Let $M$ be any compact simply-connected oriented $d$-dimensional smooth manifold and let $𝔽$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $H{H}^{*}(S^{*}\left(M\right),S^{*}\left(M\right))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}\left(LM\right)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $H{C}_{-}^{*}\left(S^{*}\left(M\right)\right)$...

We first prove that every countably presented module is a pure epimorphic image of a countably generated pure-projective module, and by using this we prove that if every countably generated pure-projective module is pure-injective then every module is pure-injective, while if in any countably generated pure-projective module every countably generated pure-projective pure submodule is a direct summand then every module is pure-projective.

In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal...

Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}\left(q\right)\bowtie k\left[G\right]$ between the generalized Taft algebra $H_{m,d}\left(q\right)$ and group algebra $k\left[G\right]$ is actually the smash product $H_{m,d}\left(q\right)♯k\left[G\right]$. 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 $H_{m,d}\left(q\right)\bowtie k[C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM▹◂k\left(G\right)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_{𝕆}$ as transversal in an order 128 group $X$ with subgroup ${\mathbb{Z}}_2^3$ and hence obtain a Hopf quasigroup $k{G}_{𝕆}>◂k\left({\mathbb{Z}}_2^3\right)$ as a particular case of our construction.

On définit une structure de bigèbre différentielle graduée sur la somme directe des complexes cellulaires des permutoèdres, qui contient une sous-bigèbre différentielle graduée dont le complexe sous-jacent est la somme directe des complexes cellulaires des polytopes de Stasheff. Ceci étend des constructions de Malvenuto et Reutenauer et de Loday et Ronco pour les sommets des mêmes polytopes.

Here we introduce the k-bi-ideals in semirings and the intra k-regular semirings. An intra k-regular semiring S is a semiring whose additive reduct is a semilattice and for each a ∈ S there exists x ∈ S such that a + xa²x = xa²x. Also it is a semiring in which every k-ideal is semiprime. Our aim in this article is to characterize both the k-regular semirings and intra k-regular semirings using of k-bi-ideals.

Commutative congruence-simple semirings were studied in [2] and [7] (but see also [1], [3]--[6]). The non-commutative case almost (see [8]) escaped notice so far. Whatever, every congruence-simple semiring is bi-ideal-simple and the aim of this very short note is to collect several pieces of information on these semirings.

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...