Rings with involution whose symmetric elements are central
In a ring R with involution whose symmetric elements S are central, the skew-symmetric elements K form a Lie algebra over the commutative ring S. The classification of such rings which are 2-torsion free is equivalent to the classification of Lie algebras K over S equipped with a bilinear form f tha...
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| Format: | Article |
| Language: | English |
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Wiley
1980-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171280000178 |
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| author | Taw Pin Lim |
| author_facet | Taw Pin Lim |
| author_sort | Taw Pin Lim |
| collection | DOAJ |
| description | In a ring R with involution whose symmetric elements S are central, the skew-symmetric elements K form a Lie algebra over the commutative ring S. The classification of such rings which are 2-torsion free is equivalent to the classification of Lie algebras K over S equipped with a bilinear form f that is symmetric, invariant and satisfies [[x,y],z]=f(y,z)x−f(z,x)y. If S is a field of char ≠2, f≠0 and dimK>1 then K is a semisimple Lie algebra if and only if f is nondegenerate. Moreover, the derived algebra K′ is either the pure quaternions over S or a direct sum of mutually orthogonal abelian Lie ideals of dim≤2. |
| format | Article |
| id | doaj-art-5f56f1046f0c43f3a62ea20d30b09a65 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1980-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-5f56f1046f0c43f3a62ea20d30b09a652025-02-03T05:48:05ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251980-01-013224725310.1155/S0161171280000178Rings with involution whose symmetric elements are centralTaw Pin Lim0Department of Actuarial Mathematics, University of Manitoba, Winnipeg R3T 2N2, Manitoba, CanadaIn a ring R with involution whose symmetric elements S are central, the skew-symmetric elements K form a Lie algebra over the commutative ring S. The classification of such rings which are 2-torsion free is equivalent to the classification of Lie algebras K over S equipped with a bilinear form f that is symmetric, invariant and satisfies [[x,y],z]=f(y,z)x−f(z,x)y. If S is a field of char ≠2, f≠0 and dimK>1 then K is a semisimple Lie algebra if and only if f is nondegenerate. Moreover, the derived algebra K′ is either the pure quaternions over S or a direct sum of mutually orthogonal abelian Lie ideals of dim≤2.http://dx.doi.org/10.1155/S0161171280000178ring with involutionsymmetric and skew-symmetric elementsLie algebrasymmetric and invariant bilinear formCartan's criterion of semisimplicity of Lie algebraspure quaternionsmutually orthogonal abelian Lie ideals. |
| spellingShingle | Taw Pin Lim Rings with involution whose symmetric elements are central International Journal of Mathematics and Mathematical Sciences ring with involution symmetric and skew-symmetric elements Lie algebra symmetric and invariant bilinear form Cartan's criterion of semisimplicity of Lie algebras pure quaternions mutually orthogonal abelian Lie ideals. |
| title | Rings with involution whose symmetric elements are central |
| title_full | Rings with involution whose symmetric elements are central |
| title_fullStr | Rings with involution whose symmetric elements are central |
| title_full_unstemmed | Rings with involution whose symmetric elements are central |
| title_short | Rings with involution whose symmetric elements are central |
| title_sort | rings with involution whose symmetric elements are central |
| topic | ring with involution symmetric and skew-symmetric elements Lie algebra symmetric and invariant bilinear form Cartan's criterion of semisimplicity of Lie algebras pure quaternions mutually orthogonal abelian Lie ideals. |
| url | http://dx.doi.org/10.1155/S0161171280000178 |
| work_keys_str_mv | AT tawpinlim ringswithinvolutionwhosesymmetricelementsarecentral |