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 tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.
Generalizing the notion of a Koszul algebra, a graded k-algebra A is K2 if its Yoneda algebra is generated as an algebra in cohomology degrees 1 and 2. We prove a strong theorem about K2 factor algebras of Koszul algebras and use that theorem to show the Stanley-Reisner face ring of a simplicial complex is K2 whenever the Alexander dual simplicial complex is (sequentially) Cohen-Macaulay.
Let $A = \bigoplus_{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra such that $A_0$ is an arbitrary finite-dimensional algebra satisfying some splitting condition. In this paper we develop a generalized Koszul theory generalizing many classical results, and use it to classify algebras stratified for all preorders.
Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomo…
In this paper, we introduce the notion of Koszul-Vinberg-Nijenhuis structures on a left-symmetric algebroid as analogues of Poisson-Nijenhuis structures on a Lie algebroid, and show that a Koszul-Vinberg-Nijenhuis structure gives rise to a hierarchy of Koszul-Vinberg structures. We introduce the notions of ${\rm KVΩ}$-structures, pseudo-Hessian-Nijenhuis structures and complementary symmetric $2$-tensors for Koszul-Vinberg structures on left-symmetric algebroids, which are analogues of ${\rm PΩ}$-structures, symplectic-Nijenhuis structures and complementary $2$-forms for Poisson structures. We…
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 give a universal description of the Koszul dual algebra under a new algebraic structure. For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.
In this paper, we introduce the notion of bigraft algebra, generalizing the notions of left and right graft algebras. We give a combinatorial description of the free bigraft algebra generated by one generator and we endow this algebra with a Hopf algebra structure, and a pairing. Next, we study the Koszul dual of the bigraft operad and we give a combinatorial description of the free dual bigraft algebra generated by one generator. With the help of a rewriting method, we prove that the bigraft operad is Koszul. Finally, we define the notion of infinitesimal bigraft bialgebra and we prove a rigi…
We discuss the notion of Poincaré duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincaré duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.
WWe describe the Koszul dual of the operad Quad of quadri-algebras, show the koszularity of Quad and give the formal series of Quad and its dual, which proves a conjecture due to Aguiar and Loday. A notion of quadri-bialgebra is also introduced, with applications to the Hopf algebras FQSym and WQSym.
In this article we introduce the notion of multi-Koszul algebra for the case of a locally finite dimensional nonnegatively graded connected algebra, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. It also extends and generalizes the definition recently introduced by the author and A. Rey. In order to simplify the exposition we consider the minimal graded projective resolution of the algebra A as a bimodule, which may be used to compute the corresponding …
In 2010, Cassidy and Vancliff extended the notion of a quadratic form on n generators to the noncommutative setting. In this article, we suggest a notion of rank for such noncommutative quadratic forms, where n = 2 or 3. Since writing an arbitrary quadratic form as a sum of squares fails in this context, our methods entail rewriting an arbitrary quadratic form as a sum of products. In so doing, we find analogs for 2 x 2 minors and determinant of a 3 x 3 matrix in this noncommutative setting.
Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $ϕ:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^ϕ\!M$ be the abelian group $M$ with the $R$-module structure induced by the Frobenius endomorphism. The $R$-module ${}^ϕ\!M$ has a natural grading given by $\text{deg} x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this paper, we prove that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\text{reg}_R\,{}^ϕ\!M <\infty$. This result supplies …
We discuss the consequences of the Poincaré duality, versus AS- Gorenstein property, for Koszul algebras (homogeneous and non homogeneous). For homogeneous Koszul algebras, the Poincaré duality property implies the existence of twisted potentials which characterize the corresponding algebras while in the case of quadratic linear Koszul algebras, the Poincaré duality is needed to get a good generalization of universal enveloping algebras of Lie algebras. In the latter case we describe and discuss the corresponding generalization of Lie algebras. We also give a short review of the notion of Kosz…
A deformation $U$, of a graded $K$-algebra $A$ is said to be of PBW type if $grU$ is $A$. It has been shown for Koszul and $N$-Koszul algebras that the deformation is PBW if and only if the relations of $U$ satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a {\it finite} and explicitly determined linear algebra problem. We extend these results to an arbitrary graded $K$-algebra, using the notion of central extensions of algebras and a homological constant attached to $A$ which we call the {\it complexity} of $A$.
The notion of a $\mathcal{K}_2$-algebra was recently introduced by Cassidy and Shelton as a generalization of the notion of a Koszul algebra. The Yoneda algebra of any connected graded algebra admits a canonical $A_{\infty}$-algebra structure. This structure is trivial if the algebra is Koszul. We study the $A_{\infty}$-structure on the Yoneda algebra of a $\mathcal{K}_2$-algebra. For each non-negative integer $n$ we prove the existence of a $\mathcal{K}_2$-algebra $B$ and a canonical $A_{\infty}$-algebra structure on the Yoneda algebra of $B$ such that the higher multiplications $m_i$ are non…
Analogous to a recent result of N. Kowalzig and U. Krähmer for twisted Calabi-Yau algebras, we show that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, thus generalizing a result of T.Tradler for finite dimensional symmetric algebras. We give a criterion to determine when a Frobenius algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf a…
In our preceding paper we have introduced the notion of an $s$-homogeneous triple. In this paper we use this technique to study connected $s$-homogeneous algebras with two relations. For such algebras, we describe all possible pairs $(A,M)$, where $A$ is the $s$-Veronese ring and $M$ is the $(s,1)$-Veronese bimodule of the $s$-homogeneous dual algebra. For each such a pair we give an intrinsic characterization of algebras corresponding to it. Due to results of our previous work many pairs determine the algebra uniquely up to isomorphism. Using our partial classification, we show that, to check…
We introduce the notion of bi-monoid in general monoidal category generalizing by this the notion of bialgebra. In the case of bimodules over a noncommutative algebra, we obtain a compatibility condition between ring and coring whenever both structures admit the same underlying bimodule.
In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let $\mathcal{A}$ be a $3$-dimensional DG Sklyanin algebra with $\mathcal{A}^{\#}=S_{a,b,c}$, where $(a,b,c)\in \Bbb{P}_k^2-\mathfrak{D}$ and $$\mathfrak{D}=\{(1,0,0), (0,1,0),(0,0,1)\}\sqcup\{(a,b,c)|a^3=b^3=c^3\}.$$ We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for $\mathcal{A}$ to be Calabi-Yau, Koszul, Gorenstein and homologically smooth, respectively…
In this paper we introduce the notions of partial bimodule algebra and partial bico- module algebra. We also deal with the existence of globalizations for these structures, generalizing related results appeared in [2, 4]. As an application we construct the partial (L;R)-smash product, extending the corresponding global notion appeared in [14] to the context of partial Hopf actions.
Let $k$ be a field and $X$ be a set of $n$ elements. We introduce and study a class of quadratic $k$-algebras called \emph{quantum binomial algebras}. Our main result shows that such an algebra $A$ defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual $A^{!}$ is Frobenius of dimension $n,$ with a \emph{regular socle} and for each $x,y \in X $ an equality of the type $xyy=αzzt,$ where $α\in k \setminus\{0\},$ and $z,t \in X$ is satisfied in $A$. We prove the equivalence of the notions \emph{a binomial skew polynomial ring} and \emph{a binomial solution o…
We introduce partial (co)actions of a Hopf algebra $H$ on an algebra. To this end, we introduce first the notion of lax coring, generalizing Wisbauer's notion of weak coring. We also have the dual notion of lax ring. Several duality results are given, and we develop Galois theory for partial $H$-comodule algebras.
In 1702.04949 noncommutative frames were introduced, generalizing the usual notion of frames of open sets of a topological space. In this paper we extend this notion to noncommutative Grothendieck topologies and their associated noncommutative toposes of sheaves of sets.
