In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a toral Lie algebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are diagonalizable

Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...

 (or semisimple). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent

Nilpotent

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....

 elements. Every toral Lie algebra is abelian; thus, its members are simultaneously diagonalizable.

Semisimple and reductive Lie algebras

A subalgebra H of a semisimple Lie algebra L is called toral if the adjoint representation

Adjoint endomorphism

In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups....

 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 a 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

Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras...

of L restricted to H is nondegenerate.

For more general Lie algebras, a Cartan algebra may differ from a maximal toral algebra.