Rings with involution whose symmetric elements are central
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Date
1980-1-1
Authors
Lim, Taw Pin
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Abstract
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.
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Taw Pin Lim, “Rings with involution whose symmetric elements are central,” International Journal of Mathematics and Mathematical Sciences, vol. 3, no. 2, pp. 247-253, 1980. doi:10.1155/S0161171280000178