Extended affine root system. IV: Simply-laced elliptic Lie algebras
For Part III, cf. ibid. 33, 301--329 (1997; Zbl 0901.20016), Part V, see CRM Proc. Lect. Notes. 30, 185--222 (2001; Zbl 1015.11013). The notion of generalized root system was introduced by the first author [Part I, Publ. Res. Inst. Math. Sci. 21, 75--179 (1985; Zbl 0573.17012)]. If the Witt index of the root lattice is 0 or 1, then the root system is either finite, affine or hyperbolic. If the Witt index is 2, the root system is called an elliptic root system. In this paper the authors study three Lie algebras associated with a simply-laced elliptic root system \(R\): (1) a certain subalgebra of the quotient Lie algebra of the lattice vertex algebra of \textit{R. Borcherds} [Proc. Natl. Acad. Sci. USA 83, 3068--3071 (1986; Zbl 0613.17012)] for the elliptic root lattice \(Q(R)\), (2) a Lie algebra presented by the Chevalley generators and the generalized Serre relations attached to the elliptic Dynkin diagram, (3) an amalgamation of an affine Heisenberg algebra and an affine Kac-Moody algebra. The main result is that these three Lie algebras are isomorphic. A good survey of the theory of generalized root systems can be found in the introduction of this paper.
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