Gontcharov, Aleksandr2013-09-102013-09-1020132013http://hdl.handle.net/10393/26086http://dx.doi.org/10.20381/ruor-3198We 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.enLie algebrasrepresentation theoryBezout domainsBezout Ringscentral extensionsconjugacy problemdirect limit lie algebrasmaximal toral subalgebracartan subalgebrafinitely generated bezout domainThesis