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). 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.

In semisimple and reductive Lie algebras

A subalgebra $\mathfrak{𝔥}$ of a semisimple Lie algebra $\mathfrak{𝔤}$ is called toral if the adjoint representation of $\mathfrak{𝔥}$ on $\mathfrak{𝔤}$, $\mathrm{ad}(\mathfrak{𝔥})\subset \mathfrak{𝔤}\mathfrak{𝔩}(\mathfrak{𝔤})$ 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, 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 $\mathfrak{𝔤}$ restricted to $\mathfrak{𝔥}$ is nondegenerate. For more general Lie algebras, a Cartan subalgebra may differ from a maximal toral subalgebra. In a finite-dimensional semisimple Lie algebra $\mathfrak{𝔤}$ over an algebraically closed field of a characteristic zero, a toral subalgebra exists. In fact, if $\mathfrak{𝔤}$ has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence, $\mathfrak{𝔤}$ 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

References

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
• Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7