Non-commutative Determinants and Quaternionic Monge-Ampère Equations

  • Semyon Alesker
Part of the Trends in Mathematics book series (TM)


First we give a survey of the notions and the properties of non-commutative determinants of Moore and Dieudonne, especially in the quaternionic case, with a particular emphasis to applications in quaternionic analysis. Then we review the theory of plurisubharmonic functions and Monge-Ampere equations in quaternionic variables, following [4] and [5].


Moore determinant Monge-Ampére equation plurisubharmonic function 

Mathematics Subject Classification (2000)

15A33 32U05 35J60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W.W. Adams, C.A. Berenstein, P. Loustaunau, I. Sabadini, I, D.C. Struppa, Regular functions of several quaternionic variables and the Cauchy-Fleeter complex. J. Geom. Anal. 9 (1999), no. 1, 1–15.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A.D. Aleksandrov, Die gemischte Diskriminanten und die gemischte Volumina. Math. Sbornik 3 (1938), 227–251.zbMATHGoogle Scholar
  3. [3]
    A.D. Aleksandrov, Dirichlet’s problem for the equation Det \(\varphi ({z_1},...,{z_n},z,{x_1},...,{x_n}) \),I. (Russian) Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1958), no. 1, 5–24.MathSciNetGoogle Scholar
  4. [4]
    S. Alesker, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables. Bull. Sci. Math. 127 (2003), no. 1, 1–35. also: math.CV/0104209MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Alesker, Quaternionic Monge-Ampère equations. J. Geom. Anal. 13 (2003), no. 2, 183–216. also: math.CV/0208005.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    E. Artin, Geometric algebra. Interscience Publishers, Inc., New York-London, 1957.zbMATHGoogle Scholar
  7. [7]
    H. Aslaksen,Quaternionic determinants. Math. Intelligencer 18 (1996), no. 3, 57–65.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Aubin, Equations du type de Monge-Ampère sur les variétés kähleriennes compactes. C.R. Acad. Sci. Paris 283 (1976), 119–121.MathSciNetzbMATHGoogle Scholar
  9. [9]
    T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 252. Springer-Verlag, New York, 1982.Google Scholar
  10. [10]
    E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37 (1976), no. 1, 1–44.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F. Brackx, R. Delanghe, F. Sommen, Clifford analysis. Research Notes in Mathematics, 76. Pitman (Advanced Publishing Program), Boston, MA, 1982.Google Scholar
  12. [12]
    L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L. Caffarelli, J.J. Kohn, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (1985), no. 2, 209–252.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S.Y. Cheng, S.-T. Yau, On the regularity of the Monge-Ampère equation det(∂ 2 u/∂xi) = F(x, u). Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S.Y. Cheng, S.-T. Yau, The real Monge-Ampére equation and affine fiat structures. Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980), 339–370, Science Press, Beijing, 1982.Google Scholar
  16. [16]
    S.S. Chern, H.I. Levine, L. Nirenberg, Intrinsic norms on a complex manifold. Global Analysis (Papers in Honor of K. Kodaira), 119–139 Univ. Tokyo Press, Tokyo, 1969.Google Scholar
  17. [17]
    J. Dieudonné, Les déterminants sur un corps non commutatif. (French) Bull. Soc. Math. France 71 (1943), 27–45.MathSciNetzbMATHGoogle Scholar
  18. [18]
    R. Fueter, Die Funktionentheotie der Differentialgleichungen △u = 0 und △△u = 0 mit vier reelen Variablen. Comment. Math. Helv. 7 (1935), 307–330.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    R. Fueter, Über die analytische Darstellung der regulären Functionen einer Quaternionenvariablen. Comment. Math. Helv. 8 (1936), 371–378.MathSciNetCrossRefGoogle Scholar
  20. [20]
    I. Gelfand, V. Retakh, Determinants of matrices over noncommutative rings. (Russian) Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 13–25, 96; translation in Funct. Anal. Appl. 25 (1991), no. 2, 91–102.Google Scholar
  21. [21]
    I. Gelfand, V. Retakh, R.L. Wilson, Quaternionic quasideterminants and determinants. math.QA/0206211.Google Scholar
  22. [22]
    I. Gelfand, S. Gelfand, V. Retakh, R.L. Wilson, Quasideterminants. math.QA/0208146.Google Scholar
  23. [23]
    F. Gürsey, H.C. Tze, Complex and quaternionic analyticity in chiral and gauge theories. I. Ann. Physics 128 (1980), no. 1, 29–130.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    G. Henkin, Private communication.Google Scholar
  25. [25]
    D. Joyce, Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000.Google Scholar
  26. [26]
    N.V. Krylov, Smoothness of the value function for a controlled diffusion process in a domain, Izvestiya Akademii Nauk SSSR, seriya matematicheskaya, Vol. 53, No. 1 (1989), 66–96 in Russian; English translation in Math. USSR Izvestija, Vol. 34, No. 1 (1990)Google Scholar
  27. [27]
    J.C. Maxwell, The Scientific Letters and Papers of James Clerk Maxwell, Volume II, 1862–1873, edited by P. M. Harman, Cambridge Univ. Press, 1995.Google Scholar
  28. [28]
    G.C. Moisil, Sur les quaternions monogènes. Bull. Sci. Math. 55 (1931), 168–194.zbMATHGoogle Scholar
  29. [29]
    E.H. Moore, On the determinant of an hermitian matrix of quaternionic elements. Bull. Amer. Math. Soc. 28 (1922), 161–162.zbMATHGoogle Scholar
  30. [30]
    V.P. Palamodov, Holomorphic synthesis of monogenic functions of several quaternionic variables. J. Anal. Math. 78 (1999), 177–204.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    D. Pertici, Quaternion regular functions and domains of regularity. Boll. Un. Mat. Ital. B (7) 7 (1993), no. 4, 973–988.MathSciNetzbMATHGoogle Scholar
  32. [32]
    A.V. Pogorelov, The regularity of the generalized solutions of the equation det \(({\partial ^2}u/\partial {x^i}\partial {x^j}) = \varphi ({x^1},{x^2},...,{x^n}) > 0 \) (Russian) Dokl. Akad. Nauk SSSR 200 (1971), 534–537.MathSciNetGoogle Scholar
  33. [33]
    A.V. Pogorelov, The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation. (Russian) Dokl. Akad. Nauk SSSR 201 (1971), 790–793.MathSciNetGoogle Scholar
  34. [34]
    A.V. Pogorelov, A regular solution of the n-dimensional Minkowski problem. Dokl. Akad. Nauk SSSR 199, 785–788 (Russian); translated as Soviet Math. Dokl. 12 (1971), 1192–1196.MathSciNetzbMATHGoogle Scholar
  35. [35]
    A.V. Pogorelov, Mnogomernoe uravnenie Monzha-Ampera det \(\left| {\left| {{z_{ij}}} \right|} \right| = \emptyset ({z_1},...,{z_n},z,{x_1},...,{x_n}) \). (Russian) [The multidimensional Monge-Ampèreequation det \(\left| {\left| {{{z}_{{ij}}}} \right|} \right| = \phi ({{z}_{1}},...,{{z}_{n}},z,{{x}_{1}},...,{{x}_{n}}) \) “Nauka”, Moscow, 1988.Google Scholar
  36. [36]
    J. Rauch, B.A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation. Rocky Mountain J. Math. 7 (1977), no. 2, 345–364.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Sudbery, Quaternionic analysis. Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 199–224.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    S.T. Yau, On Calabi’s conjecture and some new results in algebraic geometry. Proceedings of the National Academy of Sciences of the U.S.A. 74 (1977), 1798–1799.CrossRefzbMATHGoogle Scholar
  39. [39]
    S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Semyon Alesker
    • 1
  1. 1.Department of MathematicsTel Aviv University, Ramat AvivTel AvivIsrael

Personalised recommendations