Complex forms of quaternionic symmetric spaces. (English) Zbl 1079.53067
Kowalski, Oldřich (ed.) et al., Complex, contact and symmetric manifolds. In honor of L. Vanhecke. Selected lectures from the international conference “Curvature in Geometry” held in Lecce, Italy, June 11–14, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-3850-4/hbk). Progress in Mathematics 234, 265-277 (2005).
In the present paper the author gives a complete classification of complex forms \(L/V\) of quaternionic symmetric spaces \(G/K\). The case where \(G\) is a classical group and \(\text{rank} (L)= \text{rank} (G)\) follows from the matrix considerations. The computer program LiE is used for the exceptional groups. Some examples of its applications show that the complexifications \(L_ {\mathbb C}\) and \(K_ {\mathbb C}\) are conjugate in \(G_ {\mathbb C}\). They fill the cases for \(G\) exceptional and \(\text{rank} (L)= \text{rank} (G)\), and the remaining exceptional equal rank cases are worked out in the subsequent section. Finally, the cases when \(\text{rank} (L)< \text{rank} (G)\) are considered.
For the entire collection see [Zbl 1062.53001].
For the entire collection see [Zbl 1062.53001].
Reviewer: Witold Mozgawa (Lublin)
MSC:
53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |
53C35 | Differential geometry of symmetric spaces |
53-04 | Software, source code, etc. for problems pertaining to differential geometry |