On the Conjugacy of Maximal Toral Subalgebras of Certain Infinite-Dimensional Lie Algebras

Abstract

We will extend the conjugacy problem of maximal toral subalgebras for Lie algebras of the form $\g{g} \otimes_k R$ by considering $R=k[t,t^{-1}]$ and $R=k[t,t^{-1},(t-1)^{-1}]$, where $k$ is an algebraically closed field of characteristic zero and $\g{g}$ is a direct limit Lie algebra. In the process, we study properties of infinite matrices with entries in a B\'{e}zout domain and we also look at how our conjugacy results extend to universal central extensions of the suitable direct limit Lie algebras.