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][2] 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 \( {\mathfrak {g}} \) ,\( {\displaystyle \operatorname {ad} ({\mathfrak {h}})\subset {\mathfrak {gl}}({\mathfrak {g}})} \) 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.[3] 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 algebra may differ from a maximal toral algebra.
In a finite-dimensional semisimple Lie algebra \( {\mathfrak {g}} \) over an algebraically closed field of a characteristic zero, a toral subalgebra exists.[1] 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
Humphreys, Ch. II, § 8.1.
Proof (from Humphreys): Let \( x\in {\mathfrak {h}} \). Since \( \operatorname {ad} (x) \) is diagonalizable, it is enough to show the eigenvalues of a \( {\displaystyle \operatorname {ad} _{\mathfrak {h}}(x)} \) are all zero. Let \( {\displaystyle y\in {\mathfrak {h}}} \) be an eigenvector of \( {\displaystyle \operatorname {ad} _{\mathfrak {h}}(x)} \) with eigenvalue \( \lambda \]) . Then x is a sum of eigenvectors of \( {\displaystyle \operatorname {ad} _{\mathfrak {h}}(y)} \) and then \( {\displaystyle -\lambda y=\operatorname {ad} _{\mathfrak {h}}(y)x} \) is a linear combination of eigenvectors of \( {\displaystyle \operatorname {ad} _{\mathfrak {h}}(y)} \) with nonzero eigenvalues. But, unless \( \lambda =0 \) we have that \( {\displaystyle -\lambda y} \) is an eigenvector of \( {\displaystyle \operatorname {ad} _{\mathfrak {h}}(y)} \) with eigenvalue zero, a contradiction. Thus, \( \lambda =0 \). \( \square \)
Humphreys, Ch. IV, § 15.3. Corollary
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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