Related papers: Projection algebras and free projection- and idemp…
The category of all idempotent generated semigroups with a prescribed structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has an initial object called the free idempotent generated semigroup over $\mathcal{E}$,…
Let $G$ be a connected semisimple algebraic group with Lie algebra $g$ and $P$ a parabolic subgroup of $G$ with $Lie(P)=p$. The parabolic contraction of $g$ is the semi-direct product of $p$ and a $p$-module $g/p$ regarded as an abelian…
We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by…
We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham--Houghton graphs. We show that each proper ideal of…
In this paper we consider simplicial families, that is, simplicial objects indexed by a simplicial set. We develop a method to construct family hypercover refinements of a cover family based on the notion of \emph{n-spans} that we introduce…
This paper exhibits fundamental structure underlying Lie algebra homology with coefficients in tensor products of the adjoint representation, mostly focusing upon the case of free Lie algebras. The main result yields a DG category that is…
We compute the two-cocycles (or multipliers) of the free nilpotent groups of class $2$ and rank $n$ and give conditions for simplicity of the corresponding twisted group $C^*$-algebras. These groups are representation groups for…
We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.
This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_{\alpha}\}_{\alpha\in\Lambda}$, with the left regular free semigroup algebra $\mathfrak{L}_{\Lambda}$ and the…
We introduce a generalization of parametrized Rota-Baxter algebras, which includes family and matching Rota-Baxter algebras. We study the structure needed on the set $\Omega$ of parameters in order to obtain that free Rota-Baxter algebras…
Every semigroup containing an ideal subgroup is called a homogroup, and it is a grouplike if and only if it has only one central idempotent. On the other hand, a class of algebraic structures covering group-$e$-semigroups…
We study the algebraic structure of the Poisson algebra P(O) of polynomials on a coadjoint orbit O of a semisimple Lie algebra. We prove that P(O) splits into a direct sum of its center and its derived ideal. We also show that P(O) is…
We give two new constructions of the harmonic algebra of a lattice polytope $P$, a bigraded algebra whose character is the $q$-Ehrhart series of $P$ defined by Reiner and Rhoades. First, we show that the harmonic algebra is the associated…
The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the…
Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra A if it is isomorphic to a twisted group algebra k^\alpha G for some class \alpha in H^2(G,k^\times), where the action of G on k^\times is…
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of…
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
In this paper we provide a theory of chain projection ordered categories and generalize that of chain projection ordered groupoids developed by East and Azeef Muhammed recently. By using chain projection ordered categories, we obtain a…
To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of…