Irreducible graded \(A_ 1^{(1)}\)-modules
Consider the algebra \({\mathfrak g} = sl(2, \mathbb{C})\) with a standard basis \(e,f,h\), the current algebra \(\widehat {\mathfrak g} = {\mathfrak g} \otimes \mathbb{C} [t,t^{-1}]\), and its one-dimensional central extension \(\widetilde {\mathfrak g} = \widetilde {\mathfrak g} \oplus \mathbb{C} z\). We introduce the \(\mathbb{Z}^ 2\)-gradation in \(\widehat {\mathfrak g}\) by \(\deg e = (1,0)\), \(\deg f = (-1,0)\), \(\deg h = \deg z = (0,0)\), \(\deg x \otimes t^ n = \deg x + (n,n)\); let \(K\) be the category of \(\mathbb{Z}^ 2\)-graded \(\widetilde {\mathfrak g}\)-modules. In particular, \(K\) contains Verma modules, generalized Verma modules, and their irreducible factor modules. In this note we construct new series of irreducible \(\mathbb{Z}^ 2\)-graded \(\widetilde {\mathfrak g}\)-modules and analogues of Verma modules as well.
scientific article; zbMATH DE number 560999