We study and classify those tame irreducible elliptic quasi-simple Lie algebras which are simply laced and of rank l 3. The first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates. When the type is D or E the coordinates are Laurent polynomials in & variables, while for type A the coordinates can be any quantum torus in & variables. The next step is to study the universal central extension as well as the derivation algebra of the core. These are related to the first Connes cyclic homology group of the coordinates. The final step is to use this information to give constructions of Lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras.
In this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal Lie algebras. To accomplish this we develop a general theory of ޚ n -graded Lie algebras with polynomial multiplication. We construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics.ᮊ 1999 Academic Press
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