Uniqueness of braided tensor structures and the Huang--Lepowsky structure on affine vertex operator algebras in the classical Lie types and G_2

Abstract

We prove that the unitary modular tensor category structure previously constructed for all Lie types, on the module category of the affine vertex operator algebra V_{g_k} at positive integer level, coincides with the Huang-Lepowsky braided tensor structure for all the classical Lie types and G_2. This answers the problem of proving the Finkelberg equivalence theorem directly, without the equivalence of Kazhdan-Lusztig. Simply-laced and non-simply-laced types are treated uniformly: our framework rests on Wenzl's work on the unitary structure of the quantum group fusion categories, which applies to all simple Lie algebras, whereas the original work of Kazhdan and Lusztig explicitly considered the simply-laced case. The proof uses two abstract uniqueness results, proved here: in a semisimple pre-tensor category with a generating object satisfying a duality property with respect to a braiding, the associativity and braiding morphisms are determined by their restrictions to a small collection of objects. The duality property is verified for the classical types and G_2 by appealing to known generalized quantum Schur-Weyl duality. This completes our previous work, where the global quantum gauge group A_W(g,q), an intrinsic construction for braided tensor C*-categories in the sense of the Doplicher-Roberts duality problem, was canonically constructed and used to endow the Zhu algebra of V_{g_k} with what we called a unitary coboundary weak quasi-Hopf structure with 3-coboundary associator, via an isometric analytic Drinfeld twist along Wenzl's continuous de-quantization curve, for all Lie types. The appeal to the braid group duality, made here for the first time in this context, unifies the Doplicher-Roberts construction of a compact gauge group in higher-dimensional algebraic quantum field theory with the Huang-Lepowsky braided tensor structure and quantum gauge groups.