Complete classification of H-type algebras: I

2015/12/10 by Kenrô Furutani, Furutani, Kenro, Irina Markina +1
Mathematics · #17B30 #17B60 #17B70 #22E15 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1512.03469

openalex publication_date 2015/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathscr N be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product ⟨. ,.⟩ and let \mathscr N=V⊕_⊥Z, where Z is the centre of the Lie algebra and V its orthogonal complement with respect to the scalar product. We study the classification of the Lie algebras for which the space V arises as a representation space of a Clifford algebra \Cl(\mathbb R^r,s) and the representation map J\colon \Cl(\mathbb R^r,s)→(V) is related to the Lie algebra structure by ⟨ J_zv,w⟩=⟨ z,[v,w]⟩ for all z∈ \mathbb R^r,s and v,w∈ V. The classification is based on the range of parameters r and s and is completed for the Clifford modules V, having minimal possible dimension, that are not necessary irreducible. We find the necessary condition for the existence of a Lie algebra isomorphism according to the range of integer parameters 0≤ r,s

Citations

• Integral Structures on H-type Lie Algebras

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