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Author: search.filters.author.Guccione J.J.Language: search.filters.language.English

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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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    (Co)homology of Crossed Products by Weak Hopf Algebras
    (Academic Press Inc., 2023-05-15)
    We obtain a mixed complex simpler than the canonical one that computes the cyclic type homologies of a crossed product with invertible cocycle A×ρfH, of a weak module algebra A by a weak Hopf algebra H. This complex is endowed with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in [12]. When f takes its values in a separable subalgebra of A that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.