On Shirshov’s Papers for Lie Algebras

  • Leonid A. Bokut
Part of the Contemporary Mathematicians book series (CM)


Shirshov published six papers on Lie algebras in which he found the following results (in order of publication. 1953–1962):
  • • Some years before Witt [84], the “Shirshov-Witt theorem” [1].

  • • Some years before Lazard [62], the “Lazard-Shirshov elimination process” [1]. This is often called “Lazard elimination”; see for example [79].

  • • The first example of a Lie ring that is not representable into any associative ring [2]; see also P. Cartier [37] and P.M. Cohn [43].

  • • In the same year as Chen-Fox-Lyndon [38], the “Lyndon-Shirshov basis” of a free Lie algebra (Lyndon-Shirshov Lie words) [6]. This is often called the “Lyndon basis”; see for example [63], [79] [64].

  • • Independently of Lyndon [65], the “Lyndon-Shirshov (associative) words” [6] They are often called “Lyndon words”; see for example [63]. In the literature they are also often called “(Shirshov’s) regular words” or “Lyndon-Shirshov words”; see for example [42], [24], [23], [85], [76], [14].

  • • The algorithmic criterion to recognize Lie polynomials in a free associative algebra over any commutative ring [6]. The algorithm is based on the property that the maximal (in deg-lex ordering) associative word of any Lie polynomial is an associative Lyndon-Shirshov word. The Friedrichs criterion [45] follows from the Shirshov algorithmic criterion (see [6]).

  • • In the same year as Chen-Fox-Lyndon [38], the “central result on Lyndon-Shirshov words”: any word is a unique non-decreasing product of Lyndon-Shirshov words [6]. This is often called the “Lyndon theorem” or the “Chen-Fox-Lyndon theorem”.

  • • The reduction algorithm for Lie polynomials: the elimination of the maximal Lyndon-Shirshov Lie word of a Lie polynomial in a Lyndon-Shirshov Lie word [6]. The algorithm is based on the Special Bracketing Lemma [6, Lemma 4], which in turn depends on the “central result on Lyndon-Shirshov words” above.

  • • The theorem that any Lie algebra of countable dimension is embeddable into two-generated Lie algebra with the same number of defining relations [6].

  • • Some years before Viennot [82], the “Hall-Shirshov bases” of a free Lie algebra [7]: a series of bases that contains the Hall basis and the Lyndon-Shirshov basis and depends on an ordering of basic Lie words such that [w]=[[u][v]]>[v]. They are often called “Hall sets”; see for example [79].

  • • Some years before Hironaka [53] and Buchberger [35], [36], the “Gröbner-Shirshov basis theory” for Lie polynomials (Lie algebras) explicitly and for noncommutative polynomials (associative algebras) implicitly [9]. This theory includes the definition of composition (s-polynomial), the reduction algorithm, the algorithm for producing a Gröbner-Shirshov basis (this is an infinite algorithm of Knuth-Bendix type [35]; see also the software implementations in [48], [87], [15], and the “Composition-Diamond Lemma”. Shirshov’s “Composition-Diamond Lemma” for associative algebras was formulated explicitly in [25] and rediscovered by G. Bergman [78] under the name “non-commutative Gröbner basis theory”. The analogous theory for polynomials (commutative algebras) was found by B. Buchberger [35], [36] under the name “Gröbner basis theory”; similar ideas for (commutative) formal series were found by H. Hironaka [53] under the name “standard basis theory”.

  • • The “Freiheitssatz” and the decidability of the word problem for one-relator Lie algebras [9].

  • • The first linear basis of the free product of Lie algebras [10].

  • • The first example showing that an analogue of the Kurosh subgroup theorem is not valid for subalgebras of the free product of Lie algebras [10].


Word Problem Associative Algebra Free Product Free Associative Algebra Lyndon Word 
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  1. [1]
    Shirshov, A.I., Subalgebras of free Lie algebras. (Russian) Mat. Sb., N. Ser. 33(75), 441–452 (1953).Google Scholar
  2. [2]
    Shirshov, A.I. On representation of Lie rings in associative rings (Russian) Usp. Mat. Nauk 8, No. 5 (57), 173–175 (1953).MATHGoogle Scholar
  3. [3]
    Shirshov, A.I., Certain problems of the theory of non-associative algebras. Thesis, Moscow State University, 1953.Google Scholar
  4. [4]
    Shirshov, A.I., Subalgebras of free commutative and free anti-commutative algebras. Mat. Sbornik. 34(76) (1954), 81–88.MathSciNetGoogle Scholar
  5. [5]
    Shirshov, A.I., Some theorems on embedding for rings (Russian) Mat. Sb., N. Ser. 40(82), 65–72 (1956).Google Scholar
  6. [6]
    Shirshov, A.I., On free Lie rings (Russian) Mat. Sb., N. Ser. 45(87), 113–122 (1958).Google Scholar
  7. [7]
    Shirshov, A.I., On bases of a free Lie algebra. (Russian) Algebra Logika, 1, No.1, 14–19 (1962).MATHGoogle Scholar
  8. [8]
    Shirshov, A.I., Certain algorithmic problems for ε-algebras. (Russian) Sib. Mat. Zh. 3, 132–137 (1962).MATHGoogle Scholar
  9. [9]
    Shirshov, A.I., Certain algorithmic problems for Lie algebras. (Russian) Sib. Mat. Zh. 3, 292–296 (1962). English translation: Shirshov, A.I., Certain algorithmic problems for Lie algebras. (English) ACM SIGSAM Bull. 33, No. 2, 3–6 (1999).MATHGoogle Scholar
  10. [10]
    Shirshov, A.I., On the conjecture of the theory of Lie algebras. (Russian) Sib. Mat. Zh. 3, 297–301 (1962).MATHGoogle Scholar
  11. [11]
    A.I. Shirshov, Collected Works. Rings and Algebras. Nauka, Moscow, 1984.Google Scholar
  12. [12]
    Adjan, S.I. Defining relations and algorithmic problems for groups and semigroups. (English. Russian original) Proc. Steklov Inst. Math. 85, 152 p. (1966); translation from Tr. Mat. Inst. Steklov 85, 123 p. (1966).Google Scholar
  13. [13]
    Yu.A. Bahturin, A.A. Mikhalev, M.V. Zaicev, and V.M. Petrogradsky, Infinite Dimensional Lie Superalgebras. Walter de Gruyter Publ., Berlin, New York, 1992.Google Scholar
  14. [14]
    Bahturin, Yuri; Mikhalev, Alexander A.; Zaicev, Mikhail Infinite-dimensional Lie superalgebras. (English) Hazewinkel, M. (ed.), Handbook of algebra. Volume 2. Amsterdam: North-Holland. 579–614 (2000).Google Scholar
  15. [15]
    Backelin, Jörgen; Cojocaru, Svetlana; Ufnarovski, Victor The computer algebra package Bergman: Current state. (English) Herzog, Jürgen (eded.) et al., commutative algebra, singularities and computer algebra. Proceedings of the NATO advanced research workshop, Sinaia, Romania, September 17–22, 2002. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 115, 75–100 (2003).Google Scholar
  16. [16]
    G.M. Bergman, The Diamond Lemma for ring theory. Adv. in Math., 29(1978), 178–218.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Belyaev, V. Ya. Subrings of finitely presented associative rings. (English) Algebra Logika 17, 627–638 (1978).MATHMathSciNetGoogle Scholar
  18. [18]
    Bokut, L.A., Embedding of Lie algebras into algebraically closed Lie algebras. (Russian) Algebra Logika 1, No.2, 47–53 (1962).MATHMathSciNetGoogle Scholar
  19. [19]
    Bokut, L.A., Bases of free poly-nilpotent Lie algebras (Russian) Algebra Logika 2, No. 4, 13–19 (1963).MATHMathSciNetGoogle Scholar
  20. [20]
    L.A. Bokut, On a property of the Boone groups. Algebra i Logika Sem., 5 (1966), 5, 5–23; 6 (1967), 1, 15–24.MATHMathSciNetGoogle Scholar
  21. [21]
    L.A. Bokut, On Novikov’s groups. Algebra i Logika Sem., 6 (1967), 1, 25–38.Google Scholar
  22. [22]
    Bokut, L.A., Degrees of insolvability of the conjugacy problem for finitely presented groups (Russian) Algebra Logika 7, No.5, 4–70; No. 6, 4–52 (1968).Google Scholar
  23. [23]
    L.A. Bokut, Groups of fractions of multiplication semigroups of certain rings. I–III, Malcev’s problem. Sibir. Math. J., 10, 2, 246–286; 4, 744–799, 4, 800–819; 5, 965–1005.Google Scholar
  24. [24]
    L.A. Bokut, Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1173–1219.MATHMathSciNetGoogle Scholar
  25. [25]
    L.A. Bokut, Imbeddings into simple associative algebras. Algebra i Logika Sem., 15 (1976), 117–142.MATHMathSciNetGoogle Scholar
  26. [26]
    Bokut’, L.A., On algebraically closed and simple Lie algebras (Russian, English) Proc. Steklov Inst. Math. 148, 30–42 (1978).MathSciNetGoogle Scholar
  27. [27]
    L.A. Bokut, Yuqun Chen, Gröbner Shirshov bases for Lie algebras: after A.I. Shirshov. SEA Bull Math., 31 (2007), 811–831.MathSciNetGoogle Scholar
  28. [28]
    L.A. Bokut, Y. Fong, W.-F. Ke, Composition Diamond Lemma for associative conformal algebras. J. Algebra, 272(2004), 739–774.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    L.A. Bokut, Y. Fong, W.-F. Ke, P.S. Kolesnikov, Gröbner and Gröbner-Shirshov bases in Algebra and Conformal algebras. Fundamental and Applied Mathematics, 6(2000), N3, 669–706 (in Russian).MATHMathSciNetGoogle Scholar
  30. [30]
    L.A. Bokut, P.S. Kolesnikov, Gröbner-Shirshov bases: From Incipient to Nowdays, Proceedings of the POMI, 272(2000), 26–67.Google Scholar
  31. [31]
    L.A. Bokut, P.S. Kolesnikov, Gröbner-Shirshov bases: conformal algebras and pseudoalgebras, Journal of Mathematicfal Sciences, 131(5)(2005), 5962–6003.MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    L.A. Bokut, and G.P. Kulin, Algorithmic and combinatorial algebra. Mathematics and its Applications, 255, Kluwer Academic Publishers Group, Dordrecht, 1994.MATHGoogle Scholar
  33. [33]
    L.A. Bokut, K.P. Shum, Relative Gröbner-Shirshov bases for algebras and groups. Algebra i Analiz 19 (2007), no. 6, 1–21.MathSciNetGoogle Scholar
  34. [34]
    Bourbaki, N. Elements de mathematique. Fasc. XXXVII: Groupes et algèbres de Lie. Chap. II: Algèbres de Lie libres. Chap. III: Groupes de Lie. (French) Actualites scientifiques et industrielles 1349. Paris: Hermann. 320 p. (1972).Google Scholar
  35. [35]
    B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal. (German). Ph.D. thesis, University of Innsbruck, Austria, 1965.Google Scholar
  36. [36]
    B. Buchberger, An algorithmical criteria for the solvability of algebraic systems of equations (German). Aequationes Math. 4 (1970), 374–383.MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    Cartier, P. Remarques sur le theore me de Birkhoff-Witt. (French) Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III Ser. 12, 1–4 (1958).MATHMathSciNetGoogle Scholar
  38. [38]
    K.T. Chen, R.H. Fox, and R.C. Lyndon, Free differential calculus, IV: the quotient groups of the lower central series. Annals of Mathematics 68 (1958), pp. 81–95.CrossRefMathSciNetGoogle Scholar
  39. [39]
    Yuqun Chen, Wenshu Chen and Runai Luo, Word problem for Novikov’s and Boone’s group via Gröbner-Shirshov bases, SEA Bull Math., 32(2008), 5.MathSciNetGoogle Scholar
  40. [40]
    E.S. Chibrikov, On free Lie conformal algebras. Vestnik Novosib. State Univ., Ser. “Math, Mech, Inform.”, 4 (2004), No. 1, 65–86 (in Russian).Google Scholar
  41. [41]
    Chibrikov, E.S., A right normed basis for free Lie algebras-and Lyndon-Shirshov words. J. Algebra 302, No. 2, 593–612 (2006).MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    Cohn, P.M. Universal algebra. (English) Harper’s Series in Modern Mathematics. New York-Evanston-London: Harper and Row, Publishers 1965, XV, 333p. (1965).Google Scholar
  43. [43]
    Cohn, P.M., A remark on the Birkhoff-Witt theorem. English J. Lond. Math. Soc. 38, 197–203 (1963).MATHCrossRefGoogle Scholar
  44. [44]
    Cohn, P.M. Sur le critère de Friedrichs pour les commutateurs dans une algèbre asociative libre. Comptes Rendus Acad. Science Paris, 239, 743–745 (1954).MATHGoogle Scholar
  45. [45]
    Friedrichs, K.O. Mathematical aspects of the quantum theory of fields. V. (English) Commun. Pure Appl. Math. 6, 1–72 (1953).MATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    Gainov, A.T., Free commutative and free anticommutative products of algebras. (Russian) Sib. Mat. Zh. 3, 805–833 (1962).MathSciNetGoogle Scholar
  47. [47]
    Gerasimov, V.N., Distributive lattices of subspaces and the equality problem for algebras with a single relation. Algebra Logic 15 (1976), 238–274 (1977); translation from Algebra Logika 15, 384–435 (1976).MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    Gerdt, V.P.; Kornyak, V.V., Program for constructing a complete system of relations, basis elements, and commutator table for finitely presented Lie algebras and super-algebras. (English. Russian original) Program. Comput. Softw. 23, No. 3, 164–172 (1997); translation from Programmirovanie 1997, No.3, 58–71 (1997).MATHMathSciNetGoogle Scholar
  49. [49]
    P. Hall, A contribution to the theory of groups of prime power order. Proc. London Math. Soc. Ser. 2, 36 (1933), pp. 29–95.MATHCrossRefGoogle Scholar
  50. [50]
    M. Hall, A basis for free Lie rings and higher commutators in free groups. Proc. Amer. Math. Soc. 3 (1950), pp. 575–581.CrossRefGoogle Scholar
  51. [51]
    G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. London Math. Soc. 24 (1949) 247–254.CrossRefMathSciNetGoogle Scholar
  52. [52]
    Higman, G. Subgroups of finitely presented groups. (English) Proc. R. Soc. Lond., Ser. A 262, 455–475 (1961).MATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II. Ann. Math., 79(2) (1964), pp. 109–203, 205–326.CrossRefMathSciNetGoogle Scholar
  54. [54]
    S.-J. Kang, K.-H. Lee, Gröbner-Shirshov bases for irreducible sl n+1-modules, Journal of Algebra, 232 (2000), 1–20.MATHCrossRefMathSciNetGoogle Scholar
  55. [55]
    Knuth, D.E.; Bendix, P.B., Simple word problems in universal algebras. Comput. Probl. abstract Algebra, Proc. Conf. Oxford 1967, 263–297 (1970).Google Scholar
  56. [56]
    D. Kozybaev, L. Makar-Limanov, U. Umirbaev, The Freiheitssatz and the automorphisms of free right-symmetric algebras, Asian-European. J. Math. 1 (2008), 2, 243–252.MATHMathSciNetGoogle Scholar
  57. [57]
    G.P. Kukin, On the word problem for Lie algebras. Sibirsk. Math. Zh. 18 (1977), 1194–1197.MATHMathSciNetGoogle Scholar
  58. [58]
    Kukin, G.P. Subalgebras of a free Lie sum of Lie algebras with an amalgamated subalgebra. Algebra Logic 11 (1972), 59–86.MATHMathSciNetGoogle Scholar
  59. [59]
    Kukin, G.P. On the Cartesian subalgebras of a frèe Lie sum of Lie algebras. Algebra Logika 9, 701–713 (1970).MathSciNetGoogle Scholar
  60. [60]
    Kurosh, A., Nonassociative free algebras and free products of algebras. (Russian. English summary) Mat. Sb., N. Ser. 20(62), 239–262 (1947).MathSciNetGoogle Scholar
  61. [61]
    Kurosch, A. Die Untergruppen der freien Produkte von beliegiben Gruppen (German) Math. Ann. 109, 647–660 (1934).CrossRefMathSciNetGoogle Scholar
  62. [62]
    Lazard, M. Groupes, anneaux de Lie et problème de Burnside. C.I.M.E., Gruppi, Anelli di Lie e Teoria della Coomologia 60 p. (1960). The same in: Instituto Matemático dell’Universita di Roma (1960).Google Scholar
  63. [63]
    Lothaire, M. Combinatorics on words. Foreword by Roger Lyndon. Encyclopedia of Mathematics and Its Applications, Vol. 17. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company, Advanced Book Program/World Science Division. XIX, 238 p. (1983).Google Scholar
  64. [64]
    Lothaire, M. Combinatorics on words. Foreword by Roger Lyndon. 2nd ed. Encyclopedia of Mathematics and Its Applications. 17: Cambridge: Cambridge University Press. xvii, 238 p. (1997).MATHGoogle Scholar
  65. [65]
    Lyndon, R.C. On Burnside’s problem. Trans. Am. Math. Soc. 77, 202–215 (1954).MATHCrossRefMathSciNetGoogle Scholar
  66. [66]
    Lyndon, R.C. A theorem of Friedrichs. Mich. Math. J. 3, 27–29 (1956).MATHMathSciNetGoogle Scholar
  67. [67]
    Magnus, W. Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz). J. Reine Angew. Math. 163(1930), pp. 141–165.MATHGoogle Scholar
  68. [68]
    Magnus, W. Das Identitätsproblem für Gruppen mit einer definierenden Relation. (German) Math. Ann. 106, 295–307 (1932).CrossRefMathSciNetGoogle Scholar
  69. [69]
    W. Magnus, Über Beziehungen zwischen höheren Kommutatoren. J. Reine Angew. Math 177(1937), pp. 105–115.Google Scholar
  70. [70]
    Makar-Limanov, L. Algebraically closed skew fields. J. Algebra 93, 117–135 (1985).MATHCrossRefMathSciNetGoogle Scholar
  71. [71]
    Makar-Limanov, L.G. On algebras with one relation. Usp. Mat. Nauk 30, No.2(182), 217 (1975).MATHMathSciNetGoogle Scholar
  72. [72]
    Malcev, A.I., On representation of nonassociative rings. Uspehi Mat. Nauk N.S. 7 (1952), 181–185.MathSciNetGoogle Scholar
  73. [73]
    Matiyasevich, Yu.V. Enumerable sets are diophantine. Russian original) Sov. Math., Dókl. 11, 354–358 (1970); translation from Dokl. Akad. Nauk SSSR 191, 279–282 (1970).MATHGoogle Scholar
  74. [74]
    Mikhalev, A.A., The junction lemma and the equality problem for color Lie superalgebras. Vestnik. Moskov. Univ. Ser. 1. Mat. Mekh. 1989, no. 5, 88–91. English translation: Moscow Univ. Math. Bull. 44 (1989), 87–90.Google Scholar
  75. [75]
    A.A. Mikhalev, Shirshov’s composition techniques in Lie superalgbras (non-commutative Gröbner bases). Trudy Sem. Petrovsk. 18 (1995), 277–289.MATHGoogle Scholar
  76. [76]
    A.A. Mikhalev and A.A. Zolotykh, Combinatorial Aspects of Lie Superalgebras. CRC Press, Boca Raton, New York, 1995.MATHGoogle Scholar
  77. [77]
    V.N. Latyshev, Combinatorial Theory of Rings. Standard Bases. Moscow State Univ. Publ. House, Moscow, 1988.Google Scholar
  78. [78]
    T. Mora, Gröbner bases for non-commutative polynomial rings. Lecture Notes in Comput. Sci. 229 (1986), 353–362.MathSciNetGoogle Scholar
  79. [79]
    C. Reutenauer. Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.MATHGoogle Scholar
  80. [80]
    C. Reutenauer. Dimensions and characters of the derived series of the free Lie algebra. In M. Lothaire, Mots, Melanges offerts a M.-P. Schützenberger, pp. 171–184. Hermes, Paris.Google Scholar
  81. [81]
    Shelah, Saharon On a problem of Kurosh, Jonsson groups, and applications. (English) Word problems II, Stud. Logic Found. Math. Vol. 95, 373–394 (1980).Google Scholar
  82. [82]
    Viennot, Gerard. Algèbres de Lie libres et monoides libres. Bases des algèbres de Lie libres et factorisations des monoides libres. (French) Lecture Notes in Mathematics. 691. Berlin-Heidelberg-New York: Springer-Verlag. 124 p. (1978)Google Scholar
  83. [83]
    E. Witt, Treue Darstellungen Lieschen Ringe. J. Reine Angew. Math. 177(1937), pp. 152–160.Google Scholar
  84. [84]
    E. Witt, Subrings of free Lie rings Math. Zeit., 64(1956), 195–216.MATHMathSciNetGoogle Scholar
  85. [85]
    V.A. Ufnarovski, Combinatorial and Asymptotic Methods in Algebra. Encyclopaedia Math. Sci. 57 (1995), 1–196.Google Scholar
  86. [86]
    A.I. Zhukov, Reduced systems of defining relations in non-associative algebras Mat. Sb., N. Ser., 27(69) (1950), 267–280.Google Scholar
  87. [87]
    Zolotykh, A.A.; Mikhalev, A.A. Algorithms for construction of standard Gröbner-Shirshov bases of ideals of free algebras over commutative rings. (English. Russian original) Program. Comput. Softw. 24, No.6, 271–272 (1998); translation from Programmirovanie 1998, No.6, 10–11 (1998).MATHMathSciNetGoogle Scholar

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  • Leonid A. Bokut

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