8-dimensional hypercomplex nilmanifolds
by
Anna Fino
Universita' di Torino, Italy
Coauthors: Isabel Dotti (FaMAF, Universidad Nacional de Cordoba, Argentina)

Oral Communication

Let N be a simply connected nilpotent Lie group of dimension 8 endowed with an invariant hypercomplex structure, that is a pair {J₁, J₂ } of anticommuting complex structures. There is no classification of real nilpotent Lie algebras of dimension 8. By imposing the condition of admitting a hypercomplex structure, strong restrictions appear. For example, the corresponding Lie group is 2-step nilpotent and its first Betti number b₁ (N) > 3. When b₁ (N) = 4, 5 we classify, in this work, all simply connected nilpotent 8-dimensional Lie groups with an invariant hypercomplex structure. If b₁ (N) = 6, 7 the hypercomplex structure is abelian and the classification of the corresponding groups in this case was given in [I. Dotti, A. Fino, Ann. Glob. Anal. and Geom. 18 (2000), 47-50] obtaining either N₁ = R³ ×H (4, R) (with b₁ (N) = 7) or N₂ = R² ×H(1, C) (with
b₁ (N) = 6) or N₃ = R ×H₃ (1) (with b₁ (N) = 5), where H (4, R), H(1, C) and H₃ (1) are the real, complex and quaternionic Heisenberg group of dimension 5, 6, 7, respectively. In the general classification we find that N₃ admits also non abelian hypercomplex structures. Among the groups with b₁ (N) = 4 admitting hypercomplex structures we obtained the 2-step nilpotent Lie group N (U(2), C²) constructed using the standard representation of U (2) on C². This group has a canonical metric which is naturally reductive and hyperhermitian with respect to the hypercomplex structure found.

Date received: May 27, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-67.