Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor ${ℍ}_C:C-Comod\to H_C-Comod$ that restricts to a representation equivalence ${ℍ}_C:C-comod\to H_C-como{d}_{sp}^{•}$, where $H_C$ is a coradical square complete hereditary bipartite...

Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor ${Hom}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-plethory is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A\left[X\right]$ is an initial object in the category of such structures. The $D$-algebra $Int\left(D\right)$ has such a structure if $D=A$ is a domain such that the natural $D$-algebra homomorphism ${\theta }_n:{{⨂}_D}_{i=1}^{n}Int\left(D\right)\to Int\left(D^n\right)$ is an isomorphism for...

A class of stratified posets $I{*}_{\varrho }$ is investigated and their incidence algebras $KI{*}_{\varrho }$ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on $I{*}_{\varrho }$ we associate with $I{*}_{\varrho }$ a bound quiver (Q, Ω) in such a way that $KI{*}_{\varrho }\simeq K(Q,\Omega )$. We show that the fundamental group of (Q, Ω) is the free group with two free generators if $I{*}_{\varrho }$ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition $I_{\varrho }$ is three-partite a fundamental domain $I^{*+\times }$ of this covering is...

We continue our study of the category of Doi Hom-Hopf modules introduced in [Colloq. Math., to appear]. We find a sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. We also obtain a condition for a monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Moreover, we introduce morphisms between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras, which give rise to functors between the category of Doi Hom-Hopf...

By analogy with the projective, injective and flat modules, in this paper we study some properties of $C$-Gorenstein projective, injective and flat modules and discuss some connections between $C$-Gorenstein injective and $C$-Gorenstein flat modules. We also investigate some connections between $C$-Gorenstein projective, injective and flat modules of change of rings.

We describe the stable module categories of the self-injective finite-dimensional algebras of finite representation type over an algebraically closed field which are Calabi-Yau (in the sense of Kontsevich).

For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl${}_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for $W$. The rays...

A multiplicative functional on a graded connected Hopf algebra is called the character. Every character decomposes uniquely as a product of an even character and an odd character. We apply the character theory of combinatorial Hopf algebras to the Hopf algebra of simple graphs. We derive explicit formulas for the canonical characters on simple graphs in terms of coefficients of the chromatic symmetric function of a graph and of canonical characters on quasi-symmetric functions. These formulas and...