Hiroshima Mathematical Journal

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

Hironao Kato

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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

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

First available in Project Euclid: 30 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Left invariant flat projective structure prehomogeneous vector space


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

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