Advertisement

Perspectives on Shirshov’s Height Theorem

  • Alexei Belov-Kanel
  • Louis H. Rowen
Part of the Contemporary Mathematicians book series (CM)

Abstract

In this survey we consider the impact of Shirshov’s Height Theorem on algebra. In order to avoid duplication, we often refer to Kemer’s survey article [Kem09] in this volume for further details. Proofs of various quoted results are given in the book [BBL97], and in the authors’ book [BR05].

Keywords

Polynomial Identity Free Algebra Hilbert Series Multilinear Polynomial Exponential Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Am57]
    Amitsur, S.A., A generalization of Hilbert’s Nullstellensatz, Proc. Amer. Math. Soc. 8 (1957), 649–656.MATHCrossRefMathSciNetGoogle Scholar
  2. [An92]
    Anan’in, A.Z., The representability of finitely generated algebras with chain condition, Arch. Math. 59 (1992), 1–5.CrossRefMathSciNetGoogle Scholar
  3. [Ba87]
    Bakhturin, Yu.A., Identical relations in Lie algebras. Translated from the Russian by Bakhturin VNU Science Press, b.v., Utrecht, (1987).Google Scholar
  4. [Bel88a]
    Belov, A.Ya., On Shirshov bases in relatively free algebras of complexityn, Mat. Sb. 135 (1988), no. 3, 373–384.Google Scholar
  5. [Bel88b]
    Belov, A.Ya., The height theorem for Jordan and Lie PI-algebras, in: Tez. Dokl. Sib. Shkoly po Mnogoobr. Algebraicheskih Sistem, Barnaul (1988), pp. 12–13.Google Scholar
  6. [Bel89]
    Belov, A.Ya., Estimations of the height and Gelfand-Kirillov dimension of associative PI-algebras, In: Tez. Dokl. po Teorii Koletz, Algebr i Modulei. Mezhdunar. Konf. po Algebre Pamyati A.I.Mal’tzeva, Novosibirsk (1989), p. 21.Google Scholar
  7. [Bel92]
    Belov, A.Ya., Some estimations for nilpotency of nil-algebras over a field of an arbitrary characteristic and height theorem, Commun. Algebra 20 (1992), no. 10, 2919–2922.MATHCrossRefMathSciNetGoogle Scholar
  8. [Bel97]
    Belov, A.Ya., Rationality of Hilbert series with respect to free algebras, Russian Math. Surveys 52 (1997), no. 10, 394–395.MATHCrossRefMathSciNetGoogle Scholar
  9. [Bel00]
    Belov, A.Ya., Counterexamples to the Specht problem, Sb. Math. 191 (3–4) (2000), 329–340.CrossRefMathSciNetGoogle Scholar
  10. [Bel02]
    Belov, A.Ya., Algebras with polynomial idenitities: Representations and combinatorial methods, Doctor of Science Dissertation, Moscow (2002).Google Scholar
  11. [Bel07]
    Belov, A.Ya., Burnside-type problems, and theorems on height and independence (Russian), Fundam. Prikl. Mat. 13 (2007), no. 5, 19–79.Google Scholar
  12. [BBL97]
    Belov, A.Ya., Borisenko, V.V., and Latyshev, V.N., Monomial algebras. Algebra 4, J. Math. Sci. (New York) 87 (1997), no. 3, 3463–3575.MATHCrossRefMathSciNetGoogle Scholar
  13. [BC00]
    Belov, A.Ya. and Chilikov, A.A. Exponential Diophantine equations in rings of positive characteristic (Russian) Fundam. Prikl. Mat. 6(3), 649–668, (2000).MATHMathSciNetGoogle Scholar
  14. [BR05]
    Belov, A.Ya. and Rowen, L.H. Computational aspects of polynomial identities. Research Notes in Mathematics 9 AK Peters, Ltd., Wellesley, MA, 2005.MATHGoogle Scholar
  15. [BRV06]
    Kanel-Belov, A.Ya., Rowen, L.H., and Vishne, U., Normal bases of PI-algebras Adv. in Appl. Math. 37 (2006), no. 3, 378–389.MATHCrossRefMathSciNetGoogle Scholar
  16. [Ber93]
    Berele, A., Generic verbally prime PI-algebras and their GK-dimensions, Comm. Algebra 21 (1993), no. 5, 1487–1504.MATHCrossRefMathSciNetGoogle Scholar
  17. [Bog01]
    Bogdanov I., Nagata-Higman’s theorem for hemirings, Fundam. Prikl. Mat. 7 (2001), no. 3, 651–658 (in Russian).MATHMathSciNetGoogle Scholar
  18. [BLH88]
    Bokut’, L.A., L’vov, I.V., and Harchenko, V.K., Noncommutative rings, In: Sovrem. Probl. Mat. Fundam. Napravl: Vol. 18, Itogi Nauki i Tekhn., All-Union Institute for Scientificand Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1988), 5–116.Google Scholar
  19. [Br82]
    Braun, A., The radical in a finitely generated PI-algebra, Bull. Amer. Math. Soc. 7 (1982), no. 2, 385–386.MATHCrossRefMathSciNetGoogle Scholar
  20. [Br84]
    Braun, A., The nilpotence of the radical in a finitely generated PI-ring, J. Algebra 89 (1984), 375–386.MATHCrossRefMathSciNetGoogle Scholar
  21. [Che88]
    Chekanu, G.P., Local finiteness of algebras. (Russian) Mat. Issled. 105, Moduli, Algebry, Topol. (1988), 153–171, 198.MathSciNetGoogle Scholar
  22. [Che95]
    Chekanu, G.P., Independence and quasiregularity in algebras, Dokl. Akad. Nauk 337 (1994), no. 3, 316–319; translation: Russian Acad. Sci. Dokl. Math. 50 (1995), no. 1, 84–89.Google Scholar
  23. [Che96]
    Chekanu, G.P., Independence and quasiregularity in algebras I. (Moldavian) Izv. Akad. Nauk Respub. Moldova Mat. 1996, no. 3, 29–39, 120, 122.MathSciNetGoogle Scholar
  24. [ChUf85]
    Chekanu, G.P., and Ufnarovski’i, V.A., Nilpotent matrices Mat. Issled. no. 85, Algebry, Kotsa i Topologi (1985), 130–141, 155.MathSciNetGoogle Scholar
  25. [DPR61]
    Davis M., Putnam, H., and Robsinson, J. The decision problem for exponential differential equations, Annals of Math. 74 425–436, (1961).CrossRefGoogle Scholar
  26. [Dne93]
    Dniester Notebook (Dnestrovskaya tetrad), Sobolev Institute of Mathematics, Novosibirsk 1993.Google Scholar
  27. [Do94]
    Donkin, S., Polynomial invariants of representations of quivers, Comment. Math. Helv. 69 (1994), no. 1, 137–141.MATHCrossRefMathSciNetGoogle Scholar
  28. [Dr84a]
    Drensky, V., On the Hilbert series of relatively free algebras, Comm. in Algebra 12 no. 19 (1984), 2335–2347.MATHCrossRefMathSciNetGoogle Scholar
  29. [Dr84b]
    Drensky, V., Codimensions of T-ideals and Hilbert series of relatively free algebras, J. Algebra 91 no. 1 (1984), 1–17.MATHCrossRefMathSciNetGoogle Scholar
  30. [Dr00]
    Drensky, V., Free Algebras and PI-algebras: Graduate Course in Algebra, Springer-Verlag, Singapore (2000).MATHGoogle Scholar
  31. [DrFor04]
    Drensky, V. and Formanek, E., Polynomials Identity Rings, CRM Advanced Courses in Mathematics, Birkhäuser, Basel (2004).Google Scholar
  32. [For84]
    Formanek, E., Invariants and the ring of generic matrices, J. Algebra 89 (1984), no. 1, 178–223.MATHCrossRefMathSciNetGoogle Scholar
  33. [Gol64]
    Golod, E.S., On nil-algebras and residually finite p-groups, Izv. Akad. Nauk SSSR 28 (1964), no. 2, 273–276.MathSciNetGoogle Scholar
  34. [Gri99]
    Grishin, A.V., Examples of T-spaces and T-ideals in Characteristic 2 without the Finite Basis Property, Fundam. Prikl. Mat. 5 (1) (1999), no. 6, 101–118 (in Russian).MATHMathSciNetGoogle Scholar
  35. [GuKr02]
    Gupta, C.K. and Krasilnikov, A.N., A simple example of a non-finitely based system of polynomial identities, Comm. Algebra 36 (2002), 4851–4866.CrossRefMathSciNetGoogle Scholar
  36. [Hel74]
    Helling, H., Eine Kennzeichnung von Charakteren auf Gruppen und Assoziativen Algebren, Comm. in Alg. 1 (1974), 491–501.MATHCrossRefMathSciNetGoogle Scholar
  37. [Hig56]
    Higman, G., On a conjecture of Nagata, Proc. Cam. Phil. Soc. 52 (1956), 1–4.MATHCrossRefMathSciNetGoogle Scholar
  38. [Ilt91]
    Iltyakov, A.V., Finitesness of basis identities of a finitely generated alternative PI-algebra, Sibir. Mat. Zh. 31 (1991), no. 6, 87–99; English, translation: Sib. Math J. 31 (1991), 948–961.MathSciNetGoogle Scholar
  39. [Ilt03]
    Iltyakov, A.V., Polynomial identities of Finite Dimensional Lie Algebras nonograph (2003).Google Scholar
  40. [Kap49]
    Kaplansky, I., Groups with representations of bounded degree, Canadian J. Math. 1 (1949), 105–112.MATHMathSciNetGoogle Scholar
  41. [Kap50]
    Kaplansky, I., Topological representation of algebras. II, Trans. Amer. Math. Soc 66 (1949), 464–491.MATHCrossRefMathSciNetGoogle Scholar
  42. [Kem80]
    Kemer, A.R., Capell identities and the nilpotence of the radical of a finitely generated PI-algebra, Soviet Math. Dokl. 22 (3) (1980), 750–753.MATHMathSciNetGoogle Scholar
  43. [Kem87]
    Kemer, A.R., Finite basability of identities of associative algebras (Russian), Algebra i Logika 26 (1987), 597–641; English translation: Algebra and Logic 26 (1987), 362–397.MATHMathSciNetGoogle Scholar
  44. [Kem88]
    Kemer, A.R., The representability of reduced-free algebras, Algebra i Logika 27 (1988), no. 3, 274–294.MathSciNetGoogle Scholar
  45. [Kem90a]
    Kemer, A.R., Identities of Associative Algebras, Transl. Math. Monogr., 87, Amer. Math. Soc. (1991).Google Scholar
  46. [Kem90b]
    Kemer, A.R., Identities of finitely generated algebras over an infinite field (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 726–753; translation in Math. USSR-Izv. 37 (1991), no. 1, 69–96.Google Scholar
  47. [Kem95]
    Kemer, A.R., Multilinear identities of the algebras over a field of oharacteristic p, Internat. J. Algebra Comput. 5 (1995), no. 2, 189–197.MATHCrossRefMathSciNetGoogle Scholar
  48. [Kem09]
    Kemer, A.R., Comments on the Shirshov’s Height Theorem, in this collection.Google Scholar
  49. [Kol81]
    Kolotov, A.T., Aperiodic sequences and growth functions in algebras, Algebra i Logika 20 (1981), no. 2, 138–154.MathSciNetGoogle Scholar
  50. [KerLe00]
    Krause, G.R. and Lenagan, T.H., Growth of Algebras and Gelfand-Kirillov Dimension, Amer. Math. Soc. Graduate Studies in Mathematics 22 (2000).Google Scholar
  51. [Kuz75]
    Kuzmin, E.N., About Nagata-Higman Theorem, Proceedings dedicated to the 60th birthday of Academician Iliev, Sofia (1975), 101–107 (in Russian).Google Scholar
  52. [Lat72]
    Latyshev, V.N., On Regev’s theorem on indentities in a tensor product of PI-algebras, Uspehi Mat. Nauk. 27 (1972), 213–214.Google Scholar
  53. [Lat88]
    Latyshev, V.N., Combinatorial Ring Theory. Standard Bases, Moscow University Press, Moscow (1988), (in Russian).MATHGoogle Scholar
  54. [Lev46]
    Levitzki, J., On a problem of Kurosch, Bull. Amer. Math. Soc. 52 (1946), 1033–1035.MATHCrossRefMathSciNetGoogle Scholar
  55. [Lv83]
    Lvov, I.V., Braun’s theorem on the radical of PI-algebras, Institute of Mathematics, Novosibirsk (1983), preprint.Google Scholar
  56. [Mar88]
    Markov, V.T., Gelfand-Kirillov dimension: nilpotence, representability, nonmatrix varieties, In: Tez. Dokl. Sib. Shkola po Mnogoobr. Algebraicheskih Sistem, Barnaul (1988), 43–45.Google Scholar
  57. [Mis90]
    Mishchenko, S.P., A variant of a theorem on height for Lie algebras (Russian) Mat. Zametki 17 (1990), no. 4, 83–89; translation in Math. Notes 47 (1990), no. 3–4, 368–372.MathSciNetGoogle Scholar
  58. [Pch84]
    Pchelintzev, S.V., The height theorem for alternate algebras, Mat. Sb. 124 (1984), no. 4, 557–567.MathSciNetGoogle Scholar
  59. [Pro73]
    Procesi, C., Rings with polynomial identities, Research Notes in Mathematics 917. Marcel Dekker, New York, 1973.Google Scholar
  60. [Pro76]
    Procesi, C., The invariant theory of n×n matrices, Advances in Math. 19 (1976), 306–381.MATHCrossRefMathSciNetGoogle Scholar
  61. [Raz74a]
    Razmyslov, Yu P., Algebra and Logic 13 (1974)., no. 3, 192–204.MATHCrossRefGoogle Scholar
  62. [Raz74b]
    Razmyslov, Yu P., Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR Izv. 8 (1974), 724–760.CrossRefGoogle Scholar
  63. [Raz89]
    Razmyslov, Yu P. Identities of Algbras and their Representations, Nauka, Moscow (1989).Google Scholar
  64. [Reg72]
    Regev, A., Existence of identities in A ⊗ B, Israel J. Math., 11 (1972), 131–152.MATHCrossRefMathSciNetGoogle Scholar
  65. [Reg84]
    Regev, A., Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 47 (1984), 246–250.MATHCrossRefMathSciNetGoogle Scholar
  66. [Row88]
    Rowen, L.H., Ring Theory II, Pure and Applied Mathematics 128 Academic Press, New York, 1988.Google Scholar
  67. [Sch76]
    Schelter, W., Integral extensions of rings satisfying a polynomial identity, J. Algebra 40 (1976), 245–257; errata op. cit. 44 (1977), 576.MATHCrossRefMathSciNetGoogle Scholar
  68. [Scg78]
    Schelter, W., Noncommutative affine PI-algebras are catenary, J. Algebra 51 (1978), 12–18.MATHCrossRefMathSciNetGoogle Scholar
  69. [Shch01]
    Shchigolev, V.V., Finite basis property of T-spaces over fields of characteristic zero, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001) no. 5, 191–224; translation: Izv. Math. 65 (2001); no. 5, 1041–1071.MathSciNetGoogle Scholar
  70. [Shir57a]
    Shirshov, A.I., On some nonassociative nil-rings and algebraic algebras, Mat. Sb. 41 (1957), no. 3, 381–394.MathSciNetGoogle Scholar
  71. [Shir57b]
    Shirshov, A.I., On rings with identity relations, Mat. Sb. 43, (1957), no. 2, 277–283.MathSciNetGoogle Scholar
  72. [Sp50]
    Specht, W., Gesetze in Ringen I, Math. Z. 52 (1950), 557–589.MATHCrossRefMathSciNetGoogle Scholar
  73. [Ufn80]
    Ufnarovski’i, V.A., On Poincaré series of graded algebras, Mat. Zametki 27 (1980), no. 1, 21–32.MathSciNetGoogle Scholar
  74. [Ufn85]
    Ufnarovski’i, V.A., The independency theorem and its consequences, Mat. Sb., 128 (1985), no. 1, 124–13.MathSciNetGoogle Scholar
  75. [Ufn89]
    Ufnarovski’i, V.A., On regular words in Shirshov sense, In: Tez. Dokl. po Teorii Koletz, Algebr i Modulei. Mezhd. Konf. po Algebre Pamyati A.I. Mal’tzeva, Novosibirsk (1989), 140.Google Scholar
  76. [Ufn90]
    Ufnarovski’i, V.A., On using graphs for computing bases, growth functions and Hilbert series of associative algebras, Mat. Sb. 180 (1990), no. 11, 1548–1550.Google Scholar
  77. [VaZel89]
    Vais, A Ja., and Zelmanov, E.I., Kemer’s theorem for finitely generated Jordan algebras, Izv. Vyssh. Uchebn. Zved. Mat. (1989), no. 6, 42–51; translation: Soviet Math. (Iz. VUZ) 33 (1989), no. 6, 38–47.Google Scholar
  78. [Zel91]
    Zelmanov, E.I., The solution of the restricted Burnside problem for groups of prime power, Mimeographed notes, Yale University (1991)Google Scholar
  79. [ZelKos88]
    Zelmanov, E.I., On nilpotence of nilalgebras, Lect. Notes Math. 1352 (1988), 227–240.CrossRefMathSciNetGoogle Scholar
  80. [Zubk96]
    Zubkov, A.N., On a generalization of the Razmyslov-Procesi theorem. (Russian) Algebra i Logika 35 (1996), no. 4, 433–457, 498; translation in Algebra and Logic 35 (1996), no. 4, 241–254.MATHMathSciNetGoogle Scholar
  81. [Zubk00]
    Zubkov, A.N., Modules with good filtratión and invariant theory. Algebra — representation theory (Constanta, 2000), 439–460, NATO Sci. Ser. II Math. Phys. Chem. 28 Kluwer Acad. Publ., Dordrecht, 2001.Google Scholar
  82. [Zubr97]
    Zubrilin, K.A., On the largest nilpotent ideal in algebras satisfying Capelli identities, Sb. Math. 188 (1997), 1203–1211.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland 2009

Authors and Affiliations

  • Alexei Belov-Kanel
  • Louis H. Rowen

There are no affiliations available

Personalised recommendations