Symmetric matrices, orthogonal Lie algebras, and Lie-Yamaguti algebras
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2013
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| _version_ | 1866911064548966400 |
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| author | Benito, Pilar Bremner, Murray Madariaga, Sara |
| author_facet | Benito, Pilar Bremner, Murray Madariaga, Sara |
| contents | On the set H_n(K) of symmetric n by n matrices over the field K we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these non-associative structures have the orthogonal Lie algebra so(n,K) as derivation algebra. This gives an embedding of so(n,K) into so(N,K) for N = n(n+1)/2 - 1. We obtain a sequence of reductive pairs (so(N,K), so(n,K)) that provides a family of irreducible Lie-Yamaguti algebras. In this paper we explain in detail the construction of these Lie-Yamaguti algebras. In the cases n < 5, we use computer algebra to determine the polynomial identities of degree < 7; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1312_5008 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Symmetric matrices, orthogonal Lie algebras, and Lie-Yamaguti algebras Benito, Pilar Bremner, Murray Madariaga, Sara Rings and Algebras Representation Theory Primary 17A30. Secondary 17A40, 17A60, 17B10, 17B60, 17C50 On the set H_n(K) of symmetric n by n matrices over the field K we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these non-associative structures have the orthogonal Lie algebra so(n,K) as derivation algebra. This gives an embedding of so(n,K) into so(N,K) for N = n(n+1)/2 - 1. We obtain a sequence of reductive pairs (so(N,K), so(n,K)) that provides a family of irreducible Lie-Yamaguti algebras. In this paper we explain in detail the construction of these Lie-Yamaguti algebras. In the cases n < 5, we use computer algebra to determine the polynomial identities of degree < 7; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product. |
| title | Symmetric matrices, orthogonal Lie algebras, and Lie-Yamaguti algebras |
| topic | Rings and Algebras Representation Theory Primary 17A30. Secondary 17A40, 17A60, 17B10, 17B60, 17C50 |
| url | https://arxiv.org/abs/1312.5008 |