Related papers: The Bracket in the Bar Spectral Sequence for an It…

Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$. The purpose…

We introduce a new Hopf algebra that operates on pairs of finite interval partitions and permutations of equal length. This algebra captures vincular patterns, which involve specifying both the permutation patterns and the consecutive…

In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. We further showed that evaluating…

Let $A$ be a Hopf algebra over a field $K$ of characteristic 0 and suppose there is a coalgebra projection $\pi$ from $A$ to a sub-Hopf algebra $H$ that splits the inclusion. If the projection is $H$-bilinear, then $A$ is isomorphic to a…

This paper is about a correspondence between monoidal structures in categories and $n$-fold loop spaces. We develop a new syntactical technique whose role is to substitute the coherence results, which were the main ingredients in the proofs…

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a *-structure, conjugate-linear on the…

We obtain a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$. This spectrum turns out to be a quotient space of the hyper-Stonean envelope of the spectrum of $A$.

Using the shuffle structure of the graphs, we introduce a new kind of the Hopf algebraic structure for tagged graphs with, or without loops. Like a quantum group structure, its product is non-commutative. With the help of the Hopf algebraic…

We characterize interpolating sequences for multiplier algebras of spaces with the complete Pick property. Specifically, we show that a sequence is interpolating if and only if it is separated and generates a Carleson measure. This…

A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical $\overline\partial$-operator, while the other operator is the adjoint action of the…

Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups $H_i(X,G)$ by considering cycles in the simplicial scheme $BG\times X (an idea suggested by Andrei Suslin). We…

We present a mechanism for the creation of gaps in the spectra of self-adjoint operators defined over a Hilbert space of functions on a graph, which is based on the process of graph decoration. The resulting Hamiltonians can be viewed as…

We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an…

It is well known that (in suitable codimension) the spaces of long knots in $\mathbb{R}^n$ modulo immersions are double loop spaces. Hence the homology carries a natural Gerstenhaber structure, given by the Gerstenhaber structure on the…

We define several versions of the cohomology ring of an associative algebra. These ring structures unify some well known operations from homological algebra and differential geometry. They have some formal resemblance with the quantum…

If $R$ is a commutative ring, $M$ a compact $R$-oriented manifold and $G$ a finite graph without loops or multiple edges, we consider the graph configuration space $M^G$ and a Bendersky-Gitler type spectral sequence converging to the…

The distance of an operation from being associative can be "measured" by its associative spectrum, an appropriate sequence of positive integers. Associative spectra were introduced in a publication by B. Cs\'ak\'any and T. Waldhauser in…

We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real…

The procedure of the holonomy-flux algebra construction along a piecewise linear path, which consists of a countably infinite number of pieces, is described in this article. The related construction approximates the continuous distribution…

We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative…