Algebras of diffeomorphisms of the N-torus

The general algebra script D(N,V) of infinitesimal diffeomorphisms of the N-torus (S¹)^N involving generators L_v,m depending on a structure vector υ∈ℂ^N and a vector-valued index m∈ℤ^N is constructed. Several results are proved for this algebra. Special cases of the algebra were previously presented in Ramos-Shrock [E. Ramos and R. E. Shrock, Int. J. Mod. Phys. A 4, 4279 (1989).] The concept of the ℚ rank of the space V of structure vectors, denoted r_ℚ (V), is defined and certain related "Cartan matrices" are introduced. It is shown that script D(N, V), as an ungraded algebra, is simple if and only if r_ℚ (V) = N, i.e., the ℚ rank of V is maximal. A classification under isomorphisms is given for the algebra and is shown to reduce to a classification of the Cartan matrices. The space of structure vectors is isomorphic to the unique "Cartan subalgebra." A number of properties concerning the central extensions of these algebras are then proved. For the case dim V=1, r _ℚ (V) = N, it is shown that the central extension is unique. For the case dim V = 1,r_ℚ (V) < N, a new and greatly enlarged central extension is constructed involving a central charge function from ℤ^N-r to ℂ (essentially equivalent to an infinite number of central charge parameters), rather than a single central charge parameter. Finally, it is proved that for dim V>2, this algebra has no nontrivial central extension.