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In the current research paper, we define and investigate the concept of BiHom-Gerstenhaber algebra. This algebraic structure is defined by two linear maps f, g, a BiHom-commutative law ′ ′ ∧ ′ ′ , a BiHom-Lie law ′ ′ [ , ] ′ ′ with degrees 0 and - 1 respectively over a graded vector space g . Additionally, it satisfies a compatibility condition called BiHom-Leibniz relation. Furthermore, we will provide an explicit construction of the associated BiHom-Gerstenhaber algebras up to homotopy. More precisely, this algebraic structure is defined by a structure of BiHom-cocommutative coalgebra (C , Δ (f , g) , f , g) , a structure of BiHom-coLie coalgebra (C , δ (f , g) , f , g) . It is equipped with a coderivation Q for two coproducts Δ (f , g) and δ (f , g) with degree 0 verifying Q 2 = 0 and a compatibility relations, called BiHom-coLeibniz relations. This bicoalgebra is also called BiHom-Gerstenhaber algebra up to homotopy or BiHom- g ∞ -algebra. [ABSTRACT FROM AUTHOR]

We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient A → B satisfying some finiteness conditions, the Frobenius tensor category D of graded B-comodules with its stable model structure induces a monoidal model structure on C . We consider the corresponding homotopy quotient γ : C → H o C and the induced quotient T → H o T for the tensor category T of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in H o T . We apply these results in the Rep(GL(m|n))-case and study its homotopy category H o T associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of H o T by the negligible morphisms is again the representation category of a supergroup scheme. [ABSTRACT FROM AUTHOR]

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects. [ABSTRACT FROM AUTHOR]

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]

Let K be an algebraically closed characteristic 0 field and let V be the K-vector space with basis { x 0 , x 1 , x 2 , ... } . In this paper we use the theory developed in [17] to associate with each power series f ∈ K [ [ X ] ] satisfying f (0) = 1 , an involutive solution s (f) : V ⊗ V → V ⊗ V , of the braid equation, and we begin the study of this correspondence. For n ≥ 2 , let Vn be the subspace of V generated by { x 0 , ... , x n − 1 } . Each one of the solution s(f) induces by restriction a solution s n (f) on V n ⊗ V n . We also begin the study of these solutions. [ABSTRACT FROM AUTHOR]

We show that the category of partial comodules over a Hopf algebra H is comonadic over Vectk and provide an explicit construction of this comonad using topological vector spaces. The case when H is finite-dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connected linear algebraic group does not admit partiality. [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 address the structure of real Lotka-Volterra coalgebras endowed with genetic realization, deepening on the role natural bases have when dealing with this problem. We also tackle the problem of determining all possible natural bases in the two-dimensional case, and solve this problem in terms of a parametrization provided by the natural bases. [ABSTRACT FROM AUTHOR]

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper. [ABSTRACT FROM AUTHOR]

We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension k ≤ k ′ is a right adjoint on the category of k -coalgebras). [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]

We introduce a theory of modules over a representation of a small category taking values in entwining structures over a semiperfect coalgebra. This takes forward the aim of developing categories of entwined modules to the same extent as that of module categories as well as the philosophy of Mitchell of working with rings with several objects. The representations are motivated by work of Estrada and Virili, who developed a theory of modules over a representation taking values in small preadditive categories, which were then studied in the same spirit as sheaves of modules over a scheme. We also describe, by means of Frobenius and separable functors, how our theory relates to that of modules over the underlying representation taking values in small K-linear categories. [ABSTRACT FROM AUTHOR]

Gendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras. [ABSTRACT FROM AUTHOR]

We introduce the notions of silting comodules and finitely silting comodules in quasi-finite category, and study some properties of them. We investigate the torsion pair and dualities which are related to finitely silting comodules, and give the equivalences among silting comodules, finitely silting comodules, tilting comodules and finitely tilting comodules. [ABSTRACT FROM AUTHOR]

We study transfer of pure-direct-objects via functors between Grothendieck categories. We consider fully faithful functors and adjoint pairs of functors. We give applications to module categories and comodule categories. [ABSTRACT FROM AUTHOR]

The motivation of this paper is to construct a deformation theory of coderivations of coassociative coalgebras. We introduce a notion of a Coder pair, that is, a coassociative coalgebra with a coderivation. Then we define a proper corepresentation of a Coder pair and study the corresponding cohomology. Finally, we show that a Coder pair is rigid, provided that the second cohomology group is trivial and point out that a deformation of finite order is extensible to higher order deformations if the obstruction class, which is defined to be in the third cohomology group, is trivial. [ABSTRACT FROM AUTHOR]

We prove that for a large class of well-behaved cocomplete categories C the weak and strong Drinfeld centers of the monoidal category E of cocontinuous endofunctors of C coincide. This generalizes similar results in the literature, where C is the category of modules over a ring A and hence E is the category of A-bimodules. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the concept of n-Gorenstein tilting comodules and study its main properties. This concept generalizes the notion of n-tilting comodules of finite injective dimensions to the case of finite Gorenstein injective dimensions. As an application of our results, we discuss the problem of existence of complements to partial n-Gorenstein tilting comodules. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Gorenstein flat comodules and give a description of colocalization of Gorenstein flat comodules. A right C-comodule M is called a Gorenstein flat comodule if there exists an exact sequence ⋯ → F 1 → F 0 → F 0 → F 1 → ⋯ of flat comodules with M ≅ Ker (F 0 → F 1) and such that Com C (- , I) leaves the sequence exact whenever I is a finite-dimensional right C-comodule and id C (I) < ∞ . Some properties of Gorenstein flat comodules are studied. [ABSTRACT FROM AUTHOR]

We introduce and study the category of twisted modules over a triangular differential graded bocs. We show that in this category idempotents split, that it admits a natural structure of a Frobenius category, that a twisted module is homotopically trivial iff its underlying complex is acyclic, and that any homotopy equivalence of differential graded bocses determines an equivalence of the corresponding homotopy categories of twisted modules. The category of modules over an A ∞ -algebra is equivalent to the category of twisted modules over a triangular differential graded bocs, so all the preceding statements lift to the former category. [ABSTRACT FROM AUTHOR]

Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann–Gerhardt–Høgenhaven–Shipley–Ziegenhagen developed a coBökstedt spectral sequence to compute the homology of coTHH for coalgebras over the sphere spectrum. We construct a relative coBökstedt spectral sequence to study coTHH of coalgebra spectra over any commutative ring spectrum R. Further, we use algebraic structures in this spectral sequence to complete some calculations of the homotopy groups of relative topological coHochschild homology. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a more general category of Θ\Theta -Yetter-Drinfeld modules (Θ∈AutH(H)\Theta \in {\rm{Aut}}\hspace{0.33em}H\left(H)) over a Hom-Hopf algebra, which unifies two different definitions of Hom-Yetter-Drinfeld category introduced by Makhlouf and Panaite, Li and Ma, respectively. We show that the category of Θ\Theta -Yetter-Drinfeld modules with a bijective antipode SS is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category. Also by the method of symmetric pairs, we prove that if a Θ\Theta -Yetter-Drinfeld category over a Hom-Hopf algebra HH is symmetric, then HH is trivial. Finally, we find a sufficient and necessary condition for a Θ\Theta -Yetter-Drinfeld category to be pseudosymmetric.

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 the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. We consider fully faithful functors as well as adjoint pairs of functors. We give several applications to Grothendieck categories and, in particular, to (graded) module and comodule categories. [ABSTRACT FROM AUTHOR]

In this paper, we first study the species of finite topological spaces recently considered by Fauvet, Foissy, and Manchon. Then, we construct a twisted pre-Lie structure on the species of connected finite topological spaces. The underlying pre-Lie structure defines a coproduct on the species of finite topological spaces different from those already defined by the authors above. In the end, we illustrate the link between the Grossman–Larson product and the proposed coproduct. [ABSTRACT FROM AUTHOR]

We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBökstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations. [ABSTRACT FROM AUTHOR]

The study of set-theoretic solutions of the Yang-Baxter equation, also known as Yang-Baxter maps, is historically a meeting ground for various areas of mathematics and mathematical physics. In this work, we study factorization problems on rational loop groups, which give rise to Yang-Baxter maps on various geometrical objects. We also study the symplectic and Poisson geometry of these Yang-Baxter maps, which we show to be integrable maps in the sense of having natural collections of Poisson commuting integrals. In a special case, the factorization problems we consider are associated with the N-soliton collision process in the n-Manakov system, and in this context we show that the polarization scattering map is a symplectomorphism. [ABSTRACT FROM AUTHOR]

Let V ∗ ⊗ V → C be a non-degenerate pairing of countable-dimensional complex vector spaces V and V ∗ . The Mackey Lie algebra g = gl M (V , V ∗) corresponding to this pairing consists of all endomorphisms φ of V for which the space V ∗ is stable under the dual endomorphism φ ∗ : V ∗ → V ∗ . We study the tensor Grothendieck category T generated by the g -modules V, V ∗ and their algebraic duals V ∗ and V ∗ ∗ . The category T is an analogue of categories considered in prior literature, the main difference being that the trivial module C is no longer injective in T . We describe the injective hull I of C in T , and show that the category T is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category I T of objects in T which are free as I-modules. Our main result is that the category I T is also Koszul, and moreover that I T is universal among abelian C -linear tensor categories generated by two objects X, Y with fixed subobjects X ′ ↪ X , Y ′ ↪ Y and a pairing X ⊗ Y → 1 where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories T and I T . [ABSTRACT FROM AUTHOR]

Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting torsion theory for coalgebras. [ABSTRACT FROM AUTHOR]

The species of finite topological spaces admits a bimonoid structure together with a compatible internal coproduct, recently defined by F. Fauvet, L. Foissy and the second author. In this article, we define a doubling of this species in two different ways. We build a bimonoid structure on the first species thus obtained, an internal comultiplication on the second one, and describe a cointeraction between them. We also investigate two related associative products obtained by dualisation. [ABSTRACT FROM AUTHOR]

We present some tools for providing situations where the generalised Rota formula of Carlier (Algebr Geometr Topol 20:169–213, 2020) applies. As an example of this, we compute the Möbius function of the incidence algebra of any directed restriction species, free operad, or more generally free monad on a finitary polynomial monad. [ABSTRACT FROM AUTHOR]

The topic of this paper is a generalization of Tannaka duality to coclosed categories. As an application we prove reconstruction theorems for coalgebras (bialgebras, Hopf algebras) in categories of topological vector spaces over a nonarchimedean field K. In particular, our results imply reconstruction and recognition theorems for categories of locally analytic representations of compact p-adic groups, which was the major motivation for this work. [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]

We introduce the analogue of Manin's universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras P and U, where U is finite dimensional, we construct a Poisson algebra B (P , U) together with a Poisson algebra homomorphism ψ B (P , U) : P → U ⊗ B (P , U) satisfying a suitable universal property. B (P , U) is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If P = U is a finite dimensional Poisson algebra then B (P) = B (P , P) admits a unique Poisson bialgebra structure such that ψ B (P) becomes a Poisson comodule algebra and, moreover, the pair (B (P) , ψ B (P) ) is the universal coacting bialgebra of P. The universal coacting Poisson Hopf algebra H (P) on P is constructed as the initial object in the category of Poisson comodule algebra structures on P by using the free Poisson Hopf algebra on a Poisson bialgebra (Agore in J Math Phys 10:083502, 2014). [ABSTRACT FROM AUTHOR]

A well-known Assem–Smal ϕ theorem on tilting modules plays an important role in tilting theory. In this paper, we prove that a version of Assem–Smal ϕ theorem still holds in the category of comodules. Following this result, we characterize the tilting torsion-free classes in the category of comodules using the precover theory. [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]

In this paper, we define a Cayley–Dickson process for k-coalgebras proving some results that describe the properties of the final coalgebra, knowing the properties of the initial one. Namely, after applying the Cayley–Dickson process for k-coalgebras to a coassociative coalgebra, we obtain a coalternative one. Moreover, the first coalgebra is cocommutative if and only if the final coalgebra is coassociative. Finally we extend these results to a more general approach of D-coalgebras, where D is a k-coalgebra, presenting a class of examples of coalternative (non-coassociative) coalgebras obtained from group D-coalgebras. [ABSTRACT FROM AUTHOR]

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, imply the existence of the (non-homogeneous) c d -index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for bounded balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete c d -index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture non-negativity of the c d -index for acyclic digraphs having a balanced linear edge labeling. [ABSTRACT FROM AUTHOR]

We study the transfer of Baer-Kaplansky classes via additive functors between preadditive categories. We show that the Baer-Kaplansky property is well behaved with respect to fully faithful functors, but not with respect to their usual generalizations such as faithful, full, separable, naturally full, Maschke or dual Maschke functors. We also present the transfer of the Baer-Kaplansky property between subclasses of static and adstatic objects induced by adjoint pairs of functors. We mainly give applications to (graded) module categories and comodule categories. [ABSTRACT FROM AUTHOR]