Complete classification of pseudo H-type Lie algebras: I
| Autor: | Irina Markina, Kenro Furutani |
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| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
High Energy Physics::Phenomenology 010102 general mathematics Clifford algebra Orthogonal complement Algebraic geometry Clifford module 01 natural sciences Nilpotent Lie algebra Differential geometry 0103 physical sciences Lie algebra 010307 mathematical physics Geometry and Topology Isomorphism 0101 mathematics Mathematics |
| Zdroj: | Geometriae Dedicata. 190:23-51 |
| ISSN: | 1572-9168 0046-5755 |
| DOI: | 10.1007/s10711-017-0225-1 |
| Popis: | 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 |
| Databáze: | OpenAIRE |
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