The structure of standard modules. I: Universal algebras and the Rogers- Ramanujan identities
A new approach to the construction and study of algebras of endomorphisms of graded vector spaces \(V=\coprod V_ i\), where i ranges over the integers and \(V_ i=0\) for i sufficiently large, is introduced. To each family \(Z_ i\) of endomorphisms of V, where deg \(Z_ i=i\), the formal Laurent series \(Z(\zeta)=\sum Z_ i \zeta^ i\) is associated. In various settings, identities of the form \(a(\zeta_ 1/\zeta_ 2)Z(\zeta_ 1)Z(\zeta_ 2)+b(\zeta_ 2/\zeta_ 1)Z(\zeta_ 2)Z(\zeta_ 1)=c(\zeta_ 1/\zeta\) \({}_ 2)Z(\zeta_ 2)+d(\zeta_ 1/\zeta_ 2)\), where a(\(\zeta)\), b(\(\zeta)\), c(\(\zeta)\), d(\(\zeta)\) are formal Laurent series with complex coefficients, are proved. This improves our Lie-theoretic understanding of the Rogers-Ramanujan identities. Also this approach gives explicit constructions of very general modules for affine Lie algebras - modules whose construction had been inaccessible before.
algebras of endomorphisms of graded vector spaces
scientific article; zbMATH DE number 3922917
scientific article