Free nilpotent and H-type Lie algebras. Combinatorial and orthogonal designs

• Kenro Furutani ^[1] ; Irina Markina ^[2] ; Alexander Vasil'ev ^[2]
  1. [1] University of Tokyo

     University of Tokyo

     Japón

     
  2. [2] University of Bergen

     University of Bergen

     Noruega

     
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 12 (December 2015), 2015, págs. 5467-5492
• Idioma: inglés
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• Resumen

  • The aim of our paper is to construct pseudo H-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a byproduct result, we recover the existence of a rational structure on pseudo H-type algebras, which implies the existence of lattices on the corresponding pseudo H-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo H-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz–Radon orthogonal matrices.