Inventiones Mathematicae Manuscript-nr. Extended Ane Root System Iv (simply-laced Elliptic Lie Algebras)

Let (R;G) be a pair consisting of an elliptic root system R with a marking G. Assume that the attached elliptic Dynkin diagram (R;G) is simply-laced (see Sect. 2). We associate three Lie algebras, explained in 1), 2) and 3) below, to the elliptic root system, and show that all three are isomorphic. The isomorphism class is called the elliptic algebra. 1) The rst one is the subalgebra e g(R) generated by the vacuum e for 2 R of the quotient Lie algebra V Q(R) =DV Q(R) of the lattice vertex algebra (studied by Borcherds) attached to the elliptic root lattice Q(R). This algebra is isomorphic to the 2-toroidal algebra and to the intersection matrix algebra proposed by Slodowy. 2) The second algebra e ( (R;G)) is presented by Chevalley generators and generalized Serre relations attached to the elliptic Dynkin diagram (R;G). ? Present address: Research Institute for Mathematical Sciences, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto, 606-8502, Japan (e-mail: [email protected]) ?? Present address: Research Institute for Mathematical Sciences Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto, 606-8502, Japan (e-mail: [email protected]) Correspondence to: K. Saito 2 K. Saito, D. Yoshii Since the Cartan matrix for the elliptic diagram has some positive o diagonal entries, the algebra is de ned not only by Kac-Moody type relations but some others. 3) The third algebra e h Z af g af is de ned as an amalgamation of a Heisenberg algebra and an a ne Kac-Moody algebra, where the amalgamation relations between the two algebras are explicitly given. This algebra admits a sort of triangular decomposition in a generalized sense. The rst algebra e g(R) does not depend on a choice of the markingG whereas the second e ( (R;G)) and the third e h Z af g af do. This means the isomorphism depend on the choice of the marking i.e. on a choice of an element of PSL(2;Z).