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The theories of (Hopf) bialgebras and weak (Hopf) bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product. As first generalisations, these notions were formulated for monoidal categories, with braidings if needed. The present authors developed a theory of bimonads and Hopf monads H on arbitrary categories A , employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras. For τ -bimonads H , properties of braided (Hopf) bialgebras were captured by requiring a Yang–Baxter operator τ : H H → H H . The purpose of this paper is to extend the features of weak (Hopf) bialgebras to this general setting including an appropriate form of the Fundamental Theorem. This subsumes the theory of braided Hopf algebras (based on weak Yang–Baxter operators) as considered by Alonso Álvarez and others. [ABSTRACT FROM AUTHOR]

An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with Ecalle's mould calculus, i.e., mould composition and contracting arborification is precised. [ABSTRACT FROM AUTHOR]

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer–Cartan algebra—the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator—to sh Lie–Rinehart algebras. To this end, we first develop a characterization of sh Lie–Rinehart algebras in terms of differential graded cocommutative coalgebras and Lie algebra twisting cochains that extends the nowadays standard characterization of an ordinary sh Lie algebra (equivalently: L ∞ algebra) in terms of its associated generalized Cartan–Chevalley–Eilenberg coalgebra. Our approach avoids any higher brackets but reproduces these brackets in a conceptual manner. The new technical tool we develop is a notion of filtered multi derivation chain algebra, somewhat more general than the standard concept of a multicomplex endowed with a compatible algebra structure. The crucial observation, just as for ordinary Lie–Rinehart algebras, is this: For a general sh Lie–Rinehart algebra, the generalized Cartan–Chevalley–Eilenberg operator on the corresponding graded algebra involves two operators, one coming from the sh Lie algebra structure and the other one from the generalized action on the corresponding algebra; the sum of the two operators is defined on the algebra while the operators are individually defined only on a larger ambient algebra. We illustrate the structure with quasi Lie–Rinehart algebras. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with the extended centroid c . Suppose that R is acted by a pointed coalgebra with group-like elements acting as automorphisms of R . A generalized polynomial with variables acted by the coalgebra is called an identity if it vanishes on R . We prove the following: (1) If c is a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R with variables acted by Frobenius automorphisms. (2) If c is not a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R . With this, we extend Yanai's result [25] to “nonlinear identities”. These are actually special instances of our Theorems 1 and 2 below respectively, which extend Kharchenko's theory of differential identities [14,15] to the context of expansion closed word sets introduced in [6] . [ABSTRACT FROM AUTHOR]

We give some formality criteria for a differential graded Lie algebra to be formal. For instance, we show that a DG-Lie algebra L is formal if and only if the natural spectral sequence computing the Chevalley–Eilenberg cohomology H CE ⁎ ( L , L ) degenerates at E 2 . [ABSTRACT FROM AUTHOR]

Compact quantum groups of face type, as introduced by Hayashi, form a class of quantum groupoids with a classical, finite set of objects. Using the notions of weak multiplier bialgebras and weak multiplier Hopf algebras (resp. due to Böhm–Gómez-Torrecillas–López-Centella and Van Daele–Wang), we generalize Hayashi's definition to allow for an infinite set of objects, and call the resulting objects partial compact quantum groups. We prove a Tannaka–Kreĭn–Woronowicz reconstruction result for such partial compact quantum groups using the notion of partial fusion C ⁎ -categories. As examples, we consider the dynamical quantum SU ( 2 ) -groups from the point of view of partial compact quantum groups. [ABSTRACT FROM AUTHOR]