Extended affine root systems IV (Simply-Laced Elliptic Lie Algebras

Abstract.. Let (R,G) be a pair consisting of an elliptic root sys-tem R with a marking G. Assume that the attached elliptic Dynkin diagram Γ(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 first one is the subalgebra g̃(R) generated by the vac-uum eα for α ∈ R of the quotient Lie algebra VQ(R)/DVQ(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 ẽ(Γ(R,G)) is presented by Chevalley generators and generalized Serre relations attached to the elliptic Dynkin diagram Γ(R,G). Since the Cartan matrix for the ellip-tic diagram has some positive off diagonal entries, the algebra is defined not only by Kac-Moody type relations but some others. 3) The third algebra h̃Zaf ∗ gaf is defined as an amalgamation of a Heisenberg algebra and an affine 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 first algebra g̃(R) does not depend on a choice of the mark-ing G whereas the second ẽ(Γ(R,G)) and the third h̃Zaf ∗ gaf do. This means the isomorphism depend on the choice of the marking i.e. on a choice of an element of PSL(2,Z)