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).[1] Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algebra is abelian;[1] 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, over an algebraically closed field of characteristic 0 is a Cartan subalgebra and vice versa.[2] 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 algebra may differ from a maximal toral algebra.

In a finite-dimensional semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$ over an algebraically closed field of a characteristic zero, a toral subalgebra exists.[1] 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

References

1. Humphreys, Ch. II, § 8.1. harvnb error: no target: CITEREFHumphreys (help)
2. Humphreys, Ch. IV, § 15.3. Corollary harvnb error: no target: CITEREFHumphreys (help)

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 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