
Journal of Lie Theory 24 (2014), No. 4, 9791011 Copyright Heldermann Verlag 2014 Existence of Lattices on General HType Groups Kenro Furutani Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 2788510, Japan furutani_kenro@ma.noda.tus.ac.jp Irina Markina Department of Mathematics, University of Bergen, P.O.Box 7803, Bergen 5020, Norway irina.markina@math.uib.no [Abstractpdf] Let $\cal N$ be a two step nilpotent Lie algebra endowed with nondegenerate 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 pseudoHtype Lie algebras, Math. Nachr. 202 (2009) 4468; and: M. Godoy Molina, A. Korolko and I. Markina, SubsemiRiemannian geometry of general $H$type groups, Bull. Sci. Math. 137 (2013) 805833], then there is a basis with respect to which the structural constants of the Lie algebra $\cal N$ are all $\pm 1$ or $0$. Keywords: Clifford module, nilpotent two step algebra, lattice, general Htype algebras. MSC: 17B30, 22E25 [ Fulltextpdf (445 KB)] for subscribers only. 