Advertisement

Annals of Global Analysis and Geometry

, Volume 53, Issue 3, pp 405–443 | Cite as

Left-symmetric algebras and homogeneous improper affine spheres

  • Daniel J. F. Fox
Article
First Online:
Received:
Accepted:
  • 71 Downloads

Abstract

The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.

Keywords

Left-symmetric algebra Affine spheres Cayley hypersurface 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Auslander, L.: Simply transitive groups of affine motions. Am. J. Math. 99(4), 809–826 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benoist, Y.: Une nilvariété non affine. J. Differ. Geom. 41(1), 21–52 (1995)CrossRefzbMATHGoogle Scholar
  3. 3.
    Burde, D.: Étale Affine Representations of the Lie Groups, Geometry and Representation Theory of Real and \(p\)-Adic Groups (Cordoba, 1995), Progr. Math., vol. 158, pp. 35–44. Birkhäuser Boston, Boston (1998)CrossRefGoogle Scholar
  4. 4.
    Choi, Y.: Nondegenerate affine homogeneous domain over a graph. J. Korean Math. Soc. 43(6), 1301–1324 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Choi, Y., Chang, K.: Hessian geometry of the homogeneous graph domain. Commun. Korean Math. Soc. 22(3), 419–428 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choi, Y., Kim, H.: A characterization of Cayley hypersurface and Eastwood and Ezhov conjecture. Int. J. Math. 16(8), 841–862 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eastwood, M., Ezhov, V.: Cayley hypersurfaces. Tr. Mat. Inst. Steklova 253, 241–244 (2006). (Kompleks. Anal. i Prilozh.)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Elduque, A., Myung, H.C.: On Transitive Left-Symmetric Algebras, Non-Associative Algebra and its Applications (Oviedo, 1993), Math. Appl., vol. 303, pp. 114–121. Kluwer Academic Publishers, Dordrecht (1994)CrossRefGoogle Scholar
  9. 9.
    Etherington, I.M.H.: Special train algebras. Q. J. Math. Oxf. Ser. 12, 1–8 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    fedja (http://mathoverflow.net/users/1131/fedja). Zeros of gradient of positive polynomials. MathOverflow, http://mathoverflow.net/q/38634 (version: 2010-09-14)
  11. 11.
    Fox, D.J.F.: Functions dividing their Hessian determinants and affine spheres. Asian J. Math. 20(3), 503–530 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goldman, W.M., Hirsch, M.W.: Affine manifolds and orbits of algebraic groups. Trans. Am. Math. Soc. 295(1), 175–198 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Helmstetter, J.: Radical d’une algèbre symétrique à gauche. Ann. Inst. Fourier (Grenoble) 29(4), 17–35 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jacobson, N.: Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, vol. 10. Interscience Publishers, New York (1962)Google Scholar
  15. 15.
    Kang, Y.-F., Bai, C.-M.: Refinement of Ado’s theorem in low dimensions and application in affine geometry. Commun. Algebra 36(1), 82–93 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kim, H.: Complete left-invariant affine structures on nilpotent Lie groups. J. Differ. Geom. 24(3), 373–394 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kim, H.: The geometry of left-symmetric algebra. J. Korean Math. Soc. 33(4), 1047–1067 (1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kim, H.: Developing maps of affinely flat Lie groups. Bull. Korean Math. Soc. 34(4), 509–518 (1997)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mizuhara, A.: On left symmetric algebras with a principal idempotent. Math. Jpn. 49(1), 39–50 (1999)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Mizuhara, A.: On simple left symmetric algebras over a solvable Lie algebra. Sci. Math. Jpn. 57(2), 325–337 (2003)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Segal, D.: The structure of complete left-symmetric algebras. Math. Ann. 293(3), 569–578 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shima, H.: Homogeneous Hessian manifolds. Ann. Inst. Fourier (Grenoble) 30(3), 91–128 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vinberg, È.B.: The theory of homogeneous convex cones. Tr. Mosk. Mat. Obs. 12, 303–358 (1963)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Vinberg, È.B., Gorbatsevich, V.V., Onishchik, A.L.: Structure of Lie Groups and Lie Algebras, Lie Groups and Lie Algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41, p. iv+248. Springer, Berlin (1994)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada a la Ingeniería Industrial, Escuela Técnica Superior de Ingeniería y Diseão IndustrialUniversidad Politécnica de MadridMadridSpain

Personalised recommendations