]. To the best of our knowledge, none of the authors mentioned Lyndon’s paper [156] as a source of ‘Lyndon words’ before 1983(!).
In the following papers the authors mentioned both (Chen et al. [72]) and (Shirshov [203]) as a source of ‘Lyndon–Shirshov basis’ and ‘Lyndon–Shirshov words’:
The authors of [196] thank Cohn for pointing out Shirshov’s paper [203]. They also formulate Shirshov’s factorization theorem [203]. They mention [72, 203] as a source of ‘LS words’. Schützenberger also mentions [197] Shirshov’s factorization theorem, but in this case he attributes it to both Chen et al. [72] and Shirshov [203]. Actually, he cites [72] by mistake, as that result is absent from the paper, see Berstel and Perrin [12].Footnote 8
Starting with the book of Lothaire, Combinatorics on words ([151], 1983), some authors called the words and basis ‘Lyndon words’ and ‘Lyndon basis’; for instance, see Reutenauer, Free Lie algebras ([190], 1993).
2 Gröbner–Shirshov bases for associative algebras
In this section we give a proof of Shirshov’s CD-lemma for associative algebras and Buchberger’s theorem for commutative algebras. Also, we give the Eisenbud–Peeva–Sturmfels lifting theorem, the CD-lemmas for modules (following Kang and Lee [124] and Chibrikov [90]), the PBW theorem and the PBW theorem in Shirshov’s form, the CD-lemma for categories, the CD-lemma for associative algebras over commutative algebras and the Rosso–Yamane theorem for .
2.1 Composition-Diamond lemma for associative algebras
Let be a field, be the free associative algebra over generated by and be the free monoid generated by , where the empty word is the identity, denoted by 1. Suppose that is a well-ordered set. Take with the leading word and , where and . We call monic if .
A well-ordering on is called a monomial ordering whenever it is compatible with the multiplication of words, that is, for all we have
A standard example of monomial ordering on is the deg-lex ordering, in which two words are compared first by the degree and then lexicographically, where is a well-ordered set.
Fix a monomial ordering on and take two monic polynomials and in . There are two kinds of compositions:
-
(i)
If is a word such that for some with then the polynomial is called the intersection composition of and with respect to .
-
(ii)
If for some then the polynomial is called the inclusion composition of and with respect to .
Then and lies in the ideal of generated by and .
In the composition , we call an ambiguity (or the least common multiple , see below).
Consider such that very is monic. Take and . Then is called trivial modulo , denoted by
if , where , , and with .
The elements , , and are called -words.
A monic set is called a GS basis in with respect to the monomial ordering if every composition of polynomials in is trivial modulo and the corresponding .
A set is called a minimal GS basis in if is a GS basis in avoiding inclusion compositions; that is, given with , we have for all .
Put
The elements of are called -irreducible or -reduced.
A GS basis in is reduced provided that for every , where whenever with and . In other words, each is an -irreducible word.
The following lemma is key for proving the CD-lemma for associative algebras.
Lemma 1
If is a GS basis in and , where and , then .
Proof
There are three cases to consider.
Case 1 Assume that the subwords and of are disjoint, say, . Then, and for some , and so . Now,
Since and , we conclude that
with and -words and satisfying
Case 2 Assume that the subword of contains as a subword. Then with and , that is, for some -word . We have
The triviality of compositions implies that
Case 3 Assume that the subwords and of have a nonempty intersection. We may assume that and with and . Then, as in Case 2, we have
Lemma 2
Consider a set of monic polynomials. For every we have
where , , and are -words. So, is a set of linear generators of the algebra .
Proof
Induct on .
Theorem 1
(The CD-lemma for associative algebras) Choose a monomial ordering on . Consider a monic set and the ideal of generated by . The following statements are equivalent:
-
(i)
is a Gröbner–Shirshov basis in .
-
(ii)
for some and .
-
(iii)
is a linear basis of the algebra .
Proof
(i)(ii). Assume that is a GS basis and take . Then, we have where , , and . Suppose that satisfy
Induct on and to show that for some . To be more precise, induct on with the lex ordering of the pairs.
If then and hence the claim holds. Assume that . Then . Lemma 1 implies that If or then the claim follows by induction on . For the case and , induct on . Thus, (ii) holds.
(ii)(iii). By Lemma 2, generates as a linear space. Suppose that in , where and . It means that in . Then for some , which contradicts (ii).
(iii)(i). Given , Lemma 2 and (iii) yield Therefore, is a GS basis.
A new exposition of the proof of Theorem 1 (CD-lemma for associative algebras).
Let us start with the concepts of non-unique common multiple and least common multiple of two words . A common multiple means that for some . Then means that some contains some as a subword: with , where and are the same subwords in both sides. To be precise,
Define the general composition of monic polynomials as
The only difference with the previous definition of composition is that we include the case of trivial . However, in this case the composition is trivial,
It is clear that if then, up to the ordering of and ,
This implies Lemma 1. The main claim (i)(ii) of Theorem 1 follows from Lemma 1.
Shirshov algorithm. If a monic subset is not a GS basis then we can add to all nontrivial compositions, making them monic. Iterating this process, we eventually obtain a GS basis that contains and generates the same ideal, . This is called the GS completion of . Using the reduction algorithm (elimination of the leading words of polynomials), we may obtain a minimal GS basis or a reduced GS basis.
The following theorem gives a linear basis for the ideal provided that is a GS basis.
Theorem 2
If is a Gröbner–Shirshov basis then, given , by Lemma 2 there exists with (if ) such that and the set is a linear basis for the ideal of .
Proof
Take . Then by the CD-lemma for associative algebras, for some and , which implies that . Put , where is the coefficient of the leading term of and or . Then and . By induction on , the set generates as a linear space. It is clear that is a linearly independent set.
Theorem 3
Choose a monomial ordering on . For every ideal of there exists a unique reduced Gröbner–Shirshov basis for .
Proof
Clearly, a Gröbner–Shirshov basis for the ideal exists; for example, we may take . By Theorem 1, we may assume that the leading terms of the elements of are distinct. Given , put
and .
For every we show that there exists an such that .
In fact, Theorem 1 implies that for some and . Suppose that . Then we have , say, . Therefore, and for some . We claim that . Otherwise, . It follows that and so we have the infinite descending chain
which contradicts the assumption that is a well ordering.
Suppose that . Then, by the argument above, there exists such that and . Since is a well ordering, there must exist such that .
Put , where is the coefficient of the leading term of . Then and .
By induction on , we know that , and hence . Moreover, Theorem 1 implies that is clearly a minimal GS basis for the ideal .
Assume that is a minimal GS basis for .
For every we have , where and . Since is a minimal GS basis, it follows that for every .
We claim that is a reduced GS basis for . In fact, it is clear that . By Theorem 1, for every we have for some .
Take two reduced GS bases and for the ideal . By Theorem 1, for every ,
for some , , and , and hence . Since , we have . It follows that , and so .
If then . By Theorem 1, for some with . This means that and . Noting that , we have either or . If then , which contradicts ; if then , which contradicts . This shows that , and then . Similarly, .
Remark 1
In fact, a reduced GS basis is unique (up to the ordering) in all possible cases below.
Remark 2
Both associative and Lie CD-lemmas are valid when we replace the base field by an arbitrary commutative ring with identity because we assume that all GS bases consist of monic polynomials. For example, consider a Lie algebra over which is a free -module with a well-ordered -basis . With the deg-lex ordering on , the universal enveloping associative algebra has a (monic) GS basis
where and in , and the CD-lemma for associative algebras over implies that and
is a -basis for .
In fact, for the same reason, all CD-lemmas in this survey are valid if we replace the base field by an arbitrary commutative ring with identity. If this is the case then claim (iii) in the CD-lemma should read: is a free -module with a -basis . But in the general case, Shirshov’s algorithm fails: if is a monic set then , the set obtained by adding to all non-trivial compositions, is not a monic set in general, and the algorithm may stop with no result.
2.2 Gröbner bases for commutative algebras and their lifting to Gröbner–Shirshov bases
Consider the free commutative associative algebra . Given a well ordering on ,
is a linear basis for .
Choose a monomial ordering on . Take two monic polynomials and in such that for some with (so, and are not coprime in ). Then is called the -polynomial of and .
A monic subset is called a Gröbner basis with respect to the monomial ordering whenever all -polynomials of two arbitrary polynomials in are trivial modulo and corresponding .
An argument similar to the proof of the CD-lemma for associative algebras justifies the following theorem due to Buchberger.
Theorem 4
(Buchberger Theorem) Choose a monomial ordering on . Consider a monic set and the ideal of generated by . The following statements are equivalent:
-
(i)
is a Gröbner basis in .
-
(ii)
for some and .
-
(iii)
is a linear basis for the algebra .
Proof
Denote by be the usual (unique) least common multiple of two commutative words :
If is a common multiple of and then .
The -polynomial of two monic polynomials and is
An analogue of Lemma 1 is valid for because if for two monic polynomials and then
Lemma 1 implies the main claim (i)(ii) of Buchberger’s theorem.
Theorem 5
Given an ideal of and a monomial ordering on , there exists a unique reduced Gröbner basis for . Moreover, if is finite then so is .
Eisenbud et al. [99] constructed a GS basis in by lifting a commutative Gröbner basis for and adding all commutators. Write and put
Consider the natural map carrying to and the lexicographic splitting of , which is defined as the -linear map
Given , we express it as , where , using an arbitrary monomial ordering on .
Following [99], define an ordering on using the ordering as follows: given , put
It is easy to check that this is a monomial ordering on and for every . Moreover, for every .
Consider an arbitrary ideal of generated by monomials. Given , denote by the set of all monomials such that neither nor lie in .
Theorem 6
([99]) Consider the orderings on and defined above. If is a minimal Gröbner basis in then is a minimal Gröbner–Shirshov basis in , where is the monomial ideal of generated by .
Jointly with Yongshan Chen [30], we generalized this result to lifting a GS basis , see Mikhalev and Zolotykh [170], to a GS basis of ) of .
Recall that for a prime number the Gauss ordering on the natural numbers is described as whenever . Let be the usual ordering on the natural numbers. A monomial ideal of is called -Borel-fixed whenever it satisfies the following condition: for each monomial generator of , if is divisible by but no higher power of then for all and .
Thus, we have the following Eisenbud–Peeva–Sturmfels lifting theorem.
Theorem 7
([99]) Given an ideal of , take and .
-
(i)
If is -Borel-fixed then a minimal Gröbner–Shirshov basis of is obtained by applying to a minimal Gröbner basis of and adding commutators.
-
(ii)
If is -Borel-fixed for some then has a finite Gröbner–Shirshov basis.
Proof
Assume that is -Borel-fixed for some . Take a generator of , where , and suppose that is the highest power of dividing . Since , it follows that for . This implies that for , and hence, every monomial in satisfies for . Thus, is a finite set, and the result follows from Theorem 6. In particular, if then .
In characteristic observe that if the field is infinite then after a generic change of variables is -Borel-fixed. Then Theorems 6 and 7 imply
Corollary 1
([99]) Consider an infinite field and an ideal . After a general linear change of variables, the ideal in has a finite Gröbner–Shirshov basis.
2.3 Composition-Diamond lemma for modules
Consider , and , . Kang and Lee define [123] the composition of and as follows.
Definition 1
-
(a)
If there exist such that with then the intersection composition is defined as .
-
(b)
If there exist , such that then the inclusion composition is defined as .
-
(c)
The composition is called right-justified whenever for some .
If , where , , , and with and for all and , then we call trivial with respect to and and write .
Definition 2
([123, 124]) A pair of monic subsets of is called a GS pair if is closed under composition, is closed under right-justified composition with respect to , and given , , and such that if is defined, we have . In this case, say that is a GS pair for the -module , where .
Theorem 8
(Kang and Lee [123, 124], the CD-lemma for cyclic modules) Consider a pair of monic subsets of , the associative algebra defined by , and the left cyclic module defined by . Suppose that is a Gröbner–Shirshov pair for the -module and . Then or , where , , and .
Applications of Theorem 8 appeared in [125–127].
Take two sets and and consider the free left -module with -basis . Then is called a double-free module. We now define the GS basis in . Choose a monomial ordering on , and a well-ordering on . Put and define an ordering on as follows: for any , ,
Given with all monic, define composition in to be only inclusion composition, which means that for some , where . If , where , , , and , then this composition is called trivial modulo .
Theorem 9
(Chibrikov [90], see also [78], the CD-lemma for modules) Consider a non-empty set with all monic and choose an ordering on as before. The following statements are equivalent:
-
(i)
is a Gröbner–Shirshov basis in .
-
(ii)
If then for some and .
-
(iii)
is a linear basis for the quotient .
Outline of the proof. Take and express it as with and . Put
where . Up to the order of and , we have .
The composition of two monic elements is
If for monic and then . This gives an analogue of Lemma 1 for modules and the implication (i)(ii) of Theorem 9.
Given , put . We can regard every left -module as a -module in a natural way: for and . Observe that is an epimorphic image of some free -module. Assume now that , where . Put
and . Then as -modules.
Theorem 10
Given a submodule of and a monomial ordering on as above, there exists a unique reduced Gröbner–Shirshov basis for .
Corollary 2
(Cohn) Every left ideal of is a free left -module.
Proof
Take a reduced Gröbner–Shirshov basis of as a -submodule of the cyclic -module. Then is a free left -module with a -basis .
As an application of the CD-lemma for modules, we give GS bases for the Verma modules over the Lie algebras of coefficients of free Lie conformal algebras. We find linear bases for these modules.
Let be a set of symbols. Take the constant locality function ; that is, for all . Put and consider the Lie algebra over a field of characteristic 0 generated by with the relations
For every , put . It is well-known that these elements generate a free Lie conformal algebra with data (see [194]). Moreover, the coefficient algebra of is just .
Suppose that is linearly ordered. Define an ordering on as
We use the deg-lex ordering on . It is clear that the leading term of each polynomial in is with
The following lemma is essentially from [194].
Lemma 3
([78]) With the deg-lex ordering on , the set is a GS basis in .
Corollary 3
([78]) A linear basis of the universal enveloping algebra of consists of the monomials
with and such that for every we have
An -module is called restricted if for all and there is some integer such that for .
An -module is called a highest weight module whenever it is generated over by a single element satisfying , where is the subspace of generated by . In this case is called a highest weight vector.
Let us now construct a universal highest weight module over , which is often called the Verma module. Take the trivial -dimensional -module generated by ; hence, for all . Clearly,
Then has the structure of the highest weight module over with the action given by multiplication on and a highest weight vector . In addition, is the universal enveloping vertex algebra of and the embedding is given by (see also [194]).
Theorem 11
([78]) With the above notions, a linear basis of consists of the elements
satisfying the condition in Corollary 3 and .
Proof
Clearly, as -modules, we have
where . In order to show that is a Gröbner–Shirshov basis, we only need to verify that , where . Take
Then since , , , and . It follows that is a Gröbner–Shirshov basis. Now, the result follows from the CD-lemma for modules.
2.4 Composition-Diamond lemma for categories
Denote by an oriented multi-graph. A path
in with edges is a partial word on with source and target . Denote by the free category generated by (the set of all partial words (paths) on with partial multiplication, the free ‘partial path monoid’ on ). A well-ordering on is called monomial whenever it is compatible with partial multiplication.
A polynomial is a linear combination of partial words with the same source and target. Then is the partial path algebra on (the free associative partial path algebra generated by ).
Given , denote by the minimal subset of that includes and is closed under the partial operations of addition and multiplication. The elements of are of the form with , , and , and all -words have the same source and target.
Both inclusion and intersection compositions are possible.
With these differences, the statement and proof of the CD-lemma are the same as for the free associative algebra.
Theorem 12
([36], the CD-lemma for categories) Consider a nonempty set of monic polynomials and a monomial ordering on . Denote by the ideal of generated by . The following statements are equivalent:
-
(i)
The set is a Gröbner–Shirshov basis in .
-
(ii)
for some and .
-
(iii)
the set is a linear basis for , which is denoted by .
Outline of the proof.
Define and the general composition for and by the same formulas as above. Under the conditions of the analogue of Lemma 1, we again have , where and This implies the analogue of Lemma 1 and the main assertion (i)(ii) of Theorem 12.
Let us present some applications of CD-lemma for categories.
For each non-negative integer , denote by the set of integers in their usual ordering. A (weakly) monotonic map is a function from to such that implies . The objects with weakly monotonic maps as morphisms constitute the category called the simplex category. It is convenient to use two special families of monotonic maps,
defined for (and for in the case of ) by
Take the oriented multi-graph with
Consider the relation consisting of:
This yields a presentation of the simplex category .
Order now as follows.
Firstly, for put iff or ( and ).
Secondly, for
(these are all possible words on , including the empty word , where ), define
Then, for put iff lexicographically.
Thirdly, for , put iff or ( and ).
Finally, for , where , and put . Then for every ,
It is easy to check that is a monomial ordering on . Then we have
Theorem 13
([36]) For and defined above, with the ordering on , the set is a Gröbner–Shirshov basis for the simplex partial path algebra .
Corollary 4
([157]) Every morphism of the simplex category has a unique expression of the form
with , , and .
The cyclic category is defined by generators and relations as follows, see [104]. Take the oriented (multi) graph with and
Consider the relation consisting of:
The category is called the cyclic category and denoted by .
Define an ordering on as follows.
Firstly, for , put iff or ( and ).
Secondly, for put iff or ( and ).
Thirdly, for
where , put
Then for every put iff lexicographically.
Fourthly, for , iff or ( and ).
Finally, for and define
Then for every put
It is also easy to verify that is a monomial ordering on which extends . Then we have
Theorem 14
([36]) Consider and defined as the above. Put and . Then
-
(1)
With the ordering on , the set is a Gröbner–Shirshov basis for the cyclic category .
-
(2)
Every morphism of the cyclic category has a unique expression of the form
with , , , and .
2.5 Composition-Diamond lemma for associative algebras over commutative algebras
Given two well-ordered sets and , put
and denote by the -space spanned by . Define the multiplication of words as
This makes an algebra isomorphic to the tensor product , called a‘double free associative algebra’. It is a free object in the category of all associative algebras over all commutative algebras (over ): every associative algebra over a commutative algebra is isomorphic to as a -algebra and a -algebra.
Choose a monomial ordering on . The following definitions of compositions and the GS basis are taken from [170].
Take two monic polynomials and in and denote by the least common multiple of and .
-
1.
Inclusion. Assume that is a subword of , say, for some . If then and if then we set . Put . Define the composition .
-
2.
Overlap. Assume that a non-empty beginning of is a non-empty ending of , say, , , and for some and . Put . Define the composition .
-
3.
External. Take a (possibly empty) associative word . In the case that the greatest common divisor of and is non-empty and both and are non-empty, put and define the composition .
A monic subset of is called a GS basis whenever all compositions of elements of , say , are trivial modulo :
where , , , and for all .
Theorem 15
(Mikhalev and Zolotykh [170, 228], the CD-lemma for associative algebras over commutative algebras) Consider a monic subset and a monomial ordering on . The following statements are equivalent:
-
(i)
The set is a Gröbner–Shirshov basis in .
-
(ii)
For every element , the monomial contains as its subword for some .
-
(iii)
The set is a linear basis for the quotient .
Outline of the proof. For
the general composition is
where are -monic with and . Moreover, whenever with , that is, is a trivial least common multiple relative to both -words and -words. This implies the analog of Lemma 1 and the claim (i)(ii) in Theorem 15.
We apply this lemma in Sect. 4.3.
2.6 PBW-theorem for Lie algebras
Consider a Lie algebra over a field with a well-ordered linear basis and multiplication table , where for every we write with . Then is called the universal enveloping associative algebra of , where .
Theorem 16
(PBW Theorem) In the above notation and with the deg-lex ordering on , the set is a Gröbner–Shirshov basis of . Then by the CD-lemma for associative algebras, the set consists of the elements
and constitutes a linear basis of .
Theorem 17
(The PBW Theorem in Shirshov’s form) Consider with and . The following statements are equivalent.
-
(i)
For the deg-lex ordering, is a GS basis of .
-
(ii)
For the deg-lex ordering, is a GS basis of .
-
(iii)
A linear basis of consists of the words , where in the lex ordering, , and every is an -irreducible associative Lyndon–Shirshov word in .
-
(iv)
A linear basis of is the set of all -irreducible Lyndon–Shirshov Lie monomials in .
-
(v)
A linear basis of consists of the polynomials , where in the lex ordering, , and every is an -irreducible non-associative Lyndon–Shirshov word in .
The PBW theorem, Theorem 33, the CD-lemmas for associative and Lie algebras, Shirshov’s factorization theorem, and property (VIII) of Sect. 4.2 imply that every LS-subword of is a subword of some .
Makar–Limanov gave [158] an interesting form of the PBW theorem for a finite dimensional Lie algebra.
2.7 Drinfeld–Jimbo algebra , Kac–Moody enveloping algebra , and the PBW basis of
Take an integral symmetrizable Cartan matrix . Hence, , for , and there exists a diagonal matrix with diagonal entries , which are nonzero integers, such that the product is symmetric. Fix a nonzero element of with for all . Then the Drinfeld–Jimbo quantum enveloping algebra is
where
and
Theorem 18
([55]) For every symmetrizable Cartan matrix , with the deg-lex ordering on , the set is a Gröbner–Shirshov basis of the Drinfeld–Jimbo algebra , where and are the Shirshov completions of and .
Corollary 5
(Rosso [195], Yamane [220]) For every symmetrizable Cartan matrix we have the triangular decomposition
with and .
Similar results are valid for the Kac–Moody Lie algebras and their universal enveloping algebras
where are the same as for ,
and .
Theorem 19
([55]) For every symmetrizable Cartan matrix , the set is a Gröbner–Shirshov basis of the universal enveloping algebra of the Kac–Moody Lie algebra .
The PBW theorem in Shirshov’s form implies
Corollary 6
(Kac [117]) For every symmetrizable Cartan matrix , we have the triangular decomposition
Poroshenko [179, 180] found GS bases for the Kac–Moody algebras of types , , , and . He used the available linear bases of the algebras [117].
Consider now
and assume that . Introduce new variables, defined by Jimbo (see [220]), which generate :
where
Order the set as follows: Recall from Yamane [220] the notation
Consider the set consisting of Jimbo’s relations:
It is easy to see that .
A direct proof [86] shows that is a GS basis for [55]. The proof is different from the argument of Bokut and Malcolmson [55]. This yields
Theorem 20
([55]) In the above notation and with the deg-lex ordering on , the set is a Gröbner–Shirshov basis of
Corollary 7
([195, 220]) For , a linear basis of consists of
with , , and .
3 Gröbner–Shirshov bases for groups and semigroups
In this section we apply the method of GS bases for braid groups in different sets of generators, Chinese monoids, free inverse semigroups, and plactic monoids in two sets of generators (row words and column words).
Given a set consider the congruence on generated by , the quotient semigroup
and the semigroup algebra . Identifying the set with , it is easy to see that
is an algebra isomorphism.
The Shirshov completion of consists of semigroup relations, . Then is a linear basis of , and so is a linear basis of . This shows that consists precisely of the normal forms of the elements of the semigroup .
Therefore, in order to find the normal forms of the semigroup , it suffices to find a GS basis in . In particular, consider a group , where and is the free group on a set . Then has a presentation
as a semigroup.
3.1 Gröbner–Shirshov bases for braid groups
Consider the Artin braid group of type (Artin [5]). We have
3.1.1 Braid groups in the Artin–Burau generators
Assume that with and well-ordered and that the ordering on is monomial. Then every word in has the form , where , , and . Define the inverse weight of the word as
and the inverse weight lexicographic ordering as
Call this ordering the inverse tower ordering for short. Clearly, it is a monomial ordering on .
When , , and is endowed with the inverse tower ordering, define the inverse tower ordering on with respect to the presentation In general, for
with -words equipped with a monomial ordering we can define the inverse tower ordering of -words.
Introduce a new set of generators for the braid group , called the Artin–Burau generators. Put
Form the sets
Then the set
generates as a semigroup.
Order now the alphabet as
and
Order -words by the deg-inlex ordering; that is, first compare words by length and then by the inverse lexicographic ordering starting from their last letters. Then we use the inverse tower ordering of -words.
Lemma 4
(Artin [6], Markov [160]) The following Artin–Markov relations hold in the braid group :
where ;
where and ;
where or , and , .
Theorem 21
([25]) The Artin–Markov relations (1)–(13) form a Gröbner–Shirshov basis of the braid group in terms of the Artin–Burau generators with respect to the inverse tower ordering of words.
It is claimed in [25] that some compositions are trivial. Processing all compositions explicitly, [82] supported the claim.
Corollary 8
(Markov–Ivanovskii [6]) The following words are normal forms of the braid group :
where all for are free irreducible words in .
3.1.2 Braid groups in the Artin–Garside generators
The Artin–Garside generators of the braid group are (Garside [103] 1969), where with .
Putting , order by the deg-lex ordering.
Denote by , for positive words in the letters , assuming that , .
Given , for denote by the result of shifting the indices of all letters in by : , and put . Define for , while and .
Theorem 22
([23, 47]) A Gröbner–Shirshov basis of in the Artin–Garside generators consists of the following relations:
where and ; moreover, begins with unless it is empty, and .
There are corollaries.
Corollary 9
The -irreducible normal form of each word of coincides with its Garside normal form [103].
Corollary 10
(Garside [103]) The semigroup of positive braids can be embedded into a group.
3.1.3 Braid groups in the Birman–Ko–Lee generators
Recall that the Birman–Ko–Lee generators of the braid group are
and we have the presentation
Denote by the Garside word, .
Define the order as iff lexicographically. Use the deg-lex ordering on .
Instead of , we write simply or . We also set
where . In this notation, we can write the defining relations of as
where either or .
Denote by , where , a positive word in satisfying . We can use any capital Latin letter with indices instead of , and appropriate indices (for instance, and with ) instead of and . Use also the following equalities in :
for , where
for , where
Theorem 23
([24]) A Gröbner–Shirshov basis of the braid group in the Birman–Ko–Lee generators consists of the following relations:
where means, as above, a word in satisfying , , and .
There are two corollaries.
Corollary 11
(Birman et al. [13]) The semigroup of positive braids in the Birman–Ko–Lee generators embeds into a group.
Corollary 12
(Birman et al. [13]) The -irreducible normal form of a word in in the Birman–Ko–Lee generators coincides with the Birman–Ko–Lee–Garside normal form , where .
3.1.4 Braid groups in the Adjan–Thurston generators
The symmetric group has the presentation
Bokut and Shiao [58] found the normal form for in the following statement: the set is a Gröbner–Shirshov normal form for in the generators relative to the deg-lex ordering, where for and .
Take with the normal form . Define the length of as and write whenever . Moreover, every has a unique expression with all . The number is called the breadth of .
Now put
where stands for a letter with index .
Then for the braid group with generators we have . Indeed, define
These mappings are homomorphism satisfying and . Hence,
Put . These generators of are called the Adjan–Thurston generators.
Then the positive braid semigroup generated by is
Assume that . Define if and only if or and . Clearly, this is a well-ordering on . We will use the deg-lex ordering on .
Theorem 24
([89]) The Gröbner–Shirshov basis of in the Adjan–Thurston generator relative to the deg-lex ordering on consists of the relations
Theorem 25
([89]) The Gröbner–Shirshov basis of in the Adjan–Thurston generator with respect to the deg-lex ordering on consists of the relations
-
(1)
-
(2)
-
(3)
-
(4)
-
(5)
=1.
Corollary 13
(Adjan–Thurston) The normal forms for are for , where is minimal in the deg-lex ordering.
3.2 Gröbner–Shirshov basis for the Chinese monoid
The Chinese monoid over a well-ordered set has the presentation , where and consists of the relations
Theorem 26
([85]) With the deg-lex ordering on , the following relations (1)–(5) constitute a Gröbner–Shirshov basis of the Chinese monoid :
-
(1)
,
-
(2)
,
-
(3)
,
-
(4)
,
-
(5)
,
where and .
Denote by the set consistsing of the words on of the form with , where
for with , and all exponents are non-negative.
Corollary 14
([71]) This is a set of normal forms of elements of the Chinese monoid .
3.3 Gröbner–Shirshov basis for free inverse semigroup
Consider a semigroup . An element is called an inverse of whenever and . An inverse semigroup is a semigroup in which every element has a unique inverse, denoted by .
Given a set , put . On assuming that , denote by . Define the formal inverses of the elements of as
It is well known that
is the free inverse semigroup (with identity) generated by .
Introduce the notions of a formal idempotent, a (prime) canonical idempotent, and an ordered (prime) canonical idempotent in . Assume that is a well-ordering on .
-
(i)
The empty word 1 is an idempotent.
-
(ii)
If is an idempotent and then is both an idempotent and a prime idempotent.
-
(iii)
If , where , are prime idempotents then is an idempotent.
-
(iv)
An idempotent is called canonical whenever avoids subwords of the form , where , both and are idempotents.
-
(v)
A canonical idempotent is called ordered if every subword of with and being idempotents satisfies , where is the first letter of .
Theorem 27
([44]) Denote by the subset of consisting two kinds of polynomials:
-
, where and are ordered prime canonical idempotents with ;
-
, where , , and are ordered prime canonical idempotents.
Then, with the deg-lex ordering on , the set is a Gröber–Shirshov basis of the free inverse semigroup .
Theorem 28
([44]) The normal forms of elements of the free inverse semigroup are
where , and avoids subwords of the form for , while are ordered canonical idempotents such that the first (respectively last) letter of , for is not equal to the first (respectively last) letter of (respectively ).
The above normal form is analogous to the semi-normal forms of Poliakova and Schein [176], 2005.
3.4 Approaches to plactic monoids via Gröbner–Shirshov bases in row and column generators
Consider the set of elements with the ordering . Schützenberger called a plactic monoid (see also Lothaire [153], Chapter 5), where consists of the Knuth relations
A nondecreasing word is called a row and a strictly decreasing word is called a column; for example, is a row and is a column.
For two rows say that dominates whenever and every letter of is greater than the corresponding letter of , where is the length of .
A (semistandard) Young tableau on (see [152]) is a word in such that dominates for all . For example,
is a Young tableau.
Cain et al. [69] use the Schensted–Knuth normal form (the set of (semistandard) Young tableaux) to prove that the multiplication table of column words, , forms a finite GS basis of the finitely generated plactic monoid. Here the Young tableaux is the output of the column Schensted algorithm applied to , but is not made explicit.
In this section we give new explicit formulas for the multiplication tables of row and column words. In addition, we give independent proofs that the resulting sets of relations are GS bases in row and column generators respectively. This yields two new approaches to plactic monoids via their GS bases.
3.4.1 Plactic monoids in the row generators
Consider the plactic monoid , where with . Denote by the set of non-negative integers. It is convenient to express the rows as , where for is the number of occurrences of the letter . For example, .
Denote by the set of all rows in and order as follows. Given , define the length of in .
Firstly, order : for every , put if and only if or and lexicographically. Clearly, this is a well-ordering on . Then, use the deg-lex ordering on .
Lemma 5
([29]) Take . For put
where (, , , and , see below) stands for a lowercase symbol, and ( , , , and , see below) for the corresponding uppercase symbol. Take and in . Put and , where
for and .
Then in and is a Young tableau on , which could have only one row, that is, . Moreover,
where .
We should emphasize that gives explicitly the product of two rows obtained by the Schensted row algorithm.
Jointly with our students Weiping Chen and Jing Li we proved [29], independently of Knuth’s normal form theorem [137], that is a GS basis of the plactic monoid algebra in row generators with respect to the deg-lex ordering. In particular, this yields a new proof of Knuth’s theorem.
3.4.2 Plactic monoids in the column generators
Consider the plactic monoid , where with . Every Young tableaux is a product of columns. For example,
is a Young tableau.
Given a column , denote by the number of occurrences of the letter in . Then for . We write . For example,
Put . For define . Order as follows: for , put if and only if lexicographically. Then, use the deg-lex ordering on .
For , put , where stands for some lowercase symbol defined above and stands for the corresponding uppercase symbol.
Lemma 6
([29]) Take , . Define and , where
for and . Then and in , and is a Young tableau on . Moreover,
where .
Equation gives explicitly the product of two columns obtained by the Schensted column algorithm.
Jointly with our students Weiping Chen and Jing Li we proved [29], independently of Knuth’s normal form theorem [137], that is a GS basis of the plactic monoid algebra in column generators with respect to the deg-lex ordering. In particular, this yields another new proof of Knuth’s theorem. Previously Cain, Gray, and Malheiro [69] established the same result using Knuth’s theorem, and they did not find explicitly.
Remark All results of [29] are valid for every plactic monoid, not necessarily finitely generated.
4 Gröbner–Shirshov bases for Lie algebras
In this section we first give a different approach to the LS basis and the Hall basis of a free Lie algebra by using Shirshov’s CD-lemma for anti-commutative algebras. Then, using the LS basis, we construct the classical theory of GS bases for Lie algebras over a field. Finally, we mention GS bases for Lie algebras over a commutative algebra and give some applications.
4.1 Lyndon–Shirshov basis and Lyndon–Shirshov words in anti-commutative algebras
A linear space equipped with a bilinear product is called an anti-commutative algebra if it satisfies the identity , and so for every .
Take a well-ordered set and denote by the set of all non-associative words. Define three orderings , , and (non-associative deg-lex) on . For put
-
(here or is empty when or ) iff one of the following holds:
-
(a)
in the lex ordering;
-
(b)
and ;
-
(c)
, , and ;
-
(a)
-
iff one of the following holds:
-
(a)
in the deg-lex ordering;
-
(b)
and ;
-
(c)
, , and ;
-
(a)
-
iff one of the following holds:
-
(a)
;
-
(b)
if , , and then or ( and ).
-
(a)
Define regular words by induction on :
-
(i)
is a regular word.
-
(ii)
is regular if both and are regular and .
Denote by whenever is regular.
The set of all regular words on constitutes a linear basis of the free anti-commutative algebra on .
The following result gives an alternative approach to the definition of LS words as the radicals of associative supports of the normal words .
Theorem 29
([37]) Suppose that is a regular word of the anti-commutative algebra . Then , where is a Lyndon–Shirshov word in and . Moreover, the set of associative supports of the words in includes the set of all Lyndon–Shirshov words in .
Fix an ordering on and choose monic polynomials and in . If there exist such that then the inclusion composition of and is defined as .
A monic subset of is called a GS basis in if every inclusion composition in is trivial modulo .
Theorem 30
(Shirshov’s CD-lemma for anti-commutative algebras, cf. [206]) Consider a nonempty set of monic polynomials with the ordering on . The following statements are equivalent:
-
(i)
The set is a Gröbner–Shirshov basis in .
-
(ii)
If then for some , where is a normal -word.
-
(iii)
The set
is a linear basis of the algebra .
Define the subset the free anti-commutative algebra as
It is easy to prove that the free Lie algebra admits a presentation as an anti-commutative algebra: .
The next result gives an alternating approach to the definition of the LS basis of a free Lie algebra as a set of irreducible non-associative words for an anti-commutative GS basis in .
Theorem 31
([37]) Under the ordering , the subset of is an anti-commutative Gröbner–Shirshov basis in . Then is the set of all non-associative LS words in . So, the LS monomials constitute a linear basis of the free Lie algebra .
Theorem 32
([34]) Define by analogy with , but using instead of . Then with the ordering the subset of is also an anti-commutative GS basis. The set amounts to the set of all Hall words in and forms a linear basis of a free Lie algebra .
4.2 Composition-Diamond lemma for Lie algebras over a field
We start with some concepts and results from the literature concerning the theory of GS bases for the free Lie algebra generated by over a field .
Take a well-ordered set with whenever , for all . Given , define the length (or degree) of to be and denote it by or , put , and introduce
Order the new alphabet as follows:
Assuming that
where , define the Shirshov elimination
We use two linear orderings on :
-
(i)
the lex ordering (or lex-antideg ordering): if and, by induction, if and then if and only if or and ;
-
(ii)
the deg-lex ordering: if or and .
Remark In commutative algebras, the lex ordering is understood to be the lex-deg ordering with the condition for .
We cite some useful properties of ALSWs and NLSWs (see below) following Shirshov [203, 204, 207], see also [209]. Property (X) was given by Shirshov [204] and Chen et al. [72]. Property (VIII) was implicitly used in Shirshov [207], see also Chibrikov [94].
We regard as the Lie subalgebra of the free associative algebra generated by with the Lie bracket . Below we prove that is the free Lie algebra generated by for every commutative ring (Shirshov [203]). For a field, this follows from the PBW theorem because the free Lie algebra has the universal enveloping associative algebra .
Given , denote by the leading word of with respect to the deg-lex ordering and write with .
Definition 3
([156, 203]) Refer to as an associative Lyndon–Shirshov word, or ALSW for short, whenever
Denote the set of all ALSWs on by .
Associative Lyndon–Shirshov words enjoy the following properties (Lyndon [156], Chen et al. [72], Shirshov [203, 204]).
(I) Put . If and then
(II) (Shirshov’s key property of ALSWs) A word is an ALSW in if and only if is an ALSW in .
Properties (I) and (II) enable us to prove the properties of ALSWs and NLSWs (see below) by induction on length.
(III) (down-to-up bracketing) , where and . In the process we use the algorithm of joining the minimal letters of , to the previous words.
(IV) If then .
(V) (for every and ).
(VI) If then an arbitrary proper prefix of cannot be a suffix of and if .
(VII) (Shirshov’s factorization theorem) Every associative word can be uniquely represented as , where and .
Actually, if we apply to the algorithm of joining the minimal letter to the previous word using the Lie product, , then after finitely many steps we obtain , with , and would be the required factorization (see an example in the Introduction).
(VIII) If an associative word is represented as in (VII) and is a LS subword of then is a subword of one of the words , .
(IX) If and are ALSWs then so is provided that .
(X) If is an ALSW and is its longest proper ALSW ending, then is an ALSW as well (Chen et al. [72], Shirshov [204]).
Definition 4
(down-to-up bracketing of ALSW, Shirshov [203]) For an ALSW , there is the down-to-up bracketing , where each time we join the minimal letter of the previous word using Lie multiplication. To be more precise, we use the induction .
Definition 5
(up-to-down bracketing of ALSW, Shirshov [204], Chen et al. [72]) For an ALSW , we define the up-to-down Lie bracketing by the induction , where as in (X).
(XI) If then .
(XII) Shirshov’s definition of a NLSW (non-associative LS word) below is the same as and ; that is, . Chen et al. [72] used .
Definition 6
(Shirshov[203]) A non-associative word in is a NLSW if
-
(i)
is an ALSW;
-
(ii)
if then both and are NLSWs (then (IV) implies that );
-
(iii)
if then .
Denote the set of all NLSWs on by .
(XIII) If and then in .
(XIV) The set is linearly independent in for every commutative ring .
(XV) is a set of linear generators in every Lie algebra generated by over an arbitrary commutative ring .
(XVI) is the free Lie algebra over the commutative ring with the -basis .
(XVII) (Shirshov’s special bracketing [203]) Consider with . Then
-
(i)
where and possibly .
-
(ii)
Express in the form where and . Replacing by , we obtain the word
which is called the Shirshov special bracketing of relative to .
-
(iii)
in with and satisfying , and hence .
Outline of the proof. Put . Then in , where and is an ALSW. Claim (i) follows from (II) by induction on length. The same applies to claim (iii).
(XVIII) (Shirshov’s Lie elimination of the leading word) Take two monic Lie polynomials and with for some . Then is a Lie polynomial with smaller leading word, and so .
(XIX) (Shirshov’s double special bracketing) Assume that with . Then there exists a bracketing such that and .
More precisely, if , and
if , where is the Shirshov factorization of and is a subword of . In both cases in , where .
(XX) (Shirshov’s algorithm for recognizing Lie polynomials, cf. the Dynkin–Specht–Wever and Friedrich algorithms). Take . Then is an ALSW and is a Lie polynomial with a smaller maximal word (in the deg-lex ordering), , where . Then . Consequently, if and only if after finitely many steps we obtain
Here can be an arbitrary commutative ring.
Definition 7
Consider with all monic. Take and . If is an ALSW then we call a special normal -word (or a special normal -word), where is defined in (XVII) (ii). A Lie -word is called a normal -word whenever . Every special normal -word is a normal -word by (XVII) (iii).
For there are two kinds of Lie compositions:
-
(i)
If for some then the polynomial is called the inclusion composition of and with respect to .
-
(ii)
If is a word satisfying for some with then the polynomial is called the intersection composition of and with respect to , and is an ALSW by (IX).
Given a Lie polynomial and , say that is trivial modulo and write whenever , where each is a normal -word and .
A set is called a GS basis in if every composition of polynomials and in is trivial modulo and .
(XXI) If is monic and is a normal -word then , where .
A proof of (XXI) follows from the CD-lemma for associative algebras since is an associative GS basis by (IV).
(XXII) Given two monic Lie polynomials and , we have
Proof
If and are intersection compositions, where , then (XIII) and (XVII) yield
where . Hence,
In the case of inclusion compositions we arrive at the same conclusion.
Theorem 33
(PBW Theorem in Shirshov’s form [56, 57], see Theorem 17) A nonempty set of monic Lie polynomials is a Gröbner–Shirshov basis in if and only if is a Gröbner–Shirshov basis in .
Proof
Observe that, by definition, for any the composition lies in if and only if it lies .
Assume that is a GS basis in . Then we can express every composition as where are normal -words and . By (XXI), we have with . Therefore, (XXII) yields Thus, is a GS basis in .
Conversely, assume that is a GS basis in . Then the CD-lemma for associative algebras implies that for some and . Then is a Lie polynomial and . Induction on yields
Theorem 34
(The CD-lemma for Lie algebras over a field) Consider a nonempty set of monic Lie polynomials and denote by the ideal of generated by . The following statements are equivalent:
-
(i)
The set is a Gröbner–Shirshov basis in .
-
(ii)
If then for some and .
-
(iii)
The set
is a linear basis for .
Proof
(i)(ii). Denote by and the ideals of and generated by respectively. Since , Theorem 33 and the CD-lemma for associative algebras imply the claim.
(ii)(iii). Suppose that in with and , that is, . Then all must vanish. Otherwise we may assume that . Then and (ii) implies that , which is a contradiction. On the other hand, by the next property (XXIII), generates as a linear space.
(iii)(i). This part follows from (XXIII).
The next property is similar to Lemma 2.
(XXIII) Given , we can express every as
with , satisfying , and are special normal -word satisfying .
(XXIV) Given a normal -word , take . Then . It follows that is equivalent to , where are special normal -words with .
Proof
Observe that for every monic Lie polynomial , the set is a GS basis in . Then (XVIII) and the CD-lemma for Lie algebras yield .
Summary of the proof of Theorem 34.
Given two ALSWs and , define the ALSW- (or for short) as follows:
Denote by the Shirshov double special bracketing of in the case that is a trivial , by and the Shrishov special bracketings of if is an inclusion or intersection respectively. Then we can define a general Lie composition for monic Lie polynomials and with and as
if is a trivial (it is ), and
if is an inclusion or intersection .
If is a Lie GS basis then is an associative GS basis. This follows from property (XVII) (iii) and justifies the claim (i)(ii) of Theorem 34.
Shirshov’s original proof of (i)(ii) in Theorem 34, (see [207, 209]), rests on an analogue of Lemma 1 for Lie algebras.
Lemma 7
([207, 209]) If are normal -words with equal leading associative words, , then they are equal , that is,
Outline of the proof. We have and . Shirshov’s (double) special bracketing lemma yields
with . The ALSW includes and as subwords, and so there is a bracketing such that
are normal - and - words with the same leading associative word . Then
Now it is enough to prove that two normal Lie -words with the same leading associative words, say , are equal :
Since , we have by the CD-lemma for associative algebras with one Lie polynomial relation . Then is a Lie polynomial with the leading associative word smaller than . Induction on finishes the proof.
4.2.1 Gröbner–Shirshov basis for the Drinfeld–Kohno Lie algebra
In this section we give a GS basis for the Drinfeld–Kohno Lie algebra .
Definition 8
Fix an integer . The Drinfeld–Kohno Lie algebra over is defined by generators for distinct indices satisfying the relations and for distinct , , , and .
Therefore, we have the presentation , where and consists of the following relations:
Order by setting if either or and . Let be the deg-lex ordering on .
Theorem 35
([80]) With {(18), (19), (20)} as before and the deg-lex ordering on , the set is a Gröbner–Shirshov basis of .
Corollary 15
The Drinfeld–Kohno Lie algebra is a free -module with -basis , where for .
Corollary 16
([100]) The Drinfeld–Kohno Lie algebra is an iterated semidirect product of free Lie algebras generated by , for .
4.2.2 Kukin’s example of a Lie algebra with undecidable word problem
Markov [161], Post [182], Turing [211], Novikov [173], and Boone [60] constructed finitely presented semigroups and groups with undecidable word problem. For groups this also follows from Higman’s theorem [115] asserting that every recursively presented group embeds into a finitely presented group. A weak analogue of Higman’s theorem for Lie algebras was proved in [21], which was enough for the existence of a finitely presented Lie algebra with undecidable word problem. In this section we give Kukin’s construction [142] of a Lie algebra for every semigroup such that if has undecidable word problem then so does .
Given a semigroup , consider the Lie algebra
with consisting of the relations
-
(1)
;
-
(2)
;
-
(3)
.
Here, stands for the left normed bracketing.
Put and denote by the deg-lex ordering on the set . Denote by the congruence on generated by . Put
-
with .
Lemma 8
([80]) In this notation, the set is a GS basis in .
Proof
For every , we can show that by induction on . All possible compositions in are the intersection compositions of (2) and , and the inclusion compositions of and .
For , we take and . Therefore, with and . We have
For , we use , where and with for . We have
Thus, is a GS basis in .
Corollary 17
(Kukin [142]) For we have
Proof
Assume that in the semigroup . Without loss of generality we may assume that and for some and . For every relations (1) yield ; consequently, and for every . This implies that in we have
where for every we put
Moreover, holds in .
Suppose that . Then both and have the same normal form in . Since is a GS basis in , we can reduce both and to the same normal form for some using only relations . This implies that in .
By the corollary, if the semigroup has undecidable word problem then so does the Lie algebra .
4.3 Composition-Diamond lemma for Lie algebras over commutative algebras
For a well-ordered set , consider the free Lie algebra with the Lie bracket .
Given a well-ordered set , the free commutative monoid generated by is a linear basis of . Regard
as a Lie subalgebra of the free associative algebra generated by over the polynomial algebra , equipped with the Lie bracket . Then constitutes a -basis of . Put . For , put and .
Denote the deg-lex orderings on and by and . Define an ordering on as follows: for , put
We can express every element as , where , , and
Then , where are polynomials in the -algebra . The leading word of in is of the form with and . The polynomial is called monic (or -monic) if the coefficient of is equal to 1, that is, . The notion of -monic polynomials is introduced similarly: and .
Recall that every ALSW admits a unique bracketing such that is a NLSW.
Consider a monic subset . Given a non-associative word on with a fixed occurrence of some and , call an -word. Define to be the -length of . Every -word is of the form with and . If then we have the special bracketing of relative to . Refer to as a special normal -word (or special normal -word).
An -word is a normal -word, denoted by , whenever . The following condition is sufficient.
-
(i)
The -length of is 1, that is, ;
-
(ii)
if is a normal -word of -length and satisfies then whenever and whenever are normal -words of -length .
Take two monic polynomials and in and put .
There are four kinds of compositions.
-
: Inclusion composition. If for some , then
-
: Intersection composition. If and with then
-
: External composition. If then for all satisfying
we have
-
: Multiplication composition. If then for every special normal -word with we have
Given a -monic subset and , which is not necessarily in , an element is called trivial modulo if can be expressed as a -linear combination of normal -words with leading words smaller than . The set is a Gröbner–Shirshov basis in if all possible compositions in are trivial.
Theorem 36
([31], the CD-lemma for Lie algebras over commutative algebras) Consider a nonempty set of monic polynomials and denote by the ideal of generated by . The following statements are equivalent:
-
(i)
The set is a Gröbner–Shirshov basis in .
-
(ii)
If then for some and .
-
(iii)
The set is a linear basis for .
Here and
Outline of the proof.
Take and write and . Define the ALSW- (or for short) as , where
Six are possible:
-
(i)
(-trivial, -trivial) (a trivial ;
-
(ii)
(-trivial, -inclusion);
-
(iii)
(-trivial, -intersection);
-
(iv)
(-nontrivial, -trivial);
-
(v)
(-nontrivial, -inclusion);
-
(vi)
(-nontrivial, -intersection).
In accordance with , six general compositions are possible.
Denote by the Shirshov double special bracketing of whenever is a -trivial , by and the Shirshov special bracketings of whenever is a lcm of -inclusion or -intersection respectively.
Define general Lie compositions for -monic Lie polynomials and with and as
Lemma 9
([31]) The general composition of -monic Lie polynomials and with and , where is a (-trivial, -trivial) , is
Proof
By (XIX), we have
The proof is complete.
A Lie GS basis need not be an associative GS basis because the PBW-theorem is not valid for Lie algebras over a commutative algebra (Shirshov [201]). Therefore, the argument for above (see Sect. 4.2) fails for .
Moreover, Shirshov’s original proof of the CD-lemma fails because the singleton is not a GS basis in general. The reason is that there exists a nontrivial composition of type (-nontrivial, -trivial).
There is another obstacle. For , every -word is a linear combination of normal -words. For this is not the case. Hence, we must use a multiplication composition such that .
Lemma 10
([31]) If every multiplication composition , , is trivial modulo , where , then every -word is a linear combination of normal -words.
In our paper with Yongshan Chen [31], we use the following definition of triviality of a polynomial modulo :
where is the Shirshov special bracketing of the ALSW with an ALSW .
The previous definition of triviality modulo is equivalent to the usual definition by Lemma 11, which is key in the proof of the CD-lemma for Lie algebras over a commutative algebra.
Lemma 11
([31]) Given a monic set with trivial multiplication compositions, take a normal -word and a special normal -word with the same leading monomial . Then they are equal modulo .
Lemmas 10 and 11 imply
Lemma 12
([31]) Given a monic set with trivial multiplication compositions, every element of the ideal generated by is a linear combination of special normal -words.
On the other hand, (XVII) and (XIX) imply the following analogue of Lemma 1 for .
Lemma 13
([31]) Given two -monic special normal -words and with the same leading associative word , their difference is equal to , where , , and . Hence, if is a GS basis then the previous special normal -words are equal modulo .
Now the claim (i)(ii) of the CD-lemma for follows.
For every Lie algebra over the commutative algebra ,
where is just with all commutators replaced with , is the universal enveloping associative algebra of .
A Lie algebra over a commutative algebra is called special whenever it embeds into its universal enveloping associative algebra. Otherwise it is called non-special.
Shirshov (1953) and Cartier (1958) gave classical examples of non-special Lie algebras over commutative algebras over , justified using ad hoc methods. Cohn (1963) suggested another non-special Lie algebra over a commutative algebra over a field of positive characteristic.
Example 1
(Shirshov (1953)) Take and
Consider , where
Then and
is a GS basis in , which implies that belongs to the linear basis of by Theorem 36, that is, in .
On the other hand, the universal enveloping algebra of has the presentation
However, the GS completion (see Mikhalev and Zolotykh [170]) of in is
Thus, is not special.
Example 2
(Cartier [70]) Take and
Consider , where
Then is not special over .
Proof
The set is a GS basis in , where
Then, and so in .
However, in
we have
Thus, .
Conjecture (Cohn [95]) Take the algebra of truncated polynomials over a field of characteristic . The algebra
called Cohn’s Lie algebra, is not special.
In we have
where is a Jacobson–Zassenhaus Lie polynomial. Cohn conjectured that in . To prove this, we must know a GS basis of up to degree in . We found it for . For example, and a GS basis of up to degree in is
Therefore, .
Similar though much longer computations show that in and in . Thus, we have
Theorem 37
([31]) Cohn’s Lie algebras , , and are non-special.
Theorem 38
([31]) Given a commutative -algebra , if is a Gröbner–Shirshov basis in such that every is -monic then is special.
Corollary 18
([31]) Every Lie -algebra with one monic defining relation is special.
Theorem 39
([31]) Suppose that is a finite homogeneous subset of . Then the word problem of is solvable for every finitely generated commutative -algebra .
Theorem 40
([31]) Every finitely or countably generated Lie -algebra embeds into a two-generated Lie -algebra, where is an arbitrary commutative -algebra.
5 Gröbner–Shirshov bases for -algebras and operads
5.1 CD-lemmas for -algebras
Some new CD-lemmas for -algebras have appeared: for associative conformal algebras [45] and -conformal algebras [43], for the tensor product of free algebras [30], for metabelian Lie algebras [75], for associative -algebras [41], for color Lie superalgebras and Lie -superalgebras [165, 166], for Lie superalgebras [167], for associative differential algebras [76], for associative Rota–Baxter algebras [32], for -algebras [33], for dialgebras [38], for pre-Lie algebras [35], for semirings [40], for commutative integro-differential algebras [102], for difference-differential modules and difference-differential dimension polynomials [225], for -differential associative -algebras [185], for commutative associative Rota–Baxter algebras [186], for algebras with differential type operators [111].
Latyshev studied general versions of GS (or standard) bases [147, 148].
Let us state the CD-lemma for pre-Lie algebras, see [35].
A non-associative algebra is called a pre-Lie (or a right-symmetric) algebra if satisfies the identity for the associator . It is a Lie admissible algebra in the sense that is a Lie algebra.
Take a well-ordered set . Order by induction on the lengths of the words and :
-
(i)
When put if and only if .
-
(ii)
When put if and only if one of the following holds:
-
(a)
;
-
(b)
if with and then or and .
-
(a)
We now quote the definition of good words (see [198]) by induction on length:
-
(1)
is a good word for any ;
-
(2)
a non-associative word is called a good word if
-
(a)
both and are good words and
-
(b)
if then .
-
(a)
Denote by whenever is a good word.
Denote by the set of all good words in the alphabet and by the free right-symmetric algebra over a field generated by . Then forms a linear basis of , see [198]. Kozybaev et al. [141] proved that the deg-lex ordering on is monomial.
Given a set of monic polynomials and , an -word is called a normal -word whenever is a good word.
Take , , and . Then there are two kinds of compositions.
-
(i)
If then is called the inclusion composition.
-
(ii)
If is not good then is called the right multiplication composition.
Theorem 41
([35], the CD-lemma for pre-Lie algebras) Consider a nonempty set of monic polynomials and the ordering defined above. The following statements are equivalent:
-
(i)
The set is a Gröbner–Shirshov basis in .
-
(ii)
If then for some and , where is a normal -word.
-
(iii)
The set is a linear basis of the algebra .
As an application, we have a GS basis for the universal enveloping pre-Lie algebra of a Lie algebra.
Theorem 42
([35]) Consider a Lie algebra with a well-ordered linear basis . Write with . Denote by . Denote by
the universal enveloping pre-Lie algebra of . The set
is a Gröbner–Shirshov basis in .
Theorems 41 and 42 directly imply the following PBW theorem for Lie algebras and pre-Lie algebras.
Corollary 19
(Segal [198]) A Lie algebra embeds into its universal enveloping pre-Lie algebra as a subalgebra of .
Recently the CD-lemmas mentioned above and other combinatorial methods yielded many applications: for groups of Novikov–Boone type [119–121] (see also [16, 17, 77, 118], for Coxeter groups [58, 150], for center-by-metabelian Lie algebras [214], for free metanilpotent Lie algebras, Lie algebras and associative algebras [112, 168, 215, 216], for Poisson algebras [159], for quantum Lie algebras and related problems [132, 135], for PBW-bases [131, 134, 158], for extensions of groups and associative algebras [73, 74], for (color) Lie ()-superalgebras [9, 48, 91, 92, 105–107, 169, 227, 228], for Hecke algebras and Specht modules [125], for representations of Ariki–Koike algebras [126], for the linear algebraic approach to GS bases [127], for HNN groups [87], for certain one-relator groups [88], for embeddings of algebras [39, 83], for free partially commutative Lie algebras [84, 181], for quantum groups of type , , and [174, 189, 221, 222], for calculations of homogeneous GS bases [145], for Picard groups, Weyl groups, and Bruck–Reilly extensions of semigroups [7, 128–130, 139], for Akivis algebras and pre-Lie algebras [79], for free Sabinin algebras [93].
5.2 CD-lemma for operads
Following Dotsenko and Khoroshkin ([98], Proposition 3), linear bases for a symmetric operad and a shuffle operad are the same provided both of them are defined by the same generators and defining relations. It means that we need CD-lemma for shuffle operads only (and we define a GS basis for a symmetric operad as a GS basis of the corresponding shuffle operad).
We express the elements of the free shuffle operad using planar trees.
Put , where is the set of -ary operations.
Call a planar tree with leaves decorated whenever the leaves are labeled by for and every vertex is labeled by an element of .
For an arrow in a decorated tree, let its value be the minimal value of the leaves of the subtree grafted to its end. A decorated tree is called a tree monomial whenever for each its internal vertex the values of the arrows beginning from it increase from the left to the right.
Denote by the set of all tree monomials with leaves and put . Given and , define the shuffle composition as
which lies in , where and the bijection
is an -shuffle, that is,
The set is freely generated by with the shuffle composition.
Denote by the -linear space spanned by . This space with the shuffle compositions is called the free shuffle operad.
Take a homogeneous subset of . For , define an -word as before.
A well ordering on is called monomial (admissible) whenever
Assume that is equipped with a monomial ordering. Then each -word is a normal -word.
For example, the following ordering on is monomial, see Proposition 5 of [98].
Every has a unique expression
where for is the unique path from the root to the leaf and the permutation lists the labels of the leaves of the underlying tree in the order determined by the planar structure, from left to right. In this case define
Assume that is a well-ordered set and use the deg-lex ordering on . Take the order on the permutations in reverse lexicographic order: if and only if is less than as numbers.
Now, given , define
An element of is called homogeneous whenever all tree monomials occurring in this element with nonzero coefficients have the same arity degree (but not necessarily the same operation degree).
For two tree monomials and , say that is divisible by whenever there exists a subtree of the underlying tree of for which the corresponding tree monomial is equal to .
A tree monomial is called a common multiple of two tree monomials and whenever it is divisible by both and . A common multiple of two tree monomials and is called a least common multiple and denoted by whenever , where for .
Take two monic homogeneous elements and of . If and have a least common multiple then .
Theorem 43
([98], the CD-lemma for shuffle operads) In the above notation, consider a nonempty set of monic homogeneous elements and a monomial ordering on . The following statements are equivalent:
-
(i)
The set is a Gröbner–Shirshov basis in .
-
(ii)
If then for some -word .
-
(iii)
The set is a -linear basis of .
As applications, the authors of [98] calculate Gröbner–Shirshov bases for some well-known operads: the operad Lie of Lie algebras, the operad As of associative algebras, and the operad PreLie of pre-Lie algebras.
Notes
Though Shirshov [207] 1962 was the first to come up with the idea of a ‘Gröbner–Shirshov basis’ for Lie and non-commutative polynomial algebras, his paper became practically unknown outside Russia. In the meantime, Buchberger’s ‘Gröbner basis’ (Thesis 1965 [65], paper 1970 [66]) for (commutative) polynomials became very popular in science. As a result, the first author suggested the name ‘Gröbner–Shirshov basis’ for non-commutative and non-associative polynomials. For (commutative) differential polynomials an analogous, or better to say, closely related ‘basis’ is called a Ritt–Kolchin characteristic set, due to Ritt [193] 1950 and Kolchin [140] 1973, and rediscovered by Wu [219] 1978.
We use the standard algebraic terminology ‘the word problem’, ‘the identity problem’, see Kharlampovich, Sapir [136] for instance.
After his Ph.D. Thesis of 1950, Zhukov moved to the present Keldysh Institute of Applied Mathematics (Moscow) to do computational mathematics. Godunov in ‘Reminiscence about numerical schemes’, arxiv.org/pdf/0810.0649, 2008, mentioned his name in relation to the creation of the famous Godunov numerical method. So, Zhukov was a forerunner of two important computational methods!
It must be pointed out that Malcev (1909–1967) inspired Shirshov’s works very much. Malcev was an official opponent (referee) of his (second) Doctor of Sciences Dissertation at MSU in 1958. The first author, Bokut, remembers this event at the Science Council Meeting, chaired by Kolmogorov, and Malcev’s words “Shirshov’s dissertation is a brilliant one!”. Malcev and Shirshov worked together at the present Sobolev Institute of Mathematics in Novosibirsk since 1959 until Malcev’s sudden death at 1967, and have been friends despite the age difference. Malcev headed the Algebra and Logic Department (by the way, the first author is a member of the department since 1960) and Shirshov was the first deputy director of the institute (whose director was Sobolev). In those years, Malcev was interested in the theory of algorithms of mathematical logic and algorithmic problems of model theory. Thus, Shirshov had an additional motivation to work on algorithmic problems for Lie algebras. Both Maltsev and Kurosh were delighted with Shirshov’s results of [207]. Malcev successfully nominated the paper for an award of the Presidium of the Siberian Branch of the Academy of Sciences (Sobolev and Malcev were the only Presidium members from the Institute of Mathematics at the time).
The Lyndon–Shirshov basis for the alphabet is different from the above Shirshov content basis starting with monomials of degree 7.
From [12]: “A famous theorem concerning Lyndon words asserts that any word can be factorized in a unique way as a non-increasing product of Lyndon words, i.e. written with . This theorem has imprecise origin. It is usually credited to Chen et al., following the paper of Schützenberger [197] in which it appears as an example of factorization of free monoids. Actually, as pointed out to one of us by Knuth in 2004, the reference [72] does not contain explicitly this statement.”
Abbreviations
- CD-lemma:
-
Composition-Diamond lemma
- GS basis:
-
Gröbner–Shirshov basis
- LS word (basis):
-
Lyndon–Shirshov word (basis)
- ALSW(X):
-
The set of all associative Lyndon–Shirshov words in
- NLSW(X):
-
The set of all non-associative Lyndon–Shirshov words in
- PBW theorem:
-
The Poincare–Birkhoff–Witt theorem
- :
-
The free monoid generated by
- :
-
The free commutative monoid generated by
- :
-
The set of all non-associative words in
- :
-
The group generated by with defining relations
- :
-
The semigroup generated by with defining relations
- :
-
A field
- :
-
A commutative algebra over with unity
- :
-
The free associative algebra over generated by
- :
-
The associative algebra over with generators and defining relations
- :
-
A Gröbner–Shirshov completion of
- :
-
The ideal generated by a set
- :
-
The maximal word of a polynomial with respect to some ordering
- :
-
The set of all monomials avoiding the subword for all
- :
-
The polynomial algebra over generated by
- :
-
The free Lie algebra over generated by
- :
-
The free Lie algebra generated by over a commutative algebra
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Acknowledgments
Authors thank Pavel Kolesnikov, Dima Piontkovskii, Yongshan Chen and Yu Li for valuable comments and help in writing some parts of the survey. They thank the referee for valuable comments and suggestions.
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Communicated by Efim Zelmanov.
Supported by the NNSF of China (11171118), the Research Fund for the Doctoral Program of Higher Education of China (20114407110007), the NSF of Guangdong Province (S2011010003374) and the Program on International Cooperation and Innovation, Department of Education, Guangdong Province (2012gjhz0007). Supported by RFBR 12-01-00329, LSS–3669.2010.1, SB RAS Integration Grant No. 2009.97 (Russia) and Federal Target Grant “Scientific and educational personnel of innovation Russia” for 2009–2013 (government contract No.02.740.11.5191).
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Bokut, L.A., Chen, Y. Gröbner–Shirshov bases and their calculation. Bull. Math. Sci. 4, 325–395 (2014). https://doi.org/10.1007/s13373-014-0054-6
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DOI: https://doi.org/10.1007/s13373-014-0054-6
Keywords
- Gröbner basis
- Gröbner–Shirshov basis
- Composition-Diamond lemma
- Congruence
- Normal form
- Braid group
- Free semigroup
- Chinese monoid
- Plactic monoid
- Associative algebra
- Lie algebra
- Lyndon–Shirshov basis
- Lyndon–Shirshov word
- PBW theorem
- -algebra
- Dialgebra
- Semiring
- Pre-Lie algebra
- Rota–Baxter algebra
- Category
- Module