Coisotropicity of Fixed Points under Torus Action on the Variety of Lagrangian Subalgebras

Let g be a complex semi-simple Lie algebra with adjoint group G. Consider the standard Manin triple structure on the double Lie algebra d:=g+g and the associated variety of Lagrangian subalgebras L(d). In this thesis, we study the variety CL(d) of coisotropic subalgebras of g, which is embedded as a subvariety of L(d). The diagonal maximal torus H_{\Delta} in G\times G acts on L(d) and CL(d). We describe the connected components of the fixed point set L(d)^{H_{\Delta}} using toric varieties, and we compute the irreducible components of the fixed point set CL(d)^{H_{\Delta}}. We provide the first examples of continuous families in CL(d)^{H_{\Delta}}. Our results may be viewed as completing and unifying the earlier works of Zambon, Kroeger and Le.