Related papers: Note on commutativity in double semigroups and two…

We show that two natural and a priori unrelated structures encapsulate the same data, namely certain commutative and associative product structures and a class of superintegrable Hamiltonian systems. More precisely, consider a Euclidean…

It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the…

We introduce a notion of ``$n$-dual'' to a simplicial vector space for $n\ge 0$. Coming with it, there is a canonical pairing, which we show to be non-degenerate up to homotopy for homotopy $n$-types. As a result this notion of duality is…

Solitonic objects play a central role in gauge and string theory (as, e.g., monopoles, black holes, D-branes, etc.). Certain string backgrounds produce a noncommutative deformation of the low-energy effective field theory, which allows for…

In the article we study the simple unital communitative three-dimensional algebras over an algebraically closed field of characteristic not equal to 2. It is proved that every simple unital communitative three-dimensional algebra of…

We construct, in locally compact, second countable, amenable groups, sets with large density that fail to have certain combinatorial properties. For the property of being a shift of a set of measurable recurrence we show that this is…

A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither…

Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value…

We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to…

In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have…

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C $ independent of $B$ via the Hodge duality $*$. We show that the doubling of…

The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such…

These lectures deal mainly with solitons in three-dimensional Moyal-deformed sigma models. The topics are: static and moving (multi-)solitons of the (integrable) Ward sigma model, with space-space and time-space noncommutativity, their…

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply…

All of 320 layer groups, distributed into 80 clusters - single/double ordinary/gray groups - are used to complete systematization of linear (in all directions) band crossings and corresponding effective Hamiltonians in high-symmetry…

A simple observation, showing that every groupoid becomes an inverse semigroup after adding one element. In such inverse semigroups all idempotents are mutually orthogonal. This fact implies that every C*-algebra of a discrete groupoid is a…

We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic…

The weak units of strict monoidal 1- and 2-categories are already defined. In this paper, we define them for group-like 1- and 2-stacks. We show that they form a contractible Picard 1- and 2-stack, respectively. We give their cohomological…

We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects…