Toral subalgebra

In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Over an algebraically closed field, every toral Lie algebra is abelian; thus, its elements are simultaneously diagonalizable.

In semisimple and reductive Lie algebras

A subalgebra ${\displaystyle {\mathfrak {h}}}$ of a semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$ is called toral if the adjoint representation of ${\displaystyle {\mathfrak {h}}}$ on ${\displaystyle {\mathfrak {g}}}$, ${\displaystyle \operatorname {ad} ({\mathfrak {h}})\subset {\mathfrak {gl}}({\mathfrak {g}})}$ is a toral subalgebra. A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra,^{[citation needed]} 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 ${\displaystyle {\mathfrak {g}}}$ restricted to ${\displaystyle {\mathfrak {h}}}$ is nondegenerate.

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

In a finite-dimensional semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$ over an algebraically closed field of a characteristic zero, a toral subalgebra exists. In fact, if ${\displaystyle {\mathfrak {g}}}$ has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence, ${\displaystyle {\mathfrak {g}}}$ must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.

See also

• Maximal torus, in the theory of Lie groups