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.