Advertisement

Lie Maps and Related Topics

Part of the Frontiers in Mathematics book series (FM)

Abstract

Every associative ring A can be turned into a Lie ring by introducing a new product [x,y] = xy −yx. So we may regard A simultaneously as an associative ring and as a Lie ring. What is the connection between the associative and the Lie structure of A? This question has been studied for more than fifty years by numerous authors, most notably by Herstein and many of his students (see, for example, [113, 114, 115]). One of the first questions that one might ask in this context is: If rings B and A are isomorphic as Lie rings, are they then also isomorphic (or at least antiisomorphic) as associative rings? In more technical terms one can rephrase this question as whether a Lie isomorphism α: BA always “arises” from an (anti)isomorphism. This is just the simplest question that one can ask in this setting. More general (and from the point of view of the theory of Lie algebras also more natural) questions concern the structure of Lie homomorphisms between various Lie subrings of associative rings. Analogous problems can be formulated for Lie derivations.

Keywords

Direct Summand Prime Ring Associative Ring Unital Ring Jordan Derivation 
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.

Literature and Comments

  1. [1]
    J. Alaminos, M. Brešar, A. R. Villena, The strong degree and the structure of Lie and Jordan derivations from von Neumann algebras, Math. Proc. Camb. Phil. Soc. 137 (2004), 441–463.MATHCrossRefGoogle Scholar
  2. [4]
    G. Ancochea, Le théorème de von Staudt en géometrie projective quaternionienne, J. Reine Angew. Math. 184 (1942), 192–198.MathSciNetGoogle Scholar
  3. [5]
    G. Ancochea, On semi-automorphisms of division algebras, Ann. Math. 48 (1947), 147–153.CrossRefMathSciNetGoogle Scholar
  4. [7]
    P. Ara, M. Mathieu, Local multipliers of C*-algebras, Springer, 2003.Google Scholar
  5. [8]
    S. A. Ayupov, Anti-automorphisms of factors and Lie operator algebras, Quart. J. Math. 46 (1995), 129–140.MATHCrossRefMathSciNetGoogle Scholar
  6. [9]
    S. A. Ayupov, Skew commutators and Lie isomorphisms in real von Neumann algebras, J. Funct. Anal. 138 (1996), 170–187.MATHCrossRefMathSciNetGoogle Scholar
  7. [10]
    S. A. Ayupov, N. A. Azamov, Commutators and Lie isomorphisms of skew elements in prime operator algebras, Comm. Algebra 24 (1996), 1501–1520.MATHCrossRefMathSciNetGoogle Scholar
  8. [11]
    S. A. Ayupov, A. Rakhimov, S. Usmanov, Jordan, real and Lie structures in operator algebras, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997.MATHGoogle Scholar
  9. [12]
    R. Banning, M. Mathieu, Commutativity preserving mappings on semiprime rings, Comm. Algebra 25 (1997), 247–265.MATHCrossRefMathSciNetGoogle Scholar
  10. [23]
    K. I. Beidar, M. Brešar, M.A. Chebotar, W. S. Martindale 3rd, On Herstein’s Lie map conjectures, I, Trans. Amer. Math. Soc. 353 (2001), 4235–4260.MATHCrossRefMathSciNetGoogle Scholar
  11. [24]
    K. I. Beidar, M. Brešar, M.A. Chebotar, W. S. Martindale 3rd, On Herstein’s Lie map conjectures, II, J. Algebra 238 (2001), 239–264.MATHCrossRefMathSciNetGoogle Scholar
  12. [25]
    K. I. Beidar, M. Brešar, M.A. Chebotar, W. S. Martindale 3rd, On Herstein’s Lie map conjectures, III, J. Algebra 249 (2002), 59–94.MATHCrossRefMathSciNetGoogle Scholar
  13. [26]
    K. I. Beidar, M. Brešar, M.A. Chebotar, W. S. Martindale 3rd, Polynomial preserving maps on certain Jordan algebras, Israel J. Math. 141 (2004), 285–313.MATHMathSciNetGoogle Scholar
  14. [29]
    K. I. Beidar, M.A. Chebotar, On functional identities and d-free subsets of rings I, Comm. Algebra 28 (2000), 3925–3951.MATHCrossRefMathSciNetGoogle Scholar
  15. [30]
    K. I. Beidar, M.A. Chebotar, On functional identities and d-free subsets of rings II, Comm. Algebra 28 (2000), 3953–3972.MATHCrossRefGoogle Scholar
  16. [31]
    K. I. Beidar, M.A. Chebotar, On surjective Lie homomorphisms onto Lie ideals of prime rings, Comm. Algebra 29 (2001), 4775–4793.MATHCrossRefMathSciNetGoogle Scholar
  17. [32]
    K. I. Beidar, M.A. Chebotar, On Lie derivations of Lie ideals of prime rings, Israel J. Math. 123 (2001), 131–148.MATHMathSciNetGoogle Scholar
  18. [34]
    K. I. Beidar, Y. Fong, On additive isomorphisms of prime rings preserving polynomials, J. Algebra 217 (1999), 650–667.MATHCrossRefMathSciNetGoogle Scholar
  19. [39]
    K. I. Beidar, W. S. Martindale 3rd, A.V. Mikhalev, Lie isomorphisms in prime rings with involution, J. Algebra 169 (1994), 304–327.MATHCrossRefMathSciNetGoogle Scholar
  20. [47]
    D. Benkoviç, D. Eremita, Characterizing left centralizers by their action on a polynomial, Publ. Math. 64 (2004), 343–351.MATHGoogle Scholar
  21. [49]
    M. I. Berenguer, A. R. Villena, Continuity of Lie derivations on Banach algebras, Proc. Edinburgh Math. Soc. 41 (1998), 625–630.MATHMathSciNetGoogle Scholar
  22. [50]
    M. I. Berenguer, A. R. Villena, Continuity of Lie mappings of the skew elements of Banach algebras with involution, Proc. Amer. Math. Soc. 126 (1998), 2717–2720.MATHCrossRefMathSciNetGoogle Scholar
  23. [51]
    M. I. Berenguer, A. R. Villena, Continuity of Lie isomorphisms of Banach algebras, Bull. London Math. Soc. 31 (1999), 6–10.53MATHCrossRefMathSciNetGoogle Scholar
  24. [58]
    M. Brešar, Commuting traces of biadditive mappings, commutativitypreserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), 525–546.CrossRefMATHMathSciNetGoogle Scholar
  25. [70]
    M. Brešar, M. Cabrera, M. Fošner, A.R. Villena, Lie triple ideals and continuity of Lie triple isomorphisms on Jordan-Banach algebras, Studia Math. 169 (2005), 207–228.MATHMathSciNetCrossRefGoogle Scholar
  26. [78]
    M. Brešar, W. S. Martindale 3rd, C. R. Miers, Maps preserving n th powers, Comm. Algebra 26 (1998), 117–138.CrossRefMATHMathSciNetGoogle Scholar
  27. [79]
    M. Brešar, C. R. Miers, Commutativity preserving mappings of von Neumann algebras, Canad. J. Math. 45 (1993), 695–708.84MATHMathSciNetGoogle Scholar
  28. [87]
    M. A. Chebotar, On Lie automorphisms of simple rings of characteristic 2, Fundam. Prikl. Mat. 2 (1996), 1257–1268 (Russian).MATHMathSciNetGoogle Scholar
  29. [90]
    M. A. Chebotar, On Lie isomorphisms in prime rings with involution, Comm. Algebra 27 (1999), 2767–2777.MATHCrossRefMathSciNetGoogle Scholar
  30. [110]
    P. de la Harpe, Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert spaces, Lecture Notes Math. 285, Springer-Verlag, 1972.Google Scholar
  31. [111]
    I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331–341.MATHCrossRefMathSciNetGoogle Scholar
  32. [112]
    I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.CrossRefMathSciNetGoogle Scholar
  33. [113]
    I. N. Herstein, Lie and Jordan structures in simple, associative rings, Bull. Amer. Math. Soc. 67 (1961), 517–531.MATHCrossRefMathSciNetGoogle Scholar
  34. [114]
    I. N. Herstein, Topics in ring theory, The University of Chicago Press, Chicago, 1969.116MATHGoogle Scholar
  35. [117]
    L.-K. Hua, On the automorphisms of a field, Proc. Nat. Acad. Sci. U. S.A. 35 (1949), 386–389.MATHCrossRefMathSciNetGoogle Scholar
  36. [121]
    N. Jacobson, C.E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502.MATHCrossRefMathSciNetGoogle Scholar
  37. [122]
    N. Jacobson, C.E. Rickart, Homomorphisms of Jordan rings of self-adjoint elements, Trans. Amer. Math. Soc. 72 (1952), 310–322.MATHCrossRefMathSciNetGoogle Scholar
  38. [124]
    B. E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Camb. Phil. Soc. 120 (1996), 455–473.MATHGoogle Scholar
  39. [126]
    I. Kaplansky, Semi-automorphisms of rings, Duke Math. J. 14 (1947), 521–527.MATHCrossRefMathSciNetGoogle Scholar
  40. [132]
    L. A. Lagutina, Jordan homomorphisms of associative algebras with involution, Algebra i Logika 27 (1988), 402–417 (Russian).MathSciNetGoogle Scholar
  41. [146]
    R. Lü, K. Zhao, Structure of Weyl type Lie algebras, J. Algebra 306 (2006), 552–565.MATHCrossRefMathSciNetGoogle Scholar
  42. [147]
    W. S. Martindale 3rd, Lie isomorphisms of primitive rings, Proc. Amer. Math. Soc. 14 (1963), 909–916.MATHCrossRefMathSciNetGoogle Scholar
  43. [148]
    W. S. Martindale 3rd, Lie derivations of primitive rings, Michigan J. Math. 11 (1964), 183–187.MATHCrossRefMathSciNetGoogle Scholar
  44. [149]
    W. S. Martindale 3rd, Jordan homomorphisms of the symmetric elements of a ring with involution, J. Algebra 5 (1967), 232–249.MATHCrossRefMathSciNetGoogle Scholar
  45. [150]
    W. S. Martindale 3rd, Lie isomorphisms of simple rings, J. London Math. Soc. 44 (1969), 213–221.MATHCrossRefMathSciNetGoogle Scholar
  46. [151]
    W. S. Martindale 3rd, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 (1969), 437–455.MATHCrossRefMathSciNetGoogle Scholar
  47. [153]
    W. S. Martindale 3rd, A note on Lie isomorphisms, Canad. Math. Bull. 17 (1974), 243–245.MATHMathSciNetGoogle Scholar
  48. [154]
    W. S. Martindale 3rd, Lie isomorphisms of the skew elements of a simple ring with involution, J. Algebra 36 (1975), 408–415.MATHCrossRefMathSciNetGoogle Scholar
  49. [155]
    W. S. Martindale 3rd, Lie isomorphisms of the skew elements of a prime ring with involution, Comm. Algebra 4 (1976), 927–977.CrossRefMathSciNetGoogle Scholar
  50. [157]
    W. S. Martindale 3rd, Jordan homomorphisms onto nondegenerate Jordan algebras, J. Algebra 133 (1990), 500–511.MATHCrossRefMathSciNetGoogle Scholar
  51. [160]
    K. McCrimmon, The Zelmanov approach to Jordan homomorphisms of associative algebras, J. Algebra 123 (1989), 457–477.MATHCrossRefMathSciNetGoogle Scholar
  52. [161]
    C. R. Miers, Lie isomorphisms of factors, Trans. Amer. Math. Soc. 147 (1970), 55–63.MATHCrossRefMathSciNetGoogle Scholar
  53. [162]
    C. R. Miers, Lie homomorphisms of operator algebras, Pacific J. Math. 38 (1971), 717–735.MATHMathSciNetGoogle Scholar
  54. [163]
    C. R. Miers, Derived ring isomorphisms of von Neumann algebras, Canad. J. Math. 25 (1973), 1254–1268.MATHMathSciNetGoogle Scholar
  55. [164]
    C. R. Miers, Lie derivations of von Neumann algebras, Duke Math. J. 40 (1973), 403–409.MATHCrossRefMathSciNetGoogle Scholar
  56. [165]
    C. R. Miers, Lie *-triple homomorphisms into von Neumann algebras, Proc. Amer. Math. Soc. 58 (1976), 169–172.MATHCrossRefMathSciNetGoogle Scholar
  57. [166]
    C. R. Miers, Lie triple derivations of von Neumann algebras, Proc. Amer. Math. Soc. 71 (1978), 57–61.MATHCrossRefMathSciNetGoogle Scholar
  58. [170]
    S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. Algebra 195 (1997), 558–579.MATHCrossRefMathSciNetGoogle Scholar
  59. [185]
    M. P. Rosen, Lie isomorphisms of a certain class of prime rings, J. Algebra 89 (1984), 291–317.MATHCrossRefMathSciNetGoogle Scholar
  60. [192]
    G. A. Swain, Lie derivations of the skew elements of prime rings with involution, J. Algebra 184 (1996), 679–704.MATHCrossRefMathSciNetGoogle Scholar
  61. [194]
    A. R. Villena, Lie derivations on Banach algebras, J. Algebra 226 (2000), 390–409.MATHCrossRefMathSciNetGoogle Scholar
  62. [199]
    E. I. Zelmanov, On prime Jordan algebras II, Siberian Math. J. 24 (1983), 89–104.MathSciNetGoogle Scholar
  63. [200]
    K. Zhao, Weyl type algebras from quantum tori, Commun. Contemp. Math. 8 (2006), 135–165.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag 2007

Personalised recommendations