Deformations of Yang-Baxter operators via n-Lie algebra cohomology

Abstract

We introduce a cohomology theory of n-ary self-distributive objects in the tensor category of vector spaces that classifies their infinitesimal deformations. For n-ary self-distributive objects obtained from n-Lie algebras we show that (n-ary) Lie cohomology naturally injects in the self-distributive cohomology and we prove, under mild additional assumptions, that the map is an isomorphism of second cohomology groups. This shows that the self-distributive deformations are completely classified by the deformations of the Lie bracket. This theory has important applications in the study of Yang-Baxter operators as the self-distributive deformations determine nontrivial deformations of the Yang-Baxter operators derived from n-ary self-distributive structures. In particular, we show that there is a homomorphism from the second self-distributive cohomology to the second cohomology of the associated Yang-Baxter operator. Moreover, we prove that when the self-distributive structure is induced by a Lie algebra with trivial center, we get a monomorphism. We construct a deformation theory based on simultaneous deformations, where both the coalgebra and self-distributive structures are deformed simultaneously. We show that when the Lie algebra has trivial cohomology (e.g. for semi-simple Lie algebras) the simultaneous deformations might still be nontrivial, producing corresponding Yang-Baxter operator deformations. We provide examples and computations in low dimensions, and we completely characterize 2-cocycles for the self-distributive objects obtained from all the nontrivial real Lie algebras of dimension 3, i.e. the Bianchi I-IX, and all the nontrivial complex Lie algebras of dimension 3.