Extended affine Lie algebras and their vertex representations (Q584371)

\textit{K. Saito} has introduced the concept of extended affine root systems to construct a flat structure for the space of the universal deformation of a simple elliptic singularity [Publ. Res. Inst. Math. Sci. 21, 75--179 (1985; Zbl 0573.17012)]. It is by definition an extension of an affine root system by one dimensional radical. In this paper, using vertex operators, the author constructs a Lie algebra which has the extended affine root system R of type \(A_{\ell}^{(1,1)}\), \(D_{\ell}^{(1,1)}\) or \(E_{\ell}^{(1,1)}\) as the set of real roots. This is done by following the idea of \textit{I. B. Frenkel} [Lect. Appl. Math. 21, 325--353 (1985; Zbl 0558.17013)], \textit{I. B. Frenkel} and \textit{V. G. Kac} [Invent. Math. 62, 23--66 (1980; Zbl 0493.17010)] and \textit{P. Goddard} and \textit{D. Olive} [Publ., Math. Sci. Res. Inst. 3, 51--96 (1985; Zbl 0556.17004)]. The algebra may be expressed in the form \(\tilde {\mathfrak g}(R)={\mathfrak g}(R_ a)\otimes \mathbb{C}[\lambda,{\bar \lambda}^1]\oplus \mathbb{C}d_1\oplus \mathbb{C}d_2,\) where \(\mathfrak g(R_ a)\) is the affine Lie algebra of type \(A_{\ell}^{(1)}\), \(D_{\ell}^{(1)}\) or \(E_{\ell}^{(1)}\). Further, the author considers the Weyl group \(W_R\) of the Lie algebra \(\tilde{\mathfrak g}(R)\) and shows that \(W_R=W_f\ltimes H^{2\ell +1}\), where \(H^{2\ell +1}\) is a Heisenberg group and \(W_f\) the subgroup of \(W_R\) generated by the fundamental reflections.