# Kazama-Suzuki coset construction and its inverse

## TL;DR

This paper generalizes the inverse Kazama-Suzuki coset construction to arbitrary Lie algebra types, establishing categorical equivalences and proving regularity and braided monoidal structures for associated vertex operator superalgebras.

## Contribution

It extends the inverse coset construction beyond type A₁ and proves regularity and monoidal category structure for the resulting superalgebras.

## Key findings

- Categorical equivalence between module categories established.
- Regularity of the coset superalgebra proven.
- Braided monoidal category structure shown for modules.

## Abstract

We study the representation theory of the Kazama-Suzuki coset vertex operator superalgebra associated with the pair of a complex simple Lie algebra and its Cartan subalgebra. In the case of type $A_{1}$, B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin introduced another coset construction, which is "inverse" of the Kazama-Suzuki coset construction. In this paper we generalize the latter coset construction to arbitrary type and establish a categorical equivalence between the categories of certain modules over an affine vertex operator algebra and the corresponding Kazama-Suzuki coset vertex operator superalgebra. Moreover, when the affine vertex operator algebra is regular, we prove that the corresponding Kazama-Suzuki coset vertex operator superalgebra is also regular and the category of its ordinary modules carries a braided monoidal category structure by the theory of vertex tensor categories.