Non-linear Lie conformal algebras with three generators

Abstract.

We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sℓ ₂. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras \(R^d_{-1} (d \in\mathbb{N})\) and the corresponding freely generated vertex algebras \(V^d_{-1}\), which includes for d = 1 the affine vertex algebra of sℓ ₂ at the critical level k = –2. We construct free-field realizations of the algebras \(V^d_{-1}\) extending the Wakimoto realization of \(\widehat{s\ell}_{2}\) at the critical level, and we compute their Zhu algebras.