Let $Λ=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $Λ$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $Λ$ over the enveloping algebra $Λ^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $Ł$ relative to the resolution $\mathbb{P}$ is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of $Ł$ with respect to a canonical basis over $k$ associated to the resolution $\mathbb{F}$. The natural map from the Hochschild cohomology to the Koszul dual of $Λ$ is shown to be surjective onto the graded centre of the Koszul dual.
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 differentials, providing associative products on (co)homology classes. The N-Koszul calculus is calculated for the truncated polynomial algebras.
It is shown that, the quasi-Koszulities of algebras and modules are Morita invariance. A finite-dimensional $K$-algebra $A$ with an action of $G$ is quasi-Koszul if and only if so is the skew group algebra $A \ast G$, where $G$ is a finite group satisfying $\char K \nmid |G|$. A finite-dimensional $G$-graded $K$-algebra $A$ is quasi-Koszul if and only if so is the smash product $A # G^*$, where $G$ is a finite group satisfying $\char K \nmid |G|$. These results are applied to prove that, if a finite-dimensional connected quiver algebra is Koszul then so are its Galois covering algebras with fi…
We give Braverman-Gaitsgory style conditions for general PBW deformations of skew group algebras formed from finite groups acting on Koszul algebras. When the characteristic divides the order of the group, this includes deformations of the group action as well as of the Koszul relations.
The main purpose of this paper is to study a concrete example of $δ$-Koszul algebras, which is related to three questions raised by Green and Marcos in [3].
The paper is devoted to graded algebras having a single homogeneous relation. Using Gerasimov's theorem, a criterion to be N-Koszul is given, providing new examples. An alternative proof of Gerasimov's theorem for N=2 is given. Some related results on Calabi-Yau algebras are proved.
The universal enveloping algebra $U(\mathfrak{tr}_n)$ of a Lie algebra associated to the classical Yang-Baxter equation was introduced in [BEER06] where it was shown to be Koszul. This algebra appears as the $A_{n-1}$ case in a general class of braided Hopf algebras in [BB09] for any complex reflection group. In this paper, we show that the algebras corresponding to the series $B_n$ and $D_n$, which are again universal enveloping algebras, are Koszul. We further show how results of [BB09] can be used to produce pairs of adjoint functors between categories of rational Cherednik algebra represen…
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.
In this paper, we establish the Composition-Diamond lemma for right-symmetric algebras. As an application, we give a Gröbner-Shirshov basis for universal enveloping right--symmetric algebra of a Lie algebra.
In this note, we prove the Koszulity of the tensor product algebra defined in the author's previous work for sl(n) and a list of fundamental weights. This is achieved by constructing a graded Morita equivalence between the modules over this algebra and a sum of blocks of category O in type A.
We show that if $Λ$ is a $n$-Koszul algebra and $E=E(Λ)$ is its Yoneda algebra, then there is a full subcategory $\mathcal{L}_E$ of the category $Gr_E$ of graded $E$-modules, which contains all the graded $E$-modules presented in even degrees, that embeds fully faithfully into the category $C(Gr_Λ)$ of cochain complexes of graded $Λ$-modules. That extends the known equivalence, for $Λ$ Koszul (i.e. for $n=2$), between $Gr_E$ and the category of linear complexes of graded $Λ$-modules
The aim of this paper is to define and study some quasi-hereditary covers for higher zigzag algebras. We will show how these algebras satisfy three different Koszul properties: they are Koszul in the classical sense, standard Koszul and Koszul with respect to the standard module $Δ$, according with the definition given in \cite{Madsen1}. This last property gives rise to a well defined duality and the $Δ$-Koszul dual will be computed as the path algebra of a quiver with relations.
There are many structures (algebras, categories, etc) with natural gradings such that the degree 0 components are not semisimple. Particular examples include tensor algebras with non-semisimple degree 0 parts, extension algebras of standard modules of standardly stratified algebras. In this thesis we develop a generalized Koszul theory for graded algebras (categories) whose degree 0 parts may be non-semisimple. Under some extra assumption, we show that this generalized Koszul theory preserves many classical results such as the Koszul duality. Moreover, it has some close relation to the classic…
We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category $\mathcal{O}$, and some quasi-hereditary algebras with Cartan decomposition in the sense of K{ö}nig.
In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$-hereditary all the r…
Generalized \dK modules are introduced to solve an open problem: the odd Ext-module $\Eod(M)$ of a \dK module $M$ over a $d$-Koszul algebra $\m$ is a \K module over the even Yoneda algebra $\Eev(\m)$.
The $N$-Koszul algebras are $N$-homogeneous algebras which satisfy an homological property. These algebras are characterised by their Koszul complex: an $N$-homogeneous algebra is $N$-Koszul if and only if its Koszul complex is acyclic. Methods based on computational approaches were used to prove $N$-Koszulness: an algebra admitting a side-confluent presentation is $N$-Koszul if and only if the extra-condition holds. However, in general, these methods do not provide an explicit contracting homotopy for the Koszul complex. In this article we present a way to construct such a contracting homotop…
We show that the Brauer algebra over the complex numbers for an integral parameter delta can be equipped with a grading, in the case of delta being non-zero turning it into a graded quasi-hereditary algebra. In which case it is Morita equivalent to a Koszul algebra. This is done by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra for some large positive integral parameter N. The parameter delta appears then in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in pa…
It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are, in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra $A$ to be piecewise-Koszul in terms of its Yoneda-Ext algebra $E(A)$, and show an $A_{\infty}$-structure on $E(A)$. Relations between Koszul algebras and piecewise-Koszul algebras are discussed. In particular, our results are related to the third question of Green-Marcos' [6].
Let $\g$ be a reductive Lie algebra and $V$ a $\g$-semisimple module. In this article, we study the category $\G$ of graded finite-dimensional representations of $\g \ltimes V$. We produce a large class of truncated subcategories, which are directed and highest weight. Suppose $V$ is finite-dimensional with weights $\wt(V)$. Let $Ψ\subset \wt(V)$ be the set of weights contained in a face $\F$ of the polytope that is the convex hull of $\wt(V)$. For each such $Ψ$, we produce quasi-hereditary Koszul algebras. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra $\s…
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 connection between the varieties and the algebras.
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 …
Let $B$ be a generalized Koszul algebra over a finite dimensional algebra $S$. We construct a bimodule Koszul resolution of $B$ when the projective dimension of $S_B$ equals 2. Using this we prove a Poincaré-Birkhoff-Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of $S_B$ is greater than 2, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Kosz…
We motivate and study the reduced Koszul map, relating the invariant bilinear maps on a Lie algebra and the third homology. We show that it is concentrated in degree 0 for any grading in a torsion-free abelian group, and in particular it vanishes whenever the Lie algebra admits a positive grading. We also provide an example of a 12-dimensional nilpotent Lie algebra whose reduced Koszul map does not vanish. In an appendix, we reinterpret the results of Neeb and Wagemann about the second homology of current Lie algebras, which are closely related to the reduced Koszul map.
Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincare-Birkhoff-Witt (PBW) deformation of the smash product algebra B#H. Many examples of these deformations are given.
Pre-Koszul and Koszul algebras were defined by Priddy. There exist some relations between these algebras and the skew PBW extensions defined. We have established conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul K-algebra then every graded skew PBW extension of R is Koszul.
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.
Vatne and Green & Marcos have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green & Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.
Let $A$ be a bi-Koszul algebra, we describe all possible $A_\infty$-algebra structures on the Ext-algebra $E(A)$, and prove that $E(A)$ must be $[m_2, m_3]$-finitely generated. An equivalent description for a connected graded algebra to be a bi-Koszul algebra is given in terms of $A_\infty$-language. The case that $E(A)$ is endowed with minimal number of multiplications is discussed for decomposition.
We prove a version of Koszul duality and the induced derived equivalence for Adams connected $A_\infty$-algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bernšte{\uı}n-Gel'fand-Gel'fand correspondence for Adams connected $A_\infty$-algebras. We give various applications. For example, a connected graded algebra $A$ is Artin-Schelter regular if and only if its Ext-algebra $\Ext^\ast_A(k,k)$ is Frobenius. This generalizes a result of Smith in the Koszul case. If $A$ is Koszul and if both $A$ and its Koszul dual…
We study a finite dimensional quadratic graded algebra R defined from a finite ranked poset. This algebra has been central to the study of the splitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconek and Wilson . The algebra A is known to be quadratic when the poset satisfies a combinatorial condition known as uniform, and R is the quadratic dual of an associated graded algebra of A. We prove that R is Koszul and the poset is uniform if and only if the poset is Cohen-Macaulay. Koszulity of R implies Koszulity of A. We also show that when R is Koszul, the cohomology of the …
From symplectic reflection algebras, some algebras are naturally introduced. We show that these algebras are non-homogeneous N-Koszul algebras, through a PBW theorem.
For any Koszul Artin-Schelter regular algebra A, we consider a version of the universal Hopf algebra aut(A) coacting on A, introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category U, equipped with a functor M to finite dimensional vector spaces such that aut(A)= coend_U(M). Using this pair (U,M) we show that aut(A) is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of U.
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.
We identify two Batalin-Vilkovisky algebra structures, one obtained by Kowalzig and Krahmer on the Hochschild cohomology of an Artin-Schelter regular algebra with semisimple Nakayama automorphism and the other obtained by Lambre, Zhou and Zimmermann on the Hochschild cohomology of a Frobenius algebra also with semisimple Nakayama automorphism, provided that these two algebras are Koszul dual to each other.
In this paper, we show that the derived Picard group of a homologically smooth Koszul connected cochain DG algebra is isomorphic to the opposite group of the derived Picard group of its finite dimensional local Ext-algebra. As applications, we compute the derived Picard groups of some important DG algebras such as trivial DG polynomial algebras, trivial DG free algebras and several non-trivial DG down-up algebras and DG free algebras.
This paper presents an infinite family of Koszul self-injective algebras whose Hochschild cohomology ring is finite-dimensional. Moreover, for each $N \geq 5$ we give an example where the Hochschild cohomology ring has dimension $N$. This family of algebras includes and generalizes the 4-dimensional Koszul self-injective local algebras of Buchweitz, Green, Madsen and Solberg, which were used to give a negative answer to Happel's question, in that they have infinite global dimension but finite-dimensional Hochschild cohomology.
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.
The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra $A$ is AS-regular, the Ext-algebra $E$ of $A$ is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence between the stable category of finitely generated left $E$-modules, and the quotient triangulated category of the full triangulated subcategory of the derived category of right DG $A$-modules cons…
This paper introduces the study of LG-quadratic quotients of exterior algebras, showing that they are Koszul, as in the commutative case. We construct an example of an LG-quadratic algebra that is not G-quadratic and another example that is Koszul but not LG-quadratic. This is only the second known Koszul algebra that is not LG-quadratic and the first that is noncommutative.
We compute the Hilbert series, and the graded vector space structure, of Ext-algebras of quotients of Koszul algebras with almost linear resolution. The example of the generic determinantal varieties is treated in detail.
We introduce the Hochschild extensions of dg algebras, which are $A_\infty$-algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric $A_\infty$-algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau comple…
Motivated by the theory of quasi-determinants, we study non-commutative algebras of quasi-Plücker coordinates. We prove that these algebras provide new examples of non-homogeneous quadratic Koszul algebras by showing that their quadratic duals have quadratic Gröbner bases.
Let $A$ be a Koszul Artin-Schelter regular algebra with Nakayama automorphism $ξ$. We show that the Yoneda Ext-algebra of the skew polynomial algebra $A[z;ξ]$ is a trivial extension of a Frobenius algebra. Then we prove that $A[z;ξ]$ is Calabi-Yau; and hence each Koszul Artin Schelter regular algebra is a subalgebra of a Koszul Calabi-Yau algebra. A superpotential $\hat{w}$ is also constructed so that the Calabi-Yau algebra $A[z;ξ]$ is isomorphic to the derivation quotient of $\hat{w}$. The Calabi-Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Arti…
We show how to study a certain associative algebra recently discovered by Iyudu and Shkarin using the Anick resolution. This algebra is a counterexample to the conjecture of Positselski on Koszul algebras of finite global dimension.
We present a new calculus which is well-adapted to quadratic algebras. This calculus consists in Koszul (co)homology, together with Koszul cup and cap products. Some applications are given. Koszul duality for Koszul (co)homology is proved for any quadratic algebra.
We formulate the (co)bar construction theory of dg $K$-(co)rings and the calculus theory of the Hochschild homology and cohomology of dg $K$-rings. As applications, we compare the Hochschild (co)homologies of a complete typical dg $K$-ring and its Koszul dual. Moreover, we show that the Koszul dual of a finite dimensional complete typical $d$-symmetric dg $K$-ring is a $d$-Calabi-Yau dg algebra whose Hochschild cohomology is a Batalin-Vilkovisky algebra. Furthermore, we prove that the Hochschild cohomologies of a finite dimensional complete typical $d$-symmetric dg $K$-ring and its Koszul dual…
In this paper, we define $A_{\infty}$-Koszul duals for directed $A_{\infty}$-categories in terms of twists in their $A_{\infty}$-derived categories. Then, we compute a concrete formula of $A_{\infty}$-Koszul duals for path algebras with directed $A_n$-type Gabriel quivers. To compute an $A_\infty$-Koszul dual of such an algebra $A$, we construct a directed subcategory of a Fukaya category which are $A_\infty$-derived equivalent to the category of $A$-modules and compute Dehn twists as twists. The formula unveils all the ext groups of simple modules of the parh algebras and their higher composi…
Let $H$ be a twisted Calabi-Yau (CY) algebra and $σ$ a 2-cocycle on $H$. Let $A$ be an $N$-Koszul twisted CY algebra such that $A$ is a graded $H^σ$-module algebra. We show that the cleft extension $A#_σH$ is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an $N$-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of $U(\mathcal{D},λ)$, a class of pointed Hopf alg…
For a quotient algebra $U$ of the tensor algebra we give explicit conditions on its relations for $U$ being a PBW-deformation of an $N$-Koszul algebra $A$. We show there is a one-one correspondence between such deformations and a class of $A_\infty$-structures on the Yoneda algebra $Ext_A^*(k,k)$ of $A$. We compute the PBW-deformations of the algebra whose relations are the anti-symmetrizers of degree $N$ and also of cubic Artin-Schelter algebras.
