JournalsprimsVol. 36 , No. 3DOI 10.2977/prims/1195142952

Extended Affine Root System IV (Simply-Laced Elliptic Lie Algebras)

  • Kyoji Saito

    Kyoto University, Japan
  • Daigo Yoshii

    Kyoto University, Japan
Extended Affine Root System IV (Simply-Laced Elliptic Lie Algebras) cover


Let (R, G) be a pair consisting of an elliptic root system R and a marking G of R. Assume that the attached elliptic Dynkin diagram Γ(R, G) is simply-laced. To the simply-laced elliptic root system, we associate three Lie algebras, explained in 1), 2) and 3) below. The main result of the present paper is to show that all three algebras are isomorphic.

  1. The first one, studied in §3, is the subalgebra g(R) generated by the highest vector for all α ∈ R in the quotient Lie algebra V__Q(R) / DV__Q(R) of the lattice vertex algebra attached to the elliptic root lattice Q(R).
  2. The second algebra e(Γell), studied in §4, is presented by the Chevalley generators and the generalized Serre relations attached to the elliptic Dynkin diagram Γell = Γ(R, G)
  3. The third algebra ℌZ'af * gaf, studied in §5, is defined as an amalgamation of an affine Heisenberg algebra and an affine Kac–Moody algebra together with the finite amalgamation relations.