Braces and the Yang-Baxter equation

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We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. We also examine structure of braces. Recall that Braces, introduced in 2005 to construct set-theoretical solutions of the Yang-Baxter equation have links with many research areas, for example: Hopf-Galois extensions, Etale representations of Lie algebras, Pre-Lie algebras, Jacobson radical rings, among other topics. Among other topics we would like investigate symmetries of the transfer matrices and quantum integrable systems constructed from braces. Another possible topic is to investigate structure of braces by analogy with noncommutative ring theory. By Lazard's correspondence braces are connected to pre-Lie algebras (equivalent to right symmetric algebras) so obtained results can be generalised to the case of pre Lie algebras. On the other hand pre-Lie algebras correspond with Etale representations of affine Lie algebras which would give another application of results that we can obtain for braces.