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In this paper, we establish a connection between evolution algebras of dimension two and Hopf algebras, via the algebraic group of automorphisms of an evolution algebra. Initially, we describe the Hopf algebra associated with the automorphism group of a 2-dimensional evolution algebra. Subsequently, for a 2-dimensional evolution algebra A over a field K, we detail the relation between the algebra associated with the (tight) universal associative and commutative representation of A, referred to as the (tight) p-algebra, and the corresponding Hopf algebra, H , representing the affine group scheme Aut (A) . Our analysis involves the computation of the (tight) p - algebra associated with any 2-dimensional evolution algebra, whenever it exists. We find that Aut (A) = 1 if and only if there is no faithful associative and commutative representation for A. Moreover, there is a faithful associative and commutative representation for A if and only if H ≇ K and char (K) ≠ 2 , or H ≇ K (ϵ) (the dual numbers algebra) and H ≇ K in case of char (K) = 2 . Furthermore, if A is perfect and has a faithful tight p-algebra, then this p-algebra is isomorphic to H (as algebras). Finally, we derive implications for arbitrary finite-dimensional evolution algebras. [ABSTRACT FROM AUTHOR]

We introduce a new class of numerical schemes which allow for low-regularity approximations to the expectation E (| u k (t , v η) | 2) , where u k denotes the k-th Fourier coefficient of the solution u of the dispersive equation and v η (x) the associated random initial data. This quantity plays an important role in physics, in particular in the study of wave turbulence where one needs to adopt a statistical approach in order to obtain deep insight into the generic long-time behaviour of solutions to dispersive equations. Our new class of schemes is based on Wick's theorem and Feynman diagrams together with a resonance-based discretisation (Bruned and Schratz in Forum Math Pi 10:E2, 2022) set in a more general context: we introduce a novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair. The character of the scheme draws its inspiration from the treatment of singular stochastic partial differential equations via regularity structures. In contrast to classical approaches, we do not discretise the PDE itself, but rather its expectation. This allows us to heavily exploit the optimal resonance structure and underlying gain in regularity on the finite dimensional (discrete) level. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalization of well-established results regarding symmetries of k -algebras, where k is a field. Traditionally, for a k -algebra A, the group of k -algebra automorphisms of A captures the symmetries of A via group actions. Similarly, the Lie algebra of derivations of A captures the symmetries of A via Lie algebra actions. In this paper, given a category C whose objects possess k -linear monoidal categories of modules, we introduce an objec Sym C (A) that captures the symmetries of A via actions of objects in C . Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected k -algebra A, some of its symmetries are naturally captured within the weak Hopf framework. [ABSTRACT FROM AUTHOR]

In the theory of quantum automorphism groups, one constructs Hopf algebras acting on an algebra K from certain algebra morphisms σ : K → M n (K) . This approach is applied to the field K = k (t) of rational functions, and it is investigated when these actions restrict to actions on the coordinate ring B = k [ t 2 , t 3 ] of the cusp. An explicit example is described in detail and shown to define a new quantum homogeneous space structure on the cusp. [ABSTRACT FROM AUTHOR]

As a self-dual framework to unify the study of quasigroups and Hopf algebras, quantum quasigroups are defined using a quantum analogue of the combinatorial approach to classical quasigroups, merely requiring invertibility of the left and right composites. In this paper, quantum quasigroups are redefined with a quantum analogue of the equational approach to classical quasigroups. Here, the left and right composites of auxiliary quantum quasigroups participate in diagrams whose commutativity witnesses the required invertibilities. Whenever the original and two auxiliary quantum quasigroups appear on an equal footing, the triality symmetry of the language of equational quasigroups is replicated. In particular, the problem arises as to when this triality emerges in the Hopf algebra context. [ABSTRACT FROM AUTHOR]

Let H be a Hopf algebra that is ℤ -graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of H to be a Zhang twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated with quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction. [ABSTRACT FROM AUTHOR]

We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford's π-biproduct. Firstly, we discuss the endomorphism monoid Endπ-Hopf(A × H, p) and the automorphism group Autπ-Hopf(A × H, p) of Radford's π-biproduct A × H = {A × Hα}α∈π, and prove that the automorphism has a factorization closely related to the factors A and H = {Hα}α∈π. What's more interesting is that a pair of maps (FL, FR) can be used to describe a family of mappings F = {Fα}α∈π. Secondly, we consider the relationship between the automorphism group Autπ-Hopf(A × H, p) and the automorphism group Aut π - Y D -Hopf (A) of A, and a normal subgroup of the automorphism group Autπ-Hopf(A × H, p). Finally, we specifically describe the automorphism group of an example. [ABSTRACT FROM AUTHOR]

Conformal blocks of q, t-deformed Virasoro and W -algebras are important special functions in representation theory with applications in geometry and physics. In the Nekrasov–Shatashvili limit t → 1 , whenever one of the representations is degenerate then conformal block satisfies a difference equation with respect to the coordinate associated with that degenerate representation. This is a stationary Schrodinger equation for an appropriate relativistic quantum integrable system. It is expected that generalization to generic t ≠ 1 is a non-stationary Schrodinger equation where t parametrizes shift in time. In this paper we make the non-stationary equation explicit for the q, t-Virasoro block with one degenerate and four generic Verma modules and prove it when three modules out of five are degenerate, using occasional relation to Macdonald polynomials. [ABSTRACT FROM AUTHOR]

We extend Willerton’s [24] graphical calculus for bimonads to comodule monads, a monadic interpretation of module categories over a monoidal category. As an application, we prove a version of Tannaka–Krein duality for these structures. [ABSTRACT FROM AUTHOR]

In this paper, all string modules of one class of basic Hopf algebras of tame type, which are denoted by H 2 are studied. Firstly, the isomorphism classes of all string modules of H 2 are classified. Then the projective class ring of H 2 is described. Finally, the McKay matrix and the corresponding McKay quiver of H 2 are discussed. [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional Hopf algebra over an algebraically closed field k with the dual Chevalley property. We prove that H is of finite corepresentation type if and only if it is coNakayama, if and only if the link quiver Q (H) of H is a disjoint union of basic cycles, if and only if the link-indecomposable component H (1) containing k 1 is a pointed Hopf algebra and the link quiver of H (1) is a basic cycle. [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]

Let K be the smallest dimensional non-pointed basic Hopf algebra. We determine all finite-dimensional Nichols algebras in and compute the liftings. The result shows that there are some new Nichols algebras of non-diagonal type and new finite-dimensional Hopf algebras without the dual Chevalley property. [ABSTRACT FROM AUTHOR]

Thanks to the topological Hopf algebra of renormalization of Green’s functions in a gauge field theory, we associate a bi-Heyting algebra to each combinatorial Dyson–Schwinger equation. This setting leads us to characterize subsystems generated by the solution spaces of quantum motions. In addition, we apply the c 2 -invariant of Feynman diagrams to build a new Heyting algebra of multiplicative groups which encodes a more general class of subsystems of the physical theory. [ABSTRACT FROM AUTHOR]

We associate to a projective n-dimensional toric variety X Δ a pair of co-commutative (but generally non-commutative) Hopf algebras H X α , H X T . These arise as Hall algebras of certain categories Coh α (X) , Coh T (X) of coherent sheaves on X Δ viewed as a monoid scheme—i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When X Δ is smooth, the category Coh T (X) has an explicit combinatorial description as sheaves whose restriction to each A n corresponding to a maximal cone σ ∈ Δ is determined by an n-dimensional generalized skew shape. The (non-additive) categories Coh α (X) , Coh T (X) are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff–Kapranov. The Hall algebras H X α , H X T are graded and connected, and so enveloping algebras H X α ≃ U (n X α) , H X T ≃ U (n X T) , where the Lie algebras n X α , n X T are spanned by the indecomposable coherent sheaves in their respective categories. We explicitly work out several examples, and in some cases are able to relate n X T to known Lie algebras. In particular, when X = P 1 , n X T is isomorphic to a non-standard Borel in gl 2 [ t , t - 1 ] . When X is the second infinitesimal neighborhood of the origin inside A 2 , n X T is isomorphic to a subalgebra of gl 2 [ t ] . We also consider the case X = P 2 , where we give a basis for n X T by describing all indecomposable sheaves in Coh T (X) . [ABSTRACT FROM AUTHOR]

We propose a quantization algebra of the Loday-Ronco Hopf algebra k [ Y ∞ ] , based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra k [ Y ∞ ] h is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion A TopRec h is a subalgebra of a quotient algebra A Reg h obtained from k [ Y ∞ ] h that nevertheless doesn't inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of A TopRec h in low degree. [ABSTRACT FROM AUTHOR]

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg–Tran–Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane. [ABSTRACT FROM AUTHOR]

The notions of a post-group and a pre-group are introduced as a unification and enrichment of several group structures appearing in diverse areas from numerical integration to the Yang–Baxter equation. First the Butcher group from numerical integration on Euclidean spaces and the -group of an operad naturally admit a pre-group structure. Next a relative Rota–Baxter operator on a group naturally splits the group structure to a post-group structure. Conversely, a post-group gives rise to a relative Rota–Baxter operator on the sub-adjacent group. Further a post-group gives a braided group and a solution of the Yang–Baxter equation. Indeed the category of post-groups is isomorphic to the category of braided groups and the category of skew-left braces. Moreover a post-Lie group differentiates to a post-Lie algebra structure on the vector space of left invariant vector fields, showing that post-Lie groups are the integral objects of post-Lie algebras. Finally, post-Hopf algebras and post-Lie Magnus expansions are utilized to study the formal integration of post-Lie algebras. As a byproduct, a post-group structure is explicitly determined on the Lie–Butcher group from numerical integration on manifolds. [ABSTRACT FROM AUTHOR]

For a split quasireductive supergroup G defined over a field, we study structure and representation of Frobenius kernels G r of G and we give a necessary and sufficient condition for G r to be unimodular in terms of the root system of G . We also establish Steinberg's tensor product theorem for G under some natural assumptions. [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 contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over D ∞ , the infinite dihedral group.We find all the irreducible Yetter-Drinfeld modules V over D ∞ , and determine which Nichols algebras B (V) of V are finite GK-dimensional. [ABSTRACT FROM AUTHOR]

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid M ⋊ H , we construct a locally convex bialgebroid Dirac (M ⋊ H) with an antipode over C c ∞ (M) , from which the groupoid M ⋊ H can be reconstructed as its spectral action Lie groupoid A G sp (Dirac (M ⋊ H)) . [ABSTRACT FROM AUTHOR]

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques. [ABSTRACT FROM AUTHOR]

We construct a class of quantum doubles D (H D n ) of pointed Hopf algebras of rank one H D . We describe the algebra structures of D (H D n ) by generators with relations. Moreover, we give the comultiplication ΔD, counit εD and the antipode SD, respectively. [ABSTRACT FROM AUTHOR]

We calculate all irreducible representations over a subfamily of pointed Hopf algebras with group-likes the dihedral group analyzing the possible decompositions of the restriction to the dihedral group and calculating the Jacobson radical of the Hopf algebra. [ABSTRACT FROM AUTHOR]

We consider a class of Nichols algebras B (L q (1 , G)) introduced in Andruskiewitsch et al. which are domains and have many favourable properties like AS-regular and strongly noetherian. We classify their finite-dimensional simple modules and their point modules. [ABSTRACT FROM AUTHOR]

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver's moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in R d from the iterated-integrals signature. In particular, we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in R d under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations. [ABSTRACT FROM AUTHOR]

We show that finitely generated cohomology is invariant under separable equivalences for all algebras. As a result, we obtain a proof of the finite generation of cohomology for finite symmetric tensor categories in characteristic zero, as conjectured by Etingof and Ostrik. Moreover, for such categories we also determine the representation dimension and the Rouquier dimension of the stable category. Finally, we recover a number of results on the cohomology of stably equivalent and singularly equivalent algebras. [ABSTRACT FROM AUTHOR]

Let 한 be a base field of characteristic p > 0 and let U be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful U-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three. [ABSTRACT FROM AUTHOR]

For a commutative ring R and a commutative cocommutative Hopf algebra H finitely generated projective as an R-module, Tilborghs in (Math J Okayama Univ 32:43–52, 1990), established an anti-isomorphism of groups between the Brauer group BD(R, H) of H-dimodule and the Brauer B D (R , H ∗) of H ∗ -dimodule algebras, where H ∗ is the linear dual of H. In this paper, we generalize this result by constructing an anti-isomorphism of groups between BD(S, H), the Brauer group of dyslectic (S, H)-dimodule algebras and B D (S op , H ∗) , the Brauer group of dyslectic (S op , H ∗) -dimodule algebras, where S is an H-commutative H-dimodule algebra and S op is the opposite algebra of S. [ABSTRACT FROM AUTHOR]

Let L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L. [ABSTRACT FROM AUTHOR]

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category M , we introduce the notion of a Σ-twisted trace on the class Proj (M) of projective objects of M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on Proj (M) and the set of natural transformations from Σ to the Nakayama functor of M . Non-degeneracy and compatibility with the module structure (when M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular. [ABSTRACT FROM AUTHOR]

Based on the Connes–Kreimer Hopf algebra of rooted trees, rooted tree maps are defined as linear maps on the noncommutative polynomial algebra Q ⟨ x , y ⟩ . It is known that they induce a large class of linear relations for multiple zeta values. In this paper, we show for any rooted tree f there exists a unique polynomial in Q ⟨ x , y ⟩ that gives the image of the rooted tree map f ~ explicitly. We also characterize the antipode maps as the conjugation by the special map τ . [ABSTRACT FROM AUTHOR]

Noncommutative Kähler structures were recently introduced as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic 1-forms) we provide simple cohomological criteria for positivity, allowing one to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds O q (G / L S) . Building on the recently established noncommutative Borel–Weil theorem, every relative line module over O q (G / L S) can be identified as positive, negative, or flat, and it is then concluded that each Kähler structure is of Fano type. [ABSTRACT FROM AUTHOR]

Denote by H a finite dimensional Hopf C ∗ -algebra, K a Hopf ∗ -subalgebra of H. Starting with the observable algebra A K in non-equilibrium Hopf spin models, in which A K carries a coaction of relative quantum double D(H, K), the field algebra F K = A K ⋊ D (H , K) ^ is obtained, where D (H , K) ^ is the dual of D(H, K). This paper shows that the Haar integral of D(H, K) admits a faithful conditional expectation Γ from F K onto A K . The index of Γ is calculated by virtue of its quasi-basis provided by the matrix units of D (H , K) ^ . [ABSTRACT FROM AUTHOR]

Let G be a group generated by a set of finite order elements. We prove that any bicrossed product Hm,d(q) ⋈ k[G] between the generalized Taft algebra Hm,d(q) and group algebra k[G] is actually the smash product Hm,d(q)♯k[G]. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of G. As an application, the classification of H m , d (q) ⋈ k [ C n 1 × C n 2 ] is completely presented by generators and relations, where Cn denotes the n-cyclic group. [ABSTRACT FROM AUTHOR]

We first describe the Sekine quantum groups A k (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of A k and describe their representation rings r (A k) . Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of r (A k) . [ABSTRACT FROM AUTHOR]

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory Φ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra H FG cut (Φ) and lattices of intermediate structures. Feynman diagrams in H FG (Φ) are applied to describe states in Φ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space S graphon Φ , M ⊆ [ 0 , ∞) ([ 0 , 1 ]) . We associate Hopf subalgebras of H FG (Φ) generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green's functions of Φ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales. [ABSTRACT FROM AUTHOR]

We study the G -extensions C of a near-group fusion category of type (Z 2 , 1) . If C is braided we prove that C can be reconstructed from pointed fusion categories by Z 2 -extensions or Z 2 -equivariantizations. Furthermore, if C is also integral, or C is equivalent as a tensor category to the category of finite dimensional representations of a semisimple Hopf algebra, we prove that C is group-theoretical, which completes the classification of these categories in the sense of Morita equivalence. [ABSTRACT FROM AUTHOR]

In this paper, we expand on the notion of combinatorial presheaf, first introduced explicitly by Aguiar and Mahajan in 2010 but already present in the literature in some other points of view. We do this by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider functions that count the number of patterns of objects and endow the linear span of these functions with a product and a coproduct. In this way, any well-behaved family of combinatorial objects that admits a notion of substructure generates a Hopf algebra, and this association is functorial. For example, the Hopf algebra on permutations studied by Vargas in 2014 and the Hopf algebra on symmetric functions are particular cases of this construction. A specific family of pattern Hopf algebras of interest are the ones arising from commutative combinatorial presheaves. This includes the presheaves on graphs, posets and generalized permutahedra. Here, we show that all the pattern Hopf algebras corresponding to commutative presheaves are free. We also study a remarkable non-commutative presheaf structure on marked permutations, i.e. permutations with a marked element. These objects have a natural product called inflation, which is an operation motivated by factorization theorems for permutations. In this paper, we find new factorization theorems for marked permutations. We use these theorems to show that the pattern Hopf algebra for marked permutations is also free, using Lyndon words techniques. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the structures of the cell modules of the complexified Green algebra R(D(H4)), where D(H4) is the Drinfel'd quantum double of Sweedler's 4-dimensional Hopf algebra H4. We show that R(D(H4)) has one infinite dimensional cell module, one 4-dimensional cell module generated by all finite dimensional indecomposable projective modules of D(H4) and infinitely many 2-dimensional cell modules. More precisely, we obtain the decompositions of all finite dimensional cell modules into the direct sum of indecomposable submodules, and show that the infinite dimensional cell module can be written as the direct sum of two infinite dimensional indecomposable submodules. [ABSTRACT FROM AUTHOR]

In this paper, we study the category of corepresentations of a monoidal comonad. We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle (coseparable) comonad, and it is a braided category if and only if the monoidal comonad admit a cobraided structure. At last, as an application, the braided structure and the semisimplicity of the Hom-comodule category of a monoidal Hom-bialgebra are discussed. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the structures of the cell modules of the complexified Green algebra R(D(H4)), where D(H4) is the Drinfel'd quantum double of Sweedler's 4-dimensional Hopf algebra H4. We show that R(D(H4)) has one infinite dimensional cell module, one 4-dimensional cell module generated by all finite dimensional indecomposable projective modules of D(H4) and infinitely many 2-dimensional cell modules. More precisely, we obtain the decompositions of all finite dimensional cell modules into the direct sum of indecomposable submodules, and show that the infinite dimensional cell module can be written as the direct sum of two infinite dimensional indecomposable submodules. [ABSTRACT FROM AUTHOR]

By means of a new notion of subforests of an angularly decorated rooted forest, we give a combinatorial construction of a coproduct on the free Rota–Baxter algebra on angularly decorated rooted forests. We show that this coproduct equips the Rota–Baxter algebra with a bialgebra structure and further a Hopf algebra structure. [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]

We introduce a notion of split extension of (non-associative) bialgebras which generalizes the notion of split extension of magmas introduced by M. Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to the notion of action of (non-associative) bialgebras. We particularize this equivalence to (non-associative) Hopf algebras by defining split extensions of (non-associative) Hopf algebras and proving that they are equivalent to actions of (non-associative) Hopf algebras. Moreover, we prove the validity of the Split Short Five Lemma for these kinds of split extensions, and we examine some examples. [ABSTRACT FROM AUTHOR]

We first prove an analogue of Lagrange theorem for global dimensions of fusion categories, then we give a complete classifications of pre-modular fusion categories of integer global dimensions less than or equal to 10. [ABSTRACT FROM AUTHOR]

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra A ∗ . We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of A ∗ and its structure as a Hopf algebra. [ABSTRACT FROM AUTHOR]