Related papers: Noncommutative Koszul Algebras from Combinatorial …
Several constructive homological methods based on noncommutative Gr\"obner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the…
The Koszul homology algebra of a commutative local (or graded) ring $R$ tends to reflect important information about the ring $R$ and its properties. In fact, certain classes of rings are characterized by the algebra structure on their…
We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted…
Koszul algebras with quadratic Groebner bases, called strong Koszul algebras, are studied. We introduce affine algebraic varieties whose points are in one-to-one correspondence with certain strong Koszul algebras and we investigate the…
By Theorem~1.12 of the paper "A Class of Representations of Hecke Algebras", if $W$ is a Coxeter group whose proper parabolic subgroups are finite, and if the module of a finite $W$-digraph $\Gamma$ is isomorphic to the module of a…
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 study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over…
We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph $\Gamma$ with a spanning tree $T$, we associate a finite dimensional Koszul algebra…
We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings…
We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between…
This is a further investigation of our approach to group actions in homological algebra in the settings of homology of {\Gamma}-simplicial groups, particularly of {\Gamma}-equivariant homology and cohomology of {\Gamma}-groups. This…
Any finite-dimensional commutative (associative) graded algebra with all nonzero homogeneous subspaces one-dimensional is defined by a symmetric coefficient matrix. This algebraic structure gives a basic kind of $A$-graded algebras…
Let $R=\oplus_{\Gamma\in\Gamma}R_{\gamma}$ be a $\Gamma$-graded $K$-algebra over a field $K$, where $\Gamma$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $\Gamma$-grading filtration $FR$ of $R$ and the…
Standard combinatorial construction, due to Kontsevich, associates to any $\ai$-algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We propose an…
For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…
We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $\Lambda$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian,…
We extend the Koszul calculus defined on quadratic algebras by Berger, Lambre and Solotar, to N-homogeneous algebras. When N>2, the Koszul cup and cap products are defined by specific expressions, and they are compatible with the Koszul…
Let $W$ be a Coxeter group whose proper parabolic subgroups are finite. According to Theorem~1.12 of [1], if the module of a finite $W$-digraph $\Gamma$ is isomorphic to the module of a $W$-graph over $Q$, then $\Gamma$ is acyclic. We…
We develop the theory of N-homogeneous algebras in a super setting, with particular emphasis on the Koszul property. To any Hecke operator on a vector superspace, we associate certain superalgebras and generalizing the ordinary symmetric…
G-algebras, or Groebner bases algebras, were considered by Levandovsky, these algebras include very important families of algebras, like the Weyl algebras and the universal enveloping algebra of a finite dimensional Lie algebra. These…