arXiv research

Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is n…

Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is e…

The paper analyzes orbits of integer tuples using braid diagrams.

Involutive Hopf monoids yield surface invariants.

Paper introduces stratified vector bundles and their properties.

Study automorphism group actions on Jacobi diagrams spaces.

New monoids tied to symmetric group and Jones/Brauer monoids discovered.

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived …

The Morse complex is shown to be an infinite functor.

New mapping class group actions on Hochschild complexes for modular categories.

We study monoids generated by Zariski-van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C^3 listed by Sekiguchi (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaussian and, hence, are neither G…

In this paper we define a monoid of pseudo braids and prove that this monoid is isomorphic to a singular braid monoid. We also prove an analogue of Markov's theorem for pseudo braids.

A Lie 2-group $G$ is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on $G$ gives rise to the Lie 2-algebra $\mathbb{X}(G)$ of multiplicative vector fields, see (Berwick-Evans -- Lerman). The monoidal structure on $G$ gives rise to…

The paper introduces a Deligne complex for Artin monoids and studies its properties.

We study the algebraic structures of the virtual singular braid monoid, $VSB_n$, and the virtual singular pure braid monoid, $VSP_n$. The monoid $VSB_n$ is the splittable extension of $VSP_n$ by the symmetric group $S_n$. We also construct a representation of $VSB_n$.

New manifolds help understand group actions on complex chains.

This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.

The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag m…

The paper discusses strictification and non-strictification of monoidal categories.

The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphi…

The paper constructs semistrict monoidal 2-categories from foam evaluations.

The main new notions are the notions of tangent-like spaces and local monoids. The main result is the pasage from a local monoid to its tangent-like space which is a local Leibniz algebra.

In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…

Extends Jones' construction to map Thompson group to pointed links.

The paper constructs Yang-Baxter solutions using categorical augmented racks.

We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.

We prove that an Artin-Tits group of type $\tilde C$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the "generated group" method. This answers, in this particular case, a general question on Artin-Tits groups, gives …

3D HQFTs constructed using graded monoidal categories.

The paper studies submonoids of singular twisted virtual braids and their properties.

Monoidal categorifies genus zero skein algebra using K-theory.

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.

The paper presents new algebraic structures on the 2-sphere using topological field theories.

Garside's results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.

Functor connects Lie groupoid algebras to bornological structures.

We give presentations, in terms of generators and relations, for the monoids of singular braids on closed surfaces. The proof of the validity of these presentations can also be applied to verify, in a new way, the presentations given by Birman for the monoids of Singular Artin braids.

Study on endomorphism and automorphism groups of specific quandles.

In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.

In this paper we study regular irreducible algebraic monoids over $\fldc$ equipped with the euclidean topology. It is shown that, in such monoids, the Green classes and the spaces of idempotents in the Green classes all have natural manifold structures. The interactions of these manifold structures and the semigroup st…

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class…

Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial $n$ may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial m…

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits monoid of spherical type (hence of each braid monoid) with respect to the standard g…

The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful…

Open 2D TFTs extend to closed theories with circle value as Hochschild homology.

We give a solution to the word problem for the singular braid monoid SB_n. The complexity of the algorithm is quadratic in the product of the word length and the number of the singular generators in the word. Furthermore we algebraically reprove a result of Fenn, Keyman and Rourke that the monoid embeds into a group an…

Aguiar and Ardila defined the Hopf monoid GP of generalized permutahedra and showed that it contains many submonoids that correspond to combinatorial objects. They also give a basic polynomial invariant of generalized permutahedra, which then specializes to the submonoids. We define the Hopf monoid of directed graphs a…