Presentations of Z-forms for the universal enveloping algebras of affine Lie algebras.

摘要

We introduce the basic concepts of generalized Cartan matrices and Kac-Moody Lie algebras. We discuss realizations of affine algebras in terms of non-twisted and twisted loop algebras of simple Lie algebras. We give a presentation of {dollar}{lcub}cal U{rcub}sbdoubz(Asb1sp{lcub}(1){rcub}){dollar} and generalize to {dollar}{lcub}cal U{rcub}sbdoubz(Asb{lcub}n{rcub}sp{lcub}(1){rcub}), {lcub}cal U{rcub}sbdoubz(Dsb{lcub}n{rcub}sp{lcub}(1){rcub}),{dollar} and {dollar}{lcub}cal U{rcub}sbdoubz(Esb{lcub}n{rcub}sp{lcub}(1){rcub}){dollar}. Finally, we construct the twisted affine algebra {dollar}Asb2sp{lcub}(2){rcub}{dollar}, and after establishing a basis for {dollar}{lcub}cal U{rcub}sbdoubz(Asb2sp{lcub}(2){rcub}){dollar}, we give a presentation of {dollar}{lcub}cal U{rcub}sbdoubz(Asb2sp{lcub}(2){rcub}){dollar}.