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Toral Lie algebra
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Toral Lie algebra : ウィキペディア英語版
Toral Lie algebra
In mathematics, a toral Lie algebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algebra is abelian; thus, its elements are simultaneously diagonalizable.
== Semisimple and reductive Lie algebras ==
A subalgebra ''H'' of a semisimple Lie algebra ''L'' is called toral if the adjoint representation of ''H'' on ''L'', ''ad''(''H'')⊂''gl''(''L'') is a toral Lie algebra. A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra, over an algebraically closed field of characteristic 0 is a Cartan subalgebra and vice versa. In particular, a maximal toral Lie subalgebra in this setting is self-normalizing, coincides with its centralizer, and the Killing form of ''L'' restricted to ''H'' is nondegenerate.
For more general Lie algebras, a Cartan algebra may differ from a maximal toral algebra.

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