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In this paper, we give the notion of (lax or weak) Hom-corings, which leads to the introduction of partial coaction of a Hom-Hopf algebra on a Hom-algebra. We construct several pairs of adjoint functors, and define Galois Hom-corings, as a special case of which Galois Hom-coalgebras are introduced. [ABSTRACT FROM AUTHOR]

Motivated by recent work on compatible mock-Lie algebras such that any linear combination of the two mock-Lie structures is a mock-Lie structure, in this paper we introduce compatible mock-Lie bialgebras as an analogue of compatible Lie bialgebras. They can also be regarded as a "compatible version" of mock-Lie bialgebras. Constructions of compatible mock-Lie bialgebras are presented via Manin triples and matched pairs. [ABSTRACT FROM AUTHOR]

This paper is to look for bi-Frobenius algebra structures on quantum complete intersections over field k. We find a class of comultiplications, such that if − 1 ∈ k , then a quantum complete intersection becomes a bi-Frobenius algebra with comultiplication of this form if and only if all the parameters qij = ±1. Also, it is proved that if − 1 ∈ k then a quantum exterior algebra in two variables admits a bi-Frobenius algebra structure if and only if the parameter q = ±1. While if − 1 ∉ k , then the exterior algebra with two variables admits no bi-Frobenius algebra structures. We prove that the quantum complete intersections admit a bialgebra structure if and only if it admits a Hopf algebra structure, if and only if it is commutative, the characteristic of k is a prime p, and every ai a power of p. This also provides a large class of examples of bi-Frobenius algebras which are not bialgebras (and hence not Hopf algebras). In commutative case, other two comultiplications on complete intersection rings are given, such that they admit non-isomorphic bi-Frobenius algebra structures. [ABSTRACT FROM AUTHOR]

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces. [ABSTRACT FROM AUTHOR]

For a bialgebra L coacting on a k -algebra A, a classical result states A is a right L-comodule algebra if and only if A is an algebra in the category M L of right L-comodules. We generalize this to the setting of weak bialgebras H. We prove that there is an isomorphism between the categories of right H-comodule algebras and the category of algebras in M H . We also recall and introduce the formulaic notion of H coacting on a k -coalgebra and on a Frobenius k -algebra, respectively, and prove analogous category isomorphisms. Many examples are provided throughout. [ABSTRACT FROM AUTHOR]

We study Dorroh extensions of algebras and Dorroh extensions of coalgebras. Their structures are described. Some properties of these extensions are presented. We also introduce the finite duals of algebras and modules which are not necessarily unital. Using these finite duals, we determine the dual relations between the two kinds of extensions. [ABSTRACT FROM AUTHOR]

In this work we explicitly calculate all the right coideal subalgebras of the Hopf algebra H which is the bosonization of the Fomin–Kirillov algebra F K 3 over the symmetric group S 3. First we consider the case of right coideal subalgebras containing the coradical k S 3. Then we also describe the objects that do not contain the coradical, completing the classification. [ABSTRACT FROM AUTHOR]

Up to Morita equivalence, every quasi‐hereditary algebra is the dual algebra of a directed bocs or a coring. From the bocs, an exact Borel subalgebra is obtained. In this paper a characterisation of exact Borel subalgebras arising in this way is given. [ABSTRACT FROM AUTHOR]

The objective of this work is to compare several approaches to the process of renormalisation in the context of rough differential equations using the substitution bialgebra on rooted trees known from backward error analysis of B‐series. For this purpose, we present a so‐called arborification of the Hoffman–Ihara theory of quasi‐shuffle algebra automorphisms. The latter are induced by formal power series, which can be seen to be special cases of the cointeraction of two Hopf algebra structures on rooted forests. In particular, the arborification of Hoffman's exponential map, which defines a Hopf algebra isomorphism between the shuffle and quasi‐shuffle Hopf algebra, leads to a canonical renormalisation that coincides with Marcus' canonical extension for semimartingale driving signals. This is contrasted with the canonical geometric rough path of Hairer and Kelly by means of a recursive formula defined in terms of the coaction of the substitution bialgebra. [ABSTRACT FROM AUTHOR]

The goal of this article is to study the coaugmented curved A∞-coalgebra structure of the Koszul codual of a filtered dg algebra over a field k. More precisely, we first extend one result of B. Keller that allowed to compute the A∞-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra to the case of any unitary dg algebra provided with a nonnegative increasing filtration whose zeroth term is k. We then show how to compute the coaugmented curved A ∞-coalgebra structure of the Koszul codual of a Poincaré-Birkhoff-Witt (PBW) deformation of an N-Koszul algebra. [ABSTRACT FROM AUTHOR]

We introduce the notion of evolution coalgebra as an algebraic structure to describe the backwards evolution of non-Mendelian genetic systems and establish their connection with evolution algebras. We study their main properties and consider evolution coalgebras endowed with genetic realization. [ABSTRACT FROM AUTHOR]

In this paper, we study A/k-bialgebras in prime characteristic. Firstly, we prove a Cartier type structure theorem for cocomplete A/k-bialgebras. Secondly, we generalize Michaelis's theorem. In particular, we prove that a restricted (k, K)-Lie algebra is proper if and only if its restricted enveloping algebra is proper. [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]

In this note, the coradical filtration of the new class of Hopf algebrasD(m,d, ξ) is determined. As consequences, the authors show that 1)D(m,d, ξ) is not pointed; 2)D(m,d, ξ) is a co-Frobenius Hopf algebra; 3) the coradical ofD(m,d, ξ) is not a Hopf subalgebra. [ABSTRACT FROM PUBLISHER]

A Hopf algebra object in Loday and Pirashvili's category of linear maps entails an ordinary Hopf algebra and a Yetter-Drinfel'd module. We equip the latter with a structure of a braided Leibniz algebra. This provides a unified framework for examples of racks in the category of coalgebras discussed recently by Carter, Crans, Elhamdadi and Saito. [ABSTRACT FROM AUTHOR]

J.-L. Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincaré–Birkhoff–Witt and the Cartier–Milnor–Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of J.-L. Loday is extended to this more general categorical framework. [ABSTRACT FROM AUTHOR]

We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show thatis a coalgebra such that the rational part of every left finitely generated-module splits off if and only ifis an upper triangular matrix coalgebra, for a serial coalgebrawhose Ext-quiver is a finite union of cycles, a finite dimensional coalgebraand a finite dimensional bicomodule. [ABSTRACT FROM AUTHOR]

In the theory of coalgebras C over a ring R, the rational functor relates the category $_{C^*}{\mathbb{M}}$ of modules over the algebra C (with convolution product) with the category $^C{\mathbb{M}}$ of comodules over C. This is based on the pairing of the algebra C with the coalgebra C provided by the evaluation map ${\rm ev}:C^*\otimes_R C\to R$. The (rationality) condition under consideration ensures that $^C{\mathbb{M}}$ becomes a coreflective full subcategory of $_{C^*}{\mathbb{M}}$. We generalise this situation by defining a pairing between endofunctors T and G on any category ${\mathbb{A}}$ as a map, natural in $a,b\in {\mathbb{A}}$, and we call it rational if these all are injective. In case T = ( T, m, e) is a monad and G = ( G, δ, ε) is a comonad on ${\mathbb{A}}$, additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T, we construct a rational functor as the functor-part of an idempotent comonad on the T-modules ${\mathbb{A}}_{T}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories. [ABSTRACT FROM AUTHOR]

In this article, we construct a lot of new examples of Hopf group coalgebras by considering Majid's bicrossproduct Hopf algebra in Turaev category. Furthermore, we find the sufficient and necessary conditions for such Majid's bicrossproduct Hopf algebra to admit quasitriangular structures in the sense of Turaev group coalgebras. [ABSTRACT FROM AUTHOR]

We propose a new dynamical reflection algebra, distinct from the previous dynamical boundary algebra and semi-dynamical reflection algebra. The associated Yang-Baxter equations, coactions, fusions, and commuting traces are derived. Explicit examples are given and quantum integrable Hamiltonians are constructed. They exhibit features similar to the Ruijsenaars-Schneider Hamiltonians. [ABSTRACT FROM AUTHOR]