## Tokyo Journal of Mathematics

### The Intersection of Fixed Point Subgroups by Involutive Automorphisms of Compact Exceptional Lie Groups

Toshikazu MIYASHITA

#### Abstract

In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group $G = F_4, E_6, E_7$. We shall find involutive automorphisms of $G$ such that the connected component of the intersection of those fixed point subgroups coincides with the maximal torus of $G$.

#### Article information

Source
Tokyo J. Math., Volume 30, Number 2 (2007), 455-463.

Dates
First available in Project Euclid: 4 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1202136688

Digital Object Identifier
doi:10.3836/tjm/1202136688

Mathematical Reviews number (MathSciNet)
MR2376521

Zentralblatt MATH identifier
1151.22017

#### Citation

MIYASHITA, Toshikazu. The Intersection of Fixed Point Subgroups by Involutive Automorphisms of Compact Exceptional Lie Groups. Tokyo J. Math. 30 (2007), no. 2, 455--463. doi:10.3836/tjm/1202136688. https://projecteuclid.org/euclid.tjm/1202136688

#### References

• M. Berger, Les espaces symétriques non compacts, Ann. Sci. Ecole Norm. Sup., 74 (1957), 85–177.
• T. Miyashita, Fixed points subgroups $G^{\sigma, \gamma}$ by two involutive automorphisms $\sigma, \gamma$ of compact exceptional Lie groups $G = F_4, E_6$ and $E_7$, Tsukuba J. Math., 27 (2003), 199–215.
• T. Miyashita and I. Yokota, 2-graded decompositions of exceptional Lie algebra $\mathfrak{g}$ and group realizations of $\mathfrak{g}_{ev}, \mathfrak{g}_0$, Part III, $G = E_8$, Japanese J. Math., 26 (2000), 31–51.
• T. Miyashita and I. Yokota, Fixed points subgroups $G^{\sigma,\sigma'}$ by two involutive automorphisms $\sigma, \sigma'$ of compact exceptional Lie groups $G = F_4, E_6$ and $E_7$, Math. J. Toyama Univ., 24 (2001), 135–149.
• T. Miyashita and I. Yokota, Fixed points subgroups $G^{\gamma,\gamma'}$ by two involutive automorphisms $\gamma, \gamma'$ of compact exceptional Lie groups $G = G_2, F_4, E_6$ and $E_7$, Yokohama Math. J., 53 (2006), 9–38.
• T. Miyashita and I. Yokota, 3-graded decompositions of exceptional Lie algebra $\mathfrak{g}$ and group realizations of $\mathfrak{g}_{ev}, \mathfrak{g}_0$ and $\mathfrak{g}_{ed}$, Part II, $G =E_7$, Part II, Case 1, J. Math. Kyoto Univ., 46-2 (2006), 383–413.
• T. Miyashita and I. Yokota, 3-graded decompositions of exceptional Lie algebra $\mathfrak{g}$ and group realizations of $\mathfrak{g}_{ev}, \mathfrak{g}_0$ and $\mathfrak{g}_{ed}$, Part II, $G =E_7$, Part II, Case 2, 3 and 4, J. Math. Kyoto Univ., 46-4 (2006), 805–832.
• I. Yokota, Realization of involutive automorphisms $\sigma$ and $G^\sigma$ of exceptional linear Lie groups $G$, Part I, $G = G_2, F_4$, and $E_6$, Tsukuba J. Math., 4 (1990), 185–223.
• I. Yokota, Realization of involutive automorphisms $\sigma$ and $G^\sigma$ of exceptional linear Lie groups $G$, Part II, $G = E_7$, Tsukuba J. Math., 4 (1990), 378–404.
• I. Yokota, 2-graded decompositions of exceptional Lie algebra $\mathfrak{g}$ and group realizations of $\mathfrak{g}_{\it ev}, \mathfrak{g}_0$, Part I, $G = G_2, F_4, E_6$, Japanese J. Math., 24 (1998), 257–296.
• I. Yokota, 2-graded decompositions of exceptional Lie algebras $\mathfrak{g}$ and group realizations of $\mathfrak{g}_{\it ev}, \mathfrak{g}_0$, Part II, $G = E_7$, Japanese J. Math., 25 (1999), 155–179.
• I. Yokota, 3-graded decompositions of exceptional Lie algebra $\mathfrak{g}$ and group realizations of $\mathfrak{g}_{ev}, \mathfrak{g}_0$ and $\mathfrak{g}_{ed}$, Part II, $G = G_2, F_4, E_6$, Part I, J. Math. Kyoto Univ., 41-3 (2001), 449–474.