Generalized AKS Scheme of Integrability via Vertex Algebra

Abstract In this paper, we define and study the classical $R$-matrix for Lie conformal algebras (LCAs), based on which we propose to construct a new LCA. We present a systematic approach for constructing the $R$-matrix for an affine Kac–Moody LCA and study the universal vertex algebra associated with the new LCA obtained via the $R$-matrix. As an application, using the classical $R$-matrix we defined, we introduce a novel integrability scheme to construct infinite-dimensional (Liouville) integrable systems via the Feigin–Frenkel center. A scheme-theoretical explanation of our equations and the case of classical $\mathscr{W}$-algebras in our theory will be discussed in subsequent papers.