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An exposition of root systems and Lie algebras (affine and elliptic). (English) Zbl 1292.17009

From the text: The aim of this paper is to give an exposition in order to give an explicit way to understand (1) a non-topological proof for the existence of a base of an affine root system (Theorem 3.1, originally given in [I. G. Macdonald, Invent. Math. 15, 91–143 (1972; Zbl 0244.17005)]), (2) a Serre-type definition of an elliptic Lie algebra g with rank \(\geq 2\) (Definition 5.1, originally given in [H. Yamane, Publ. Res. Inst. Math. Sci. 40, No. 2, 441–469 (2004; Zbl 1142.17308)]) and the fact that the non-isotropic roots form the corresponding elliptic root system and their multiplicities are one (Theorem 5.1, originally given in [Yamane, op. cit.]), and (3) a list of the multiplicities of the isotropic roots of g, proved from a viewpoint of the Saito-marking lines (Theorem 6.1, new result).
As for (2), we point out that our defining relations are closely related to defining relations, called Drinfeld realization, of the quantum affine algebras due to V. G. Drinfeld [Sov. Math., Dokl. 36, No. 2, 212–216 (1988); translation from Dokl. Akad. Nauk SSSR 269, 13–17 (1987; Zbl 0667.16004), Theorems 3 and 4].

MSC:

17B22 Root systems
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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