Double loop algebras and elliptic root systems (Q522165)

An elliptic root system is a root system defined on a real vector space with positive semi-definite bilinear form whose radical is of dimension two. The paper under review explains the algebraic structure of 2-toroidal Lie algebras in terms of the corresponding elliptic root systems and elliptic Weyl groups. The paper also shows that all these objects can be naturally constructed in terms of the space of \(\overline{\partial}\)-connections on a topologically trivial principal \(G\)-bundles over an elliptic curve, where \(G\) is the corresponding double loop group.