Derivation Lie Algebra In The Torus Algebras

In the thesis, i discuss three classes of infinite-dimensional subalgebras of derivations DerL on commutative torus L=C[t₁^±1,t₂^±1].These subalgebras are denoted as follows:1. Horizontal Vector Fields subalgebra (?)₁=Spanc{L_u,v|u,v∈Z}, Lie brackethere u,v,u₁,v₁,u₂,v₂∈Z.2. A class of complete subalgebra(?)₂=Spanc{L_i,L_u,v|i=1.2. u, v∈Z²\{(0,0)}}, Lie brackethere (u.v), (u₁,v₁),(u₂,v₂)∈Z²\{(0,0)}.3. A subalgebra as split extension of Witt algebra by its a module denoted(?)₃= Spanc{M_r,N_s|r,s∈Z}, Lie brackethere r, s∈Z.My main work include following:? Universal central extension,automorphism group and derivations of (?)₁,? Completeness and ideals of (?)₂,? Universal central extension and a class of representations of (?)₃.