Toroidal Lie algebras and vertex representations
The paper under review describes a theory of the toroidal Lie algebra; the Lie algebra \({\mathbb C}[s,s^{-1},t,t^{-1}]\otimes {\mathfrak g}\) of polynomial maps from a complex torus \({\mathbb C}^{\times}\times {\mathbb C}^{\times}\) into a finite dimensional simple Lie algebra \({\mathfrak g}\). The authors describe the universal central extension \({\mathfrak t}\) of this algebra. Unlike the affine case where \(\hat{\mathfrak g}\) is a one dimensional central extension of the loop algebra, \({\mathfrak t}\) is an infinite dimensional central extension of \({\mathbb C}[s,s^{-1},t,t^{- 1}]\otimes {\mathfrak g}.\) Next they give an abstract presentation of \({\mathfrak t}\) in terms of generators and relations involving the extended Cartan matrix of \({\mathfrak g}\). For \(X=A,D,E\), let \(Q\) be the root lattice of the corresponding Lie algebra \(\hat{\mathfrak g}=\hat {\mathfrak g}(X_{{\mathfrak t}}^{(1)})\). Construct the Fock space \(V(Q)={\mathbb C}[Q]\otimes S({\mathfrak a}_ -)\), where \({\mathfrak a}\) is the Heisenberg algebra defined on \(Q\). A representation of \(\hat{\mathfrak g}\) is generated by the Fourier components of the vertex operators \(X(\alpha,z)\), \(\alpha \in \Delta (=\) the affine root system of \(\hat{\mathfrak g})\). This Lie algebra is a non-faithful homomorphic image of \({\mathfrak t}={\mathfrak t}(X_{{\mathfrak t}}^{(1)})\). It can be made faithful by enlarging \(Q\) to a nondegenerate lattice \(\Gamma\), forming the Heisenberg algebra \({\mathfrak b}\) on \(\Gamma\) and using \(V(\Gamma,{\mathfrak b})={\mathbb C}[\Gamma]\otimes S({\mathfrak b}_-).\) Consider the \({\mathfrak t}\)-module \(V(\Gamma)={\mathbb C}[\Gamma]\otimes S({\mathfrak a}_-)\). For \(\lambda\in \Gamma\), set \(V(\lambda)=e^{\lambda+Q}\otimes S({\mathfrak a}_-)\), which is a \({\mathfrak t}\)-submodule of \(V(\Gamma\)). The authors show that \(V(\Gamma)=\oplus V(\lambda)\), where the summation runs over a complete set of representatives of \(\Gamma\) /Q, and that, for all \(\lambda\), \(V(\lambda)\) is an indecomposable \({\mathfrak t}\)-module. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.
scientific article; zbMATH DE number 4154683