Related papers: Oriented singquandles and related algebraic struct…
We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the…
In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between…
Although algebraic structures are frequently analyzed using unary and binary operations, they can also be effectively defined and unified through ternary operations. In this context, we introduce structures that contain two constants and a…
In this article, we describe how coalgebraic structures on operads induce algebraic structures on their categories of algebras and coalgebras.
We prove that on 2-connected closed oriented manifolds, the analytic and algebraic constructions of an IBL$_\infty$ structure associated to a closed oriented manifold coincide. The corresponding structure is invariant under orientation…
In generalization of knot quandles we introduce similar algebraic structures associated with arbitrary pairs consisting of a path-connected topological space and its path-connected subspace.
In this short survey we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent…
The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular…
It is shown how to endow a hierarchy of sets of binary patterns with the structure of an abstract,normed C*-algebra. In the course we also recover an intermediate connection with the words of a Dyck language and Tempereley-Lieb algebras for…
We study different algebraic structures associated to an operad and their relations: to any operad $\mathbf{P}$ is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra;…
We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from…
We study operator algebras associated to integral domains. In particular, with respect to a set of natural identities we look at the possible nonselfadjoint operator algebras which encode the ring structure of an integral domain. We show…
This paper is devoted to qualgebras and squandles, which are quandles enriched with a compatible binary/unary operation. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. Topologically,…
This paper is a survey of our recent work on operator algebras associated to dynamical systems that lead to classification results for the systems in terms of algebraic invariants of the operator algebras.
We present a new approach for constructing covariant symplectic structures for geometrical theories, based on the concept of adjoint operators. Such geometric structures emerge by direct exterior derivation of underlying symplectic…
We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make systematic use of covariance algebras…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
We review "quantum" invariants of closed oriented 3-dimensional manifolds arising from operator algebras.
Language theory, symbolic dynamics, modelisation of viral insertion into the genetic code of a host cell motivate the introduction of new types of bialgebras whose coalgebra parts are not necessarily coassociative. One of the aim of this…
We define 2-calibrated structures, which are analogs of symplectic structures in odd dimensions. We show the existence of differential topological constructions compatible with the structure.