Advertisement

A combinatorial approach to the theory of Pi-algebras and exponential growth

  • A. Giambruno

Abstract

The theory of Pi-algebras has been developed in the last decade mainly through combinatorial methods pertaining to the representation theory of the symmetric group in characteristic zero. These methods together with the study of the asymptotic behavior of a numerical sequence related to a PI-algebra have lead to the notion of the PIexponent of an algebra. Through the scale determined by the exponent many results have been obtained in the last few years shedding new light on the understanding of the polynomial identities of an algebra.

Keywords

Associative Algebra Young Diagram Characteristic Zero Combinatorial Approach Polynomial Identity 
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. [1]
    Amitsur, S.A. (1968): Rings with involution. Israel J. Math. 6, 99–106MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Amitsur, S.A. (1968): Identities in rings with involution. Israel J. Math. 7, 63–68MathSciNetCrossRefGoogle Scholar
  3. [3]
    Bahturin, Y., Giambruno, A., Zaicev, M. (1999): G-identities on associative algebras. Proc. Amer. Math. Soc. 127, 63–69MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Berele, A., Giambruno, A., Regev, A. (1996): Involution codimensions and trace codimensions of matrices are asymptotically equal. Israel J. Math. 96, 49–62MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Berele, A., Regev, A. (1983): Applications of hook Young diagrams to P.I. algebras. J. Algebra 82, 559–567MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Berele, A., Regev, A. (1995): On the codimensions of the verbally prime P.I. algebras. Israel J. Math. 91, 239–247MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Berele, A., Regev, A. (1998): Codimensions of products and intersections of verbally prime T-ideals. Israel J. Math. 103, 17–28MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Curtis, C.W., Reiner, I. (1962): Representation theory of finite groups and associative algebras. Wiley, New YorkMATHGoogle Scholar
  9. [9]
    Drensky, V. (1987): Extremal varieties of algebras. I. Serdica 13, 320–332 (Russian)MathSciNetMATHGoogle Scholar
  10. [10]
    Drensky, V. (1988): Extremal varieties of algebras. II. Serdica 14, 20–27 (Russian)MathSciNetMATHGoogle Scholar
  11. [11]
    Drensky, V., Giambruno, A. (1994): Cocharacters, codimensions and Hilbert series of the polynomial identities for 2 × 2 matrices with involution. Canad. J. Math. 46, 718–733MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Giambruno, A., Mishchenko, S. (2000): Polynomial growth of the ✱-codimensions and Young diagrams. Comm. Algebra, to appearGoogle Scholar
  13. [13]
    Giambruno, A., Mishchenko, S. (2000): On star-varieties with almost polynomial growth. Algebra Colloq., to appearGoogle Scholar
  14. [14]
    Giambruno, A., Regev, A. (1985): Wreath products and P.I. algebras. J. Pure Appl. Algebra 35, 133–149MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Giambruno, A., Zaicev, M. (1998): On codimension growth of finitely generated associative algebras. Adv. Math. 140, 145–155MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Giambruno, A., Zaicev, M. (1999): Exponential codimension growth of PI-algebras: an exact estimate. Adv. Math. 142, 221–243MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Giambruno, A., Zaicev, M. (1999): Involution codimensions of finite dimensional algebras and exponential growth. J. Algebra 222, 471–484MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Giambruno, A., Zaicev, M. (2001): A characterization of algebras with polynomial growth of the codimensions. Proc. Amer. Math. Soc. 129, 59–67MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Giambruno, A., Zaicev, M. (2000): Minimal varieties of of algebras of exponential growth. Electron. Res. Announc. Amer. Math. Soc. 6, 40–44MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Giambruno, A., Zaicev, M. (2000): Minimal varieties of exponential growth. PreprintGoogle Scholar
  21. [21]
    Giambruno, A., Zaicev, M. (2000): A characterization of varieties of associative algebras of exponent two. Serdica Math. J., 26, 245–252MathSciNetGoogle Scholar
  22. [22]
    Kemer, A.R. (1978): T-ideals with power growth of the codimensions are Specht. Sibirsk. Mat. Zh. 19, 54–69 (Russian) [English transi: Siberian Math. J. 19, 37-48 (1978)]MathSciNetMATHGoogle Scholar
  23. [23]
    Kemer, A.R. (1991): Ideals of identities of associative algebras. (Translations of Mathematical Monographs, vol. 87). American Mathematical Society, Providence, RIMATHGoogle Scholar
  24. [24]
    Krakowsky, A., Regev, A. (1973): The polynomial identities of the Grassmann algebra. Trans. Amer. Math. Soc. 181, 429–438MathSciNetGoogle Scholar
  25. [25]
    James, G., Kerber, A. (1981): The representation theory of the symmetric group. (Encyclopedia of Mathematics and its Applications, vol. 16). Addison-Wesley, LondonMATHGoogle Scholar
  26. [26]
    Latyshev, V.N. (1977): Complexity of nonmatrix varieties of associative algebras. Algebra i Logika 16, 149–183 (Russian) [English transi.: Algebra and Logic 16, 98-122 (1977)MathSciNetGoogle Scholar
  27. [27]
    Lewin, J. (1974): A matrix representation for associative algebras. I. Trans. Amer. Math. Soc. 188, 293–308MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Loday, J.L., Procesi, C. (1988): Homology of symplectic and orthogonal algebras. Adv. Math. 69, 93–108MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    Malcev, J.N. (1971): A basis for the identities of the algebra of upper triangular matrices. Algebra i Logika 10, 393–400 (Russian) [English transi.: Algebra and Logic 10, 242-247 (1971)]MathSciNetGoogle Scholar
  30. [30]
    Mishchenko, S., Valenti, A. (2000): A star-variety with almost polynomial growth. J. Algebra 223, 66–84MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Procesi, C. (1967): Non-commutative affine rings. Atti Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 8, 239–255MathSciNetMATHGoogle Scholar
  32. [32]
    Regev, A. (1972): Existence of identities in AB. Israel J. Math. 11, 131–152MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Regev, A. (1984): Codimensions and trace codimensions of matrices are asymptotically equal. Israel J. Math. 47, 246–250MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Regev, A. (1981): Asymptotic values for degrees associated with strips of Young diagrams. Adv. Math. 41, 115–136MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    Rowen, L.H. (1988): Ring theory, vols. I, II. Academic Press, Boston, MAGoogle Scholar

Copyright information

© Springer-Verlag Italia 2001

Authors and Affiliations

  • A. Giambruno

There are no affiliations available

Personalised recommendations