Constructing bases for radicals and nilradicals of Lie algebras

Abstract:

The radical and nilradical of a finite-dimensional Lie algebra $L$ are well defined unique subspaces of $L$. Nevertheless, we show the impossibility of ever finding a general algorithm that will construct finite bases for radicals (or nilradicals) of arbitrary finite-dimensional Lie algebras. Our approach involves an investigation of the relationship between radicals of associative algebras and radicals of Lie algebras. Building on a result of Richman in the constructive theory of associative algebras, we prove that bases for radicals and nilradicals of finite-dimensional Lie algebras over a discrete field $F$ can always be constructed if and only if $F$ satisfies Seidenberg’s condition P. A special case is that if we restrict ourselves to fields of characteristic zero, we can indeed always construct bases for radicals. Our proofs are entirely constructive (i.e., do not use the general law of excluded middle).