Related papers: A combinatorial ${E_\infty}$-algebra structure on …

We consider injectivity and surjectivity of some maps on the exterior algebra of isomorphic finite-dimensional vector spaces. We prove the properties of the maps in full generality, for any dimension of the vector space and any subspace. We…

We introduce a coshuffle comultiplication on the singular chain complex of configuration spaces, and we show that this structure endows the configuration space with the structure of a differential graded coalgebra (DGCoAlg). We then prove…

We present the Gerstenhaber algebra structure on the Hochschild cohomology of Koszul algebras defined by quivers and relations using the idea of homotopy liftings. E.L. Green, G. Hartman, E.N. Marcos and O. Solberg provided a canonical way…

We define the secondary Hochschild complex for an entwining structure over a commutative $k$-algebra $B$. We show that this complex carries the structure of a weak comp algebra. We obtain two distinct cup product structures for the…

We use compactifications of C*-algebras to introduce noncommutative coarse geometry. We transfer a noncommutative coarse structure on a C*-algebra with an action of a locally compact Abelian group by translations to Rieffel deformations and…

The normalized singular chains of a path connected pointed space $X$ may be considered as a connected $E_{\infty}$-coalgebra $\mathbf{C}_*(X)$ with the property that the $0^{\text{th}}$ homology of its cobar construction, which is naturally…

We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between…

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E-infinity algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category…

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a…

This paper constructs (with challenging obstacles) on the three torus with its cubical decomposition: Firstly, a combinatorial graded intersection algebra (graded by the codimension) which is commutative and associative defined by…

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The…

Let $G$ be a finite $p$-group and $\mathbb{F}$ a field of characteristic $p$. We filter the cochain complex of a free $G$-space with coefficients in $\mathbb{F}$ by powers of the augmentation ideal of $\mathbb{F} G$. We show that the cup…

We give a construction that takes a simple linear algebraic group $G$ over a field and produces a commutative, unital, and simple non-associative algebra $A$ over that field. Two attractions of this construction are that (1) when $G$ has…

We construct a combinatorial moduli space closely related to the KSV-compactification of the moduli space of bordered marked Riemann surfaces. The open part arises from symmetric metric ribbon graphs. The compactification is obtained by…

Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule…

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate…

We show that certain C*-algebras which have been studied among others by Arzumanian, Vershik, Deaconu, and Renault in connection to a measure preserving transformation of a measure space and/or to a covering map of a compact space are…

Let k be a non archimedean field. If X is a k-algebraic variety and U a locally closed semi-algebraic subset of X^{an} -- the Berkovich space associated to X -- we show that for l \neq char(\tilde{k}), the cohomology groups H^i_c (\bar{U},…

In this paper, we study compatible Leibniz algebras. We characterize compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable differential graded Lie algebra. We define a cohomology theory of compatible Leibniz algebras…

Let $P$ be an $n$-gon with $n\geq3.$ There is a formal combinatorial $A_\infty$-coalgebra structure on cellular chains $C_*(P)$ with non-vanishing higher order structure when $n\geq5$. If $X_g$ is a closed compact surface of genus $g\geq2$…