The Hopf envelope of a bialgebra is the free Hopf algebra generated by the given bialgebra. Its existence, as well as that of the cofree Hopf algebra, is a well-known fact in Hopf algebra theory, but their construction is not particularly handy or friendly. In this note, we offer a novel realisation of the Hopf envelope and of the cofree Hopf algebra of a finite-dimensional bialgebra as a particular quotient and sub-bialgebra, respectively, of the bialgebra itself. Our construction can also be extended to the infinite-dimensional case, provided that the bialgebra satisfies additional conditions, such as being right perfect as an algebra or admitting a n-antipode, the latter being a notion hereby introduced. Remarkably, the machinery we develop also allows us to give a new description of the Hopf envelope of a commutative bialgebra and of the cofree cocommutative Hopf algebra of a cocommutative bialgebra. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

A well-known result by Larson and Sweedler shows that integrals on a Hopf algebra can be obtained by applying the Structure Theorem for Hopf modules to the rational part of its linear dual. This fact can be rephrased by saying that taking the space of integrals comes from a right adjoint functor from a category of modules to the category of vector spaces. This observation inspired the categorical approach that we advocate in this work, which yields to a new notion of integrals for bialgebras in the linear setting. Despite the novelty of the construction, it returns the classical definition in the presence of an antipode. We test this new concept on bialgebras that satisfy at least one of the following properties: being coseparable as regular module coalgebras, having a one-sided antipode, being commutative, being cocommutative, or being finite-dimensional. One of the main results we obtain in this process is a dual Maschke-type theorem relating coseparability and total integrals. Remarkably, there are cases in which the space of integrals turns out to be isomorphic to that of the associated Hopf envelope. In particular, this space results to be one-dimensional for finite-dimensional bialgebras, providing an existence and uniqueness theorem for integrals in the finite-dimensional case. Furthermore, explicit computations are given for concrete examples including the polynomial bialgebra with one group-like variable, the quantum plane and the coordinate bialgebra of n-by-n matrices.

The fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath ( A⊗ H_4^op, H_4, ψ) , where H_4 is the Sweedler 4-dimensional Hopf algebra over a field k and A=Cl(α , β , γ ) is the Clifford algebra generated by two elements G , X with relations G^2=α , X^2=β and XG+GX=γ , (α , β , γ∈ k ) which becomes naturally an H_4 -comodule algebra. We show that, when char( k ) 2, this cowreath is always separable and h-separable as well.

Liftable pairs of adjoint functors between braided monoidal categories in the sense of [GV1] provide auto-adjunctions between the associated categories of bialgebras. Motivated by finding interesting examples of such pairs, we study general pre-rigid monoidal categories. Roughly speaking, these are monoidal categories in which for every object X, an object X* and a nicely behaving evaluation map from X* circle times X to the unit object exist. A prototypical example is the category of vector spaces over a field, where X* is not a categorical dual if X is not finite-dimensional. We explore the connection with related notions such as right closedness, and present meaningful examples. We also study the categorical frameworks for Turaev's Hopf group-(co)algebras in the light of pre-rigidity and closedness, filling some gaps in literature along the way. Finally, we show that braided pre-rigid monoidal categories indeed provide an appropriate setting for liftability in the sense of loc. cit. and we present an application, varying on the theme of vector spaces, showing how -in favorable cases-the notion of pre-rigidity allows to construct liftable pairs of adjoint functors when right closedness of the category is not available.

The purpose of this paper is to introduce and study BiHom-NS-algebras, which are a generalization of NS-algebras using two homomorphisms. Moreover, we discuss their relationships with twisted Rota-Baxter operators in a BiHom-associative context. Furthermore, we introduce a generalization of Nijenhuis operators that lead to BiHom-NS-algebras along BiHom-associative algebras

It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between bialgebras and (restricted) Lie algebras. Moreover, in this framework, the notions of augmented monad and combinatorial rank play a central role. In order to set these results into a wider context, we are led to substitute the monadic decomposition by what we call the adjoint decomposition. This construction has the advantage of reducing the computational complexity when compared to the first one. We connect the two decompositions by means of an embedding and we investigate its properties by using a relative version of Grothendieck fibration. As an application, in this wider setting, by using the notion of augmented monad, we introduce a notion of combinatorial rank that, among other things, is expected to give some hints on the length of the monadic decomposition.

Let 𝒜 and ℬ be monoidal categories and let R:𝒜→ℬ be a lax monoidal functor. If R has a left adjoint L, it is well-known that the two adjoints induce functors R=(R):(𝒜)→(ℬ) and L=(L):(ℬ)→(𝒜) respectively. The pair (L, R) is called liftable if the functor R has a left adjoint and if the functor L has a right adjoint. A pleasing fact is that, when 𝒜 , ℬ and R are moreover braided, a liftable pair of functors as above gives rise to an adjunction at the level of bialgebras. In this note, sufficient conditions on the category 𝒜 for R to possess a left adjoint, are given. Natively these conditions involve the existence of suitable colimits that we interpret as objects which are simultaneously initial in four distinguished categories (among which the category of epi-induced objects), allowing for an explicit construction of L , under the appropriate hypotheses. This is achieved by introducing a relative version of the notion of weakly coreflective subcategory, which turns out to be a useful tool to compare the initial objects in the involved categories. We apply our results to obtain an analogue of Sweedler’s finite dual for the category of vector spaces graded by an abelian group G endowed with a bicharacter. When the bicharacter on G is skew-symmetric, a lifted adjunction as mentioned above is explicitly described, inducing an auto-adjunction on the category of bialgebras “colored” by G.

We study BiHom–Novikov–Poisson algebras, which are twisted generalizations of Novikov–Poisson algebras and Hom–Novikov–Poisson algebras, and find that BiHom–Novikov–Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom–Novikov–Poisson algebras give rise to BiHom-Poisson algebras.

Abstract We contribute to the study of Rota–Baxter operators on types of algebras other than associative and Lie algebras. If A is an algebra of a certain type and R is a Rota–Baxter operator on A, one can define a new multiplication on A by means of R and the previous multiplication and ask under what circumstances the new algebra is of the same type as A. Our first main result deals with such a situation in the case of BiHom-Lie algebras. Our second main result is a BiHom analogue of Aguiar’s theorem that shows how to obtain a pre-Lie algebra from a Rota–Baxter operator of weight zero on a Lie algebra. The BiHom analogue does not work for BiHom-Lie algebras, but for a new concept we introduce here, called left BiHom-Lie algebra, at which we arrived by defining first the BiHom version of Leibniz algebras.

Motivated by an example related to the tensor algebra, a stronger version of the notion of separable functor, called heavily separable (h-separable for short), was introduced and investigated in [1]. Here we study h-coseparable coalgebras in monoidal categories with special concern with the monoidal category TA♯ of right transfer morphisms through an algebra A in a monoidal category. We characterize the h-separability of the forgetful functor from the category of entwined modules associated to a cowreath to the base category using suitable Casimir morphisms. Even if there are non trivial examples of h-coseparable coalgebras over a field [1, Theorem 4.4], here we provide non trivial examples of h-coseparable coalgebras in the monoidal category TA⊗Hop# where H=H4 is the Sweedler 4-dimensional Hopf algebra over a field k and A=Cl(α,β,γ) the Clifford algebra.

We introduce and study infinitesimal BiHom-bialgebras, BiHom-Novikov algebras, BiHom-Novikov-Poisson algebras, and find some relations among these concepts. Our main result is to show how to obtain a left BiHom-pre-Lie algebra from an infinitesimal BiHom-bialgebra.

Prompted by an example related to the tensor algebra, we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable. We test this notion on several functors traditionally connected to the study of separability.

The purpose of this paper is to study Rota-Baxter operators for BiHom-associative algebras. Moreover, we introduce and discuss the properties of the notions of BiHom-(tri)dendriform algebra, BiHom-Zinbiel algebra and BiHom-quadri-algebra. We construct the free Rota-Baxter BiHom-associative algebra and present some observations about categories and functors related to Rota-Baxter structures.

We introduce the concept of {sigma, tau}-Rota-Baxter operator, as a twisted version of a Rota-Baxter operator of weight zero. We show how to obtain a certain {sigma, tau}-Rota-Baxter operator from a solution of the associative (Bi)Hom-Yang-Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

We give a description of the category of restricted Lie algebras over a field of prime characteristic by means of monadic decomposition of the functor that computes the -vector space of primitive elements of a -bialgebra.

We investigate some properties of Rota-Baxter operators on BiHom-Lie algebras. Along the way, we introduce BiHom analogues of pre-Lie and Leibniz algebras.

Let $mathcal{A}$ and $mathcal{B}$ be monoidal categories and let $left( L:mathcal{B}rightarrow mathcal{A},R:mathcal{A}rightarrow mathcal{B}right) $ be a pair of adjoint functors. Supposing that $R$ is moreover a lax monoidal functor (or, equivalently, that $L$ is colax monoidal), $R$ induces a functor $overline{R}:{sf Alg}({mathcal{A}})rightarrow {sf Alg}({mathcal{B}})$ and $L$ colifts to a functor $underline{L}: {sf Coalg}({mathcal{B}})rightarrow {sf Coalg}({mathcal{A}})$, as is well-known. An adjoint pair of such functors $(L,R)$ is called if the functor $overline{R}$ has a left adjoint and if the functor $underline{L}$ has a right adjoint. A pleasing fact is that, when $mathcal{A}$ and $mathcal{B}$ are moreover endowed with a braiding, a liftable pair of functors as above gives rise to an adjunction $left({underline{overline{L}}}:{sf Bialg}(mathcal{B})rightarrow {sf Bialg}(mathcal{A}), {underline{overline{R}}}:{sf Bialg}(mathcal{A}) rightarrow {sf Bialg}(mathcal{B})right) $, provided $R$ is braided with respect to the braidings of $mathcal{A}$ and $mathcal{B}$. In this note, sufficient conditions on the category $mathcal{A}$ for $overline{R}$ to possess a left adjoint, are given. Furthermore, it is shown that a so-called pre-rigid braided monoidal category $mathcal{C}$ always allows for a liftable pair of adjoint functors $left( (-)^{*}:mathcal{C}rightarrow mathcal{C}^{rm op},(-)^{*}:mathcal{C}^{rm op} rightarrow mathcal{C}right)$ provided $overline{(-)^{*}}$ has an adjoint. Moreover, for such a category $mathcal{C}$, a duality result in the spirit of a theorem originally due to Michaelis is proved. Our results are illustrated by considering $mathcal{C}$ to be the category of vector spaces graded by an abelian group.

We prove that a finite-dimensional Hopf algebra with the dual Chevalley Property over a field of characteristic zero is quasi-isomorphic to a Radford-Majid bosonization whenever the third Hochschild cohomology group in the category of Yetter-Drinfeld modules of its diagram with coefficients in the base field vanishes. Moreover we show that this vanishing occurs in meaningful examples where the diagram is a Nichols algebra.

In this paper Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg-Moore categories. To this aim we introduce the notion of Milnor-Moore category as a monoidal category for which a Milnor-Moore type Theorem holds. We also show how to lift the property of being a Milnor-Moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples.