Rota-Baxter operators and factorization of Lie affgebras

Abstract

We introduce Rota-Baxter operators on Lie affgebras by an identity independent of the choice of origin and prove that they induce descendant Lie affgebra structures. We establish a bijection between ordered affine Lie factorizations and idempotent operators split at a chosen idempotent fixed point. A cokernel obstruction determines the existence of this point, while invariance under the structural endomorphisms characterizes splitting. We also determine when weighted descendants satisfy the affine Jacobi identity and when operators of nonzero weight give affine doubles. For canonical Lie affgebras, we describe all affine Rota-Baxter operators by Lie subalgebra decompositions and translation vectors, and classify them up to automorphism in dimensions at most two.