Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which are integrable under $\mathfrak g[[t]]$; under some assumptions on the central charge they constructed a tensor structure there and proved that the resulting tensor category is equivalent to the category of finite-dimensional representations of the corresponding quantum group (where the parameter $q$ in the quantum group depends on the central charge).

My question is the following: is there a known analog of these results for Lie super-algebras. I am particularly interested in $\mathfrak g=\text{gl}(m|n)$. Have people studied the Kazhdan-Lusztig category for Lie super-algebras?

Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which are integrable under $\mathfrak g[[t]]$; under some assumptions on the central charge they constructed a tensor structure there and proved that the resulting tensor category is equivalent to the category of finite-dimensional representations of the corresponding quantum group (where the parameter $q$ in the quantum group depends on the central charge).

My question is the following: is there a known analog of these results for Lie super-algebras? I am particularly interested in $\mathfrak g=\text{gl}(m|n)$. Have people studied the Kazhdan-Lusztig category for Lie super-algebras?