Hopf braces, related structures and their associated categories
Brais Ramos Pérez (Santiago de Compostela)

Abstract: Hopf braces are recent mathematical objects introduced by I. Angiono et al. [1] and obtained through a linearisation process from skew braces,  which give rise to non-degenerate, bijective and not necessarily involutive solutions of the Quantum Yang-Baxter Equation (see [4]), whose formulation is the following

(τ ⊗ idV ) ◦ (idV ⊗ τ ) ◦ (τ ⊗ idV) = (idV ⊗ τ ) ◦ (τ ⊗ idV ) ◦ (idV ⊗ τ ), (QYBE)

where τ : V ⊗KV → V ⊗KV is a linear map and V , a K-vector space. As was proven in [1, Corollary 2.4], cocommutative Hopf braces are also relevant from a physical standpoint because they also induce solutions of the above-mentioned equation.
This talk is devoted to the study of Hopf braces and some related structures in an arbitrary braided monoidal category C, not necessarily the category of vector spaces. The outline of the talk will be as follows:

• To introduce the category of Hopf braces within C, and to explore certain algebraic properties inherent to these structures.
• To introduce the category of brace triples in C, motivated by the following problem: how can one deform a Hopf algebra A ∈ C in order to obtain a new Hopf algebra Anew such that the pair (A, Anew) constitutes a Hopf brace in C? The complete details may be found in [2].
• To construct new Hopf braces in C via crossed products. The results obtained will be applied to derive Hopf brace structures involving the Drinfeld double of a Hopf algebra, a problem that has not previously been addressed in the literature (see [3]).

References
[1] I. Angiono, C. Galindo and L. Vendramin, Hopf braces and Yang-Baxter operators, Proc. Am. Math. Soc. 145(5) (2017) 1981-1995.
[2] J.M. Fernández Vilaboa, R. González Rodríguez and B. Ramos Pérez, Categorical isomorphisms for Hopf braces, Hacet. J. Math. Stat. (2025), in press.
[3] R. González Rodríguez and B. Ramos Pérez, About Hopf braces and crossed products, Preprint (2025), arXiv:2502.20919.
[4] L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comput. 86(307) (2017) 2519-2534.