Existence of Lattices on General H-Type Groups

Kenro Furutani¹ ; Irina Markina²

¹Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278-8510, Japan
²Department of Mathematics, University of Bergen, P.O.Box 7803, Bergen 5020, Norway

Journal of Lie Theory, Tome 24 (2014) no. 4, pp. 979-1011

Résumé

Let $\cal N$ be a two step nilpotent Lie algebra endowed with non-degenerate scalar product $\langle\cdot \,,\cdot\rangle$ and let ${\cal N}=V\oplus_{\perp}Z$, where $Z$ is the center of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We prove that if $(V,\langle\cdot\,,\cdot\rangle_V)$ is the Clifford module for the Clifford algebra ${\rm Cl} (Z,\langle\cdot\,,\cdot\rangle_Z)$ such that the homomorphism $J\colon {\rm Cl}(Z,\langle\cdot\,,\cdot\rangle_Z)\to{\rm End}(V)$ is skew symmetric with respect to the scalar product $\langle\cdot\,,\cdot\rangle_V$, or in other words the Lie algebra $\cal N$ satisfies conditions of general $H$-type Lie algebras [see P. Ciatti, Scalar products on Clifford modules and pseudo-H-type Lie algebras, Math. Nachr. 202 (2009) 44--68; and: M. Godoy Molina, A. Korolko and I. Markina, Sub-semi-Riemannian geometry of general $H$-type groups, Bull. Sci. Math. 137 (2013) 805--833], then there is a basis with respect to which the structural constants of the Lie algebra $\cal N$ are all $\pm 1$ or $0$.

Détail

Classification : 17B30, 22E25
Mots-clés : Clifford module, nilpotent two step algebra, lattice, general H-type algebras