| Abstract: | For a finite-dimensional simple Lie algebra ({mathfrak{g}}), we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra ({{widehat{mathfrak{g}}}}) at a fixed level ({ellinmathbb{N}}) with a certain tensor category of finite-dimensional ({mathfrak{g}})-modules. More precisely, the category of level ? standard ({{widehat{mathfrak{g}}}})-modules is the module category for the simple vertex operator algebra ({L_{widehat{mathfrak{g}}}(ell, 0)}), and as is well known, this category is equivalent as an abelian category to ({mathbf{D}(mathfrak{g},ell)}), the category of finite-dimensional modules for the Zhu’s algebra ({A{(L_{widehat{mathfrak{g}}}(ell, 0))}}), which is a quotient of ({U(mathfrak{g})}). Our main result is a direct construction using Knizhnik–Zamolodchikov equations of the associativity isomorphisms in ({mathbf{D}(mathfrak{g},ell)}) induced from the associativity isomorphisms constructed by Huang and Lepowsky in ({{L_{widehat{mathfrak{g}}}(ell, 0) - mathbf{mod}}}). This construction shows that ({mathbf{D}(mathfrak{g},ell)}) is closely related to the Drinfeld category of ({U(mathfrak{g})})[[h]]-modules used by Kazhdan and Lusztig to identify categories of ({{widehat{mathfrak{g}}}})-modules at irrational and most negative rational levels with categories of quantum group modules. |
