Hiroshima Mathematical Journal

Left invariant flat projective structures on Lie groups and prehomogeneous vector spaces

Hironao Kato

Full-text: Open access

Abstract

We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a result, direct sums of special linear Lie algebras $\sll(2) \oplus \sll(m_1) \oplus \cdots \oplus \sll(m_k)$ admit left invariant flat complex projective structures if the equality $4 + m_1^2 + \cdots + m_k^2 -k - 4 m_1 m_2 \cdots m_k = 0$ holds. These contain $\sll(2)$, $\sll(2) \oplus \sll(3)$, $\sll(2) \oplus \sll(3) \oplus \sll(11)$ for example.

Article information

Source
Hiroshima Math. J., Volume 42, Number 1 (2012), 1-35.

Dates
First available in Project Euclid: 30 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1333113005

Digital Object Identifier
doi:10.32917/hmj/1333113005

Mathematical Reviews number (MathSciNet)
MR2952071

Zentralblatt MATH identifier
1294.53018

Subjects
Primary: 53B1 11S90: Prehomogeneous vector spaces
Secondary: 53C10: $G$-structures

Keywords
Left invariant flat projective structure prehomogeneous vector space

Citation

Kato, Hironao. Left invariant flat projective structures on Lie groups and prehomogeneous vector spaces. Hiroshima Math. J. 42 (2012), no. 1, 1--35. doi:10.32917/hmj/1333113005. https://projecteuclid.org/euclid.hmj/1333113005


Export citation

References

  • Y. Agaoka: Invariant flat projective structures on homogeneous spaces, Hokkaido Math. J. 11 (1982), 125–172.
  • Y. Agaoka, H. Kato: Invariants and left invariant flat projective structures on Lie groups, in preparation.
  • A. $\check{\mathrm{C}}$ap, J. Slov$\acute{\mathrm{a}}$k: Parabolic Geometries I, Background and General Theory, Math. Surveys Monogr. 154, Amer. Math. Soc. (2009).
  • A. Elduque: Invariant projectively flat affine connections on Lie groups, Hokkaido Math. J. 30 (2001), 231–239.
  • V.V. Gorbatsevich, A.L. Onishchik, E.B. Vinberg: Lie Groups and Lie Algebras III, Encyclopaedia Math. Sci. 41, Springer (1994).
  • A.L. Onishchik, E.B. Vinberg: Lie Groups and Algebraic Groups, Springer-Verlag Berlin Heidelberg (1990).
  • W. M. Goldman: Geometric structures on manifolds and varieties of representations, Geometry of Group Representation, Contemporary Math. 74 (1998), 169–198.
  • H. Kato: Low dimensional Lie groups admitting left invariant flat projective or affine structures, preprint.
  • H. Kim: The geometry of left-symmetric algebra, J. Korean. Math. Soc. 33 No. 4 (1996), 1047–1067.
  • T. Kimura: Introduction to Prehomogeneous Vector Spaces, Amer. Math. Soc. (2003).
  • S. Kobayashi: Transformation Groups in Differential Geometry, Springer-Verlag Berlin Heidelberg New York (1972).
  • S. Kobayashi, T. Nagano: On projective connections, J. Math. Mech. 13 (1964), 215–235.
  • S. Kobayashi, K. Nomizu: Foundations of Differential Geometry, Vol. II, Wiley-Interscience Publication, New York (1969).
  • S. Kobayashi, T. Ochiai: Holomorphic projective structures on compact complex surfaces, Math. Ann. 249 (1980), 75–94.
  • A. Martin Mendez, J. F. Torres Lopera: Homogeneous spaces with invariant flat Cartan structures, Indiana Univ. Math. J. 56 No.3 (2007), 1233–1260.
  • M. Sato: Theory of prehomogeneous vector spaces, Sūgaku no Ayumi 15 (1970), 85–157, Notes by Takuro Shintani, in Japanese.
  • M. Sato, T. Kimura: A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.
  • S. Sternberg: Lectures on Differential Geometry, Second edition, Chelsea Publishing, New York (1983).
  • N. Tanaka: On the equivalence problems associated with a certain class of homogeneous spaces, J. Math. Soc. Japan, 17 (1965), 103–139.
  • H. Urakawa: On invariant projectively flat affine connections, Hokkaido Math. J. 28 (1999), 333–356.