Toral Lie algebra

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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[edit]

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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