A current group GX is an infinite-dimensional Lie group of smooth maps from a smooth manifold X to a finite-dimensional Lie group G, endowed with pointwise multiplication. This thesis concerns current groups G§ for compact Riemann surfaces §. We extend some results in the literature to discuss the topology of G§ where G has non-trivial fundamental group, and use these results to discuss the theory of central extensions of G§. The second object of interest in the thesis is the Jacobi group, which we think of as being associated to a compact Riemann surface of genus one. A connection is made between the Jacobi group and a certain central extension of G§. Finally, we define a generalisation of the Jacobi group that may be thought of as being associated to a compact Riemann surface of genus g ≥ 1.

Document intends to re-establish Hom-Lie algebra theory for a wider class of morphisms on the underlying coefficient algebra. A look is taken into deformed Witt and Virasoro algebras and a new direction is taken into further quasi-Hom-Lie VIrasoro-type extensions for different Witt algebras.

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\'zout domain and we also look at how our conjugacy results extend to universal central extensions of the suitable direct limit Lie algebras.

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\'zout domain and we also look at how our conjugacy results extend to universal central extensions of the suitable direct limit Lie algebras.