Algebras of diffeomorphisms of the N-torus

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The general algebra 𝒟(N,V) of infinitesimal diffeomorphisms of the N-torus (S1)N involving generators Lv,m depending on a structure vector v∈CN and a vector-valued index m∈ZN 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 rQ(V), is defined and certain related ‘‘Cartan matrices’’ are introduced. It is shown that 𝒟(N,V), as an ungraded algebra, is simple if and only if rQ(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, rQ(V)=N, it is shown that the central extension is unique. For the case dim V=1,rQ(V)<N, a new and greatly enlarged central extension is constructed involving a central charge function from ZN−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.