Journal of Lie Theory 12 (2002), No. 1, 069--079
Copyright Heldermann Verlag 2002
Integral Structures on H-type Lie Algebras
Gordon Crandall
Dept. of Mathematics, LaGuardia Community College, City University of New York, 31-10 Thomson Avenue, Long Island City, NY 11101, U.S.A.
Józef Dodziuk
Graduate Center, City University, 365 Fifth Avenue, New York, NY 10016, U.S.A.
We prove that every H-type Lie algebra possesses a basis with respect to which the structure constants are integers. Existence of such an integral basis implies via the Mal'cev criterion that all simply connected H-type Lie groups contain co-compact lattices. Since the Campbell-Hausdorff formula is very simple for two-step nilpotent Lie groups we can actually avoid invoking the Mal'cev criterion and exhibit our lattices in an explicit way. As an application, we calculate the isoperimetric dimensions of H-type groups.
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