Lie algebra all of which elements are semisimple
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.
- ^ a b Humphreys 1972, Ch. II, ยง 8.1.
- ^ Proof (from Humphreys): Let . Since is diagonalizable, it is enough to show the eigenvalues of are all zero. Let be an eigenvector of with eigenvalue . Then is a sum of eigenvectors of and then is a linear combination of eigenvectors of with nonzero eigenvalues. But, unless , we have that is an eigenvector of with eigenvalue zero, a contradiction. Thus, .