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Author: search.filters.author.Jorge A. GuccioneSubject: search.filters.subject.Non-degenerate solution

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    Solutions of the braid equation attached to power series
    (Taylor and Francis Ltd., 2024-01-01)
    Let K be an algebraically closed characteristic 0 field and let V be the K-vector space with basis (Formula presented.). In this paper we use the theory developed in [17] to associate with each power series (Formula presented.) satisfying (Formula presented.), an involutive solution (Formula presented.), of the braid equation, and we begin the study of this correspondence. For (Formula presented.), let Vn be the subspace of V generated by (Formula presented.). Each one of the solution s(f) induces by restriction a solution (Formula presented.) on (Formula presented.). We also begin the study of these solutions.
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    Set-theoretic type solutions of the braid equation
    (Academic Press Inc., 2024-04-15)
    In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX, where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q-cycle sets, q-braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting.
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    Hopf Q -braces structures on rank one pointed Hopf algebras
    (World Scientific, 2025-01-01)
    In this paper we determine all the Hopf [Formula: see text]-brace structures on rank one pointed Hopf algebras and compute the socle of each one of them. We also identify which among them are Hopf skew-braces. Then we determine when two Hopf [Formula: see text]-brace structures on rank one pointed Hopf algebras are isomorphic, and, finally, we compute all the weak braiding operators on these Hopf algebras.
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