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

  • Kyoji Saito

    Kyoto University, Japan
  • Daigo Yoshii

    Kyoto University, Japan


Let be a pair consisting of an elliptic root system and a marking of . Assume that the attached elliptic Dynkin diagram 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 generated by the highest vector for all in the quotient Lie algebra of the lattice vertex algebra attached to the elliptic root lattice .
  2. The second algebra , studied in 4, is presented by the Chevalley generators and the generalized Serre relations attached to the elliptic Dynkin diagram .
  3. The third algebra , 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.

Cite this article

Kyoji Saito, Daigo Yoshii, Extended Affine Root System IV (Simply-Laced Elliptic Lie Algebras). Publ. Res. Inst. Math. Sci. 36 (2000), no. 3, pp. 385–421

DOI 10.2977/PRIMS/1195142952