Related papers: Multiplicative structures on cones and duality
Under certain hypotheses, we prove a loop space decomposition for simply-connected Poincar\'e Duality complexes of dimension $n$ whose $(n-1)$-skeleton is a co-$H$-space. This unifies many known decompositions obtained in different contexts…
We show that the mapping cone of a morphism of differential graded Lie algebras $\chi\colon L\to M$ can be canonically endowed with an $L_\infty$-algebra structure which at the same time lifts the Lie algebra structure on $L$ and the usual…
Quantum LDPC codes have attracted intense interest due to their advantageous properties for realizing efficient fault-tolerant quantum computing. In particular, sheaf codes represent a novel framework that encompasses all well-known good…
The well known Joker $\mathcal{A}(1)$-module of Adams and Priddy is known to be realisable as the cohomology of a $1$-connected space. By attaching an extra cell we obtain an $8$-dimensional Poincar\'e duality space whose mod~$2$ cohomology…
An early result of Noncommutative Geometry was Connes' observation in the 1980's that the Dirac-Dolbeault cycle for the $2$-torus $\mathbb{T}^2$, which induces a Poincar\'e self-duality for $\mathbb{T}^2$, can be 'quantized' to give a…
We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along…
We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincar\'e…
In this paper, we develop the theory of bimodules over von Neumann algebras, with an emphasis on categorical aspects. We clarify the relationship between dualizability and finite index. We also show that, for von Neumann algebras with…
We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with…
This paper explores the properties of multiplicative Lie algebra structures on a nilpotent group of class $2$. We also present a method for determining a multiplicative Lie algebra structure on a group that serves as an extension of one Lie…
This (partially expository) paper discusses Lagrangian Floer cohomology in the context of Lefschetz fibrations, with emphasis on the algebraic structures encountered there. In addition to the well-known directed A_infinity algebras which…
We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the…
The recent investigation of the gauge structure of extended geometry is generalised to situations when ancillary transformations appear in the commutator of two generalised diffeomorphisms. The relevant underlying algebraic structure turns…
We explain a direct topological proof for the multiplicativity of Duflo isomorphism for arbitrary finite dimensional Lie algebras, and derive the explicit formula for the Duflo map. The proof follows a series of implications, starting with…
In this note we introduce and investigate the concepts of dual entwining structures and dual entwined modules. This generalizes the concepts of dual Doi-Koppinen structures and dual Doi-Koppinen modules introduced (in the infinite case over…
In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the…
Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures…
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize…
We give an explicit description of the set of all factorization structures, or twisting maps, existing between the algebras k^2 and k^2, and classify the resulting algebras up to isomorphism. In the process we relate several different…
In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincar\'e algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the…