Non-negative integral level affine Lie algebra tensor categories and their associativity isomorphisms

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 $\hat{\mathfrak{g}}$ at a fixed level $\ell \in \mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator algebra $L_{\hat{\mathfrak{g}}}(\ell ,0)$, 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 $A(L_{\hat{\mathfrak{g}}}(\ell ,0))$, 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 $L_{hat\mathfrak{g}}(\ell ,0)-\mathbf{mod}$. This construction shows that $\mathbf{D}(\mathfrak{g},\ell )$ is closely related to the Drinfeld category of $U(\mathfrak{g})[[\hslash ]]$-modules used by Kazhdan and Lusztig to identify categories of $\hat{\mathfrak{g}}$-modules at irrational and most negative rational levels with categories of quantum group modules.

49 pages

References in corpus (2)

• Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models
• Intertwining operators among modules for affine Lie algebra and lattice vertex operator algebras which respect integral forms

Cited by in corpus (3)

• Braided tensor categories of admissible modules for affine Lie algebras
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• The quantum group dual of the first-row subcategory for the generic Virasoro VOA