| Journal: | Boletín de la Sociedad Matemática Mexicana |
| Database: | PERIÓDICA |
| System number: | 000345588 |
| ISSN: | 0037-8615 |
| Authors: | Cervantes Polanco, J.R1 Sánchez Valenzuela, O.A1 |
| Institutions: | 1Universidad de Guanajuato, Facultad de Matemáticas, Guanajuato. México |
| Year: | 2010 |
| Season: | Abr |
| Volumen: | 16 |
| Number: | 1 |
| Pages: | 29-38 |
| Country: | México |
| Language: | Inglés |
| Document type: | Artículo |
| Approach: | Experimental, analítico |
| English abstract | Let V = U ⊕W be a complex vector space. Then End(U ⊕W) has natural Lie algebra and Lie superalgebra structures. With given geometries BU : U×U → C, and BW : W×W → C, a geometry B can be defined on V via BU ⊕BW. We address the question of what determines the choice in using the Lie algebra or the Lie superalgebra structure of End(U ⊕ W) by considering the linear maps that preserve B. It is found that if nontrivial maps U → W and W → U are to be included, then the Lie algebra structure of End(U ⊕W) requires geometries on U and W of the same type —that is, both orthogonal, or both symplectic, or both unitary, or both anti-unitary— whereas the Lie superalgebra requires to combine the geometry-types of U and W in such a way that one is orthogonal and the other symplectic, or one is unitary and the other anti-unitary. The question of defining a geometry B on U ⊕ W of odd degree is also addressed, and the Lie algebra and Lie superalgebra structures of the subspace of End(U ⊕ W) that preserve such a B are determined |
| Disciplines: | Matemáticas |
| Keyword: | Matemáticas puras, Algebras de Lie, Grupos ortogonales, Grupos simplécticos |
| Keyword: | Mathematics, Pure mathematics, Lie algebras, Orthogonal groups, Symplectic groups |
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