Highest weight representations of the Lie algebra \(\mathfrak{gl}_{\infty}\)
The Lie algebra \(\mathfrak{gl}_{\infty}\) is defined as the set of all infinite matrices \((a_{ij})_{i,j\in\mathbb{Z}}\) with only a finite number of nonzero coefficients. The corresponding group contains all matrices \((A_{ij})_{i,j\in\mathbb{Z}}\), where only a finite number of coefficients \(A_{ik}\) is different from \(\delta_{ij}\). Here irreducible representations of \(\mathfrak{gl}_{\infty}\) are investigated, which are highest weight representations and integrable. The representations are characterized by Gelfand-Tsetlin bases, and explicit forms of the operators are given.
scientific article; zbMATH DE number 4171170