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

TL;DR

This paper develops a cohomology theory for $n$-ary self-distributive objects in vector spaces, linking their deformations to those of $n$-Lie algebras and applying this to classify Yang-Baxter operator deformations.

Contribution

It introduces a cohomology framework for $n$-ary self-distributive objects, connecting their deformations to Lie algebra cohomology and Yang-Baxter operators.

Findings

Self-distributive deformations are classified by Lie algebra deformations.

A homomorphism from self-distributive cohomology to Yang-Baxter cohomology is established.

Complete characterization of 2-cocycles for 3-dimensional Lie algebras.

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-distribuitve 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 nontrivial 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 $3$.