Chevalley groups associated to elliptic Lie algebras
Generalizations of affine Lie algebras have been an active area of research recently and the terminology is by no means standardized. The algebras here called elliptic are also known as toroidal or extended affine algebras. The class of algebras under study in this paper is defined as follows. Let \({\mathfrak g}\) be a finite-dimensional complex simple Lie algebra. Then the associated elliptic algebra \(\hat {\mathfrak g}\) is the universal central extension of the doubly affinized algebra \({\mathfrak g} \otimes k[T^{\pm 1}, S^{\pm 1}]\), for some field \(k\). The author is concerned with the Chevalley group \(G(k[T^{\pm 1}, S^{\pm 1}])\) and the elementary subgroup \(E(k[T^{\pm 1}, S^{\pm 1}])\) generated by the root system. In contrast to the affine case, the elliptic Weyl group is not a Coxeter group and so \(E(k[T^{\pm 1}, S^{\pm 1}])\) does not have a Tits system. Most of the paper is devoted to analyzing the algebraic structure of \(E(k[T^{\pm 1}, S^{\pm 1}])\) and determining the relations of the Weyl groups of the elliptic root systems \(A_l^{(1,1)}\), \(B_l^{(1,1)}\), \(C_l^{(1,1)}\) and \(D_l^{(1,1)}\).
scientific article; zbMATH DE number 1251406