## Nagoya Mathematical Journal

### Tangent loci and certain linear sections of adjoint varieties

#### Abstract

An adjoint variety $X(\mathfrak{g})$ associated to a complex simple Lie algebra $\mathfrak{g}$ is by definition a projective variety in $\mathbb {P}_{*}(\mathfrak{g})$ obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in $\mathfrak{g}$. We first describe the tangent loci of $X(\mathfrak{g})$ in terms of $\mathfrak{sl}_{2}$-triples. Secondly for a graded decomposition of contact type $\mathfrak{g} = \oplus_{-2 \le i \le 2}\, \mathfrak{g}_{i}$, we show that the intersection of $X(\mathfrak{g})$ and the linear subspace $\mathbb {P}_*(\mathfrak{g}_{1})$ in $\mathbb {P}_{*}$$(\mathfrak{g})$ coincides with the cubic Veronese variety associated to $\mathfrak{g}$.

#### Article information

Source
Nagoya Math. J., Volume 158 (2000), 63-72.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631375

Mathematical Reviews number (MathSciNet)
MR1766575

Zentralblatt MATH identifier
0982.14028

Subjects
Secondary: 17B99: None of the above, but in this section

#### Citation

Kaji, Hajime; Yasukura, Osami. Tangent loci and certain linear sections of adjoint varieties. Nagoya Math. J. 158 (2000), 63--72. https://projecteuclid.org/euclid.nmj/1114631375

#### References

• H. Asano, On triple systems (in Japanese) , Yokohama City Univ. Ronso, Ser. Natural Sci., 27 (1975), 7–31.
• H. Asano, Symplectic triple systems and simple Lie algebras (in Japanese) , RIMS Kokyuroku, Kyoto Univ., 308 (1977), 41–54.
• D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York (1993).
• H. Freudenthal, Lie groups in the foundations of geometry , Advances in Math., 1 (1964), 145–190.
• T. Fujita and J. Roberts, Varieties with small secant varieties : The extremal case , Amer. J. Math., 103 (1981), 953–976.
• W. Fulton and J. Harris, Representation Theory : A First Course, GTM 129, Springer-Verlag, New York (1991).
• J. E. Humphreys, Introduction to Lie algebras and representation theory, GTM 9, Springer-Verlag, New York (1972).
• J. E. Humphreys, Linear Algebraic Groups, GTM 21, Springer-Verlag, New York (1975).
• H. Kaji, M. Ohno and O. Yasukura, Adjoint varieties and their secant varieties , Indag. Math., 10 (1999), 45–57.
• S. Mukai, Projective geometry of homogeneous spaces (in Japanese) , Proc. Symp. Algebraic Geometry, “Projective Varieties/Projective Geometry of Algebraic Varieties”, Waseda University, Japan (1994), 1–52.