Complete classification of pseudo H -type Lie algebras: I
 作者： Kenro Furutani,  Irina Markina 作者单位： 1Science University of Tokyo2University of Bergen 刊名： Geometriae Dedicata, 2017, Vol.190 (1), pp.23-51 来源数据库： Springer Nature Journal DOI： 10.1007/s10711-017-0225-1 关键词： Clifford module;  Nilpotent 2-step Lie algebra;  Pseudo H -type Lie algebras;  Lie algebra isomorphism;  Scalar product;  Primary 17B60;  17B30;  17B70;  22E15; 原始语种摘要： Let $${\mathscr {N}}$$ be a 2-step nilpotent Lie algebra endowed with a non-degenerate scalar product $$\langle .\,,.\rangle$$ , and let $${\mathscr {N}}=V\oplus _{\perp }Z$$ , where Z is the centre of the Lie algebra and V its orthogonal complement. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra $${{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})$$ , and the representation map $$J:{{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(V)$$ is related to the Lie algebra structure by $$\langle J_zv,w\rangle =\langle z,[v,w]\rangle$$ for all $$z\in {\mathbb {R}}^{r,s}$$ and $$v,w\in V$$ . The classification depends on parameters r and s and is completed for the Clifford modules V having minimal... possible dimension, that are not necessary irreducible. We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters $$0\le r,s<\infty$$ . We present a constructive proof for the isomorphism maps for isomorphic Lie algebras and determine the class of non-isomorphic Lie algebras.

• algebra　代数学
• classification　分类
• isomorphism　同晶型
• isomorphic　同形的
• representation　表现
• constructive　建设的
• proof　证明
• degenerate　退化
• pseudo
• parameters　参数