Related papers: Zonotopal algebra
The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras.…
We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras respectively. We then relate a construction of Efimov in the context of…
Many combinatorial optimisation problems hide algebraic structures that, once exposed, shrink the search space and improve the chance of finding the global optimal solution. We present a general framework that (i) identifies algebraic…
Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental…
We provide an internal characterization of those finite algebras (i.e., algebraic structures) $\mathbf A$ such that the number of homomorphisms from any finite algebra $\mathbf X$ to $\mathbf A$ is bounded from above by a polynomial in the…
We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is…
We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…
To any graph and smooth algebraic curve $C$ one may associate a "hypercurve" arrangement and one can study the rational homotopy theory of the complement $X$. In the rational case ($C=\mathbb{C}$), there is considerable literature on the…
This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on…
We initiate the systematic study of endomorphism algebras of permutation modules and show they are obtainable by a descent from a certain "generic" Hecke algebra, infinite-dimensional in general, coming from the universal enveloping algebra…
This article presents a new set representation named the hybrid zonotope that is equivalent to the union of $2^N$ constrained zonotopes -- convex polytopes -- through the addition of $N$ binary zonotope factors. The major contribution of…
The machinery of zonotopal algebra is linked with two particular polytopes: the Stanley-Pitman polytope and the regular simplex $\mathfrak{Sim}_n(t_1,...,t_n)$ with parameters $t_1,...,t_n\in \mathbb{R}_+^n$, defined by the inequalities…
We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is…
This paper provides an extensive study of the homotopy theory of types of algebras with units, like unital associative algebras or unital commutative algebras for instance. To this purpose, we endow the Koszul dual category of curved…
We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer. This result raises…
The Orlik-Solomon algebra ${\cal A}(G)$ of a matroid $G$ is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
Polytopes from subgraph statistics are important in applications and conjectures and theorems in extremal graph theory can be stated as properties of them. We have studied them with a view towards applications by inscribing large explicit…
We present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies, and the higher operations on the Yoneda algebra. We…
We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first…