Koszul Algebras Research Articles

Drinfeld twists of Koszul algebras

Given a Hopf algebra H and a counital 2-cocycle μ on H, Drinfeld introduced a notion of twist which deforms an H-module algebra A into a new algebra A μ . We show that when A is a quadratic algebra, and H acts on A by degree-preserving endomorphisms, then the twist A μ is also quadratic. Furthermore, if A is a Koszul algebra, then A μ is a Koszul algebra. As an application, we prove that the twist of the q-quantum plane by the quasitriangular structure of the quantum enveloping algebra U q ( s l 2 ) is a quadratic algebra equal to the q − 1 -quantum plane.

Graded trace generating functions of Aut(ΓKn) acting on A(ΓKn)

In this paper, we describe the algebras [Formula: see text] associated with the Hasse graph of the poset of the faces of the complete graph at n vertices [Formula: see text]. Present the relations that define this algebra, calculate the Hilbert series of quadratic grade algebra [Formula: see text] and the graded trace generating functions of [Formula: see text] acting on [Formula: see text] and show that [Formula: see text] is a Koszul algebra. The methodology adopted in this paper consists of building the algebra [Formula: see text] using [I. Gelfand, V. Retakh, S. Serconek and R. L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. 11(2) (2005) 281], giving a presentation based on what was established by [V. Retakh, S. Serconek and R. L. Wilson, On a class of koszul algebras associated to directed graphs, J. Algebra 304(2) (2006) 1114–1129] in terms of the vertices of the graph [Formula: see text]. After that, we use [V. Retakh, S. Serconek and R. L. Wilson, Hilbert series of algebras associated to directed graphs, J. Algebra 312(1) (2007) 142–151] to determine the Hilbert series of [Formula: see text] and use [9] again to show that this algebra is a Koszul algebra. We use [J. Caldeira, A. De Souza Lima and J. Eder Salvador De Vasconcelos, Representations of automorphism groups of algebras associated to star polygons, J. Algebra Appl. 18(10) (2019) 1950197; C. Duffy, Representations of [Formula: see text] acting on homogeneous components of [Formula: see text] and [Formula: see text], Adv. Appl. Math. 42(1) (2009) 94–122] to determine the generating functions of the graduated trace of [Formula: see text] acting on [Formula: see text].

The Ext-algebra and associated monomial algebra of a finite dimensional K-algebra

It is well known that if the ext-algebra E(A) of an algebra A is generated in degrees 0 and 1, A must be a Koszul algebra. We seek to generalize this notion. To do so, we set A=KQ/I where Q is a finite quiver and I is an admissible ideal. We use a construction from [13], [1], and [11] to form a family of elements in KQ which yields a projective resolution of A¯, called the AGS resolution. We denote this family {fij} and consider the case where the AGS resolution is minimal. Then we consider the associated monomial algebra of A, found in [8] and [9], which we denote AMON. We prove that if the AGS resolution is minimal and E(AMON) is finitely generated, then E(A) is finitely generated. Next we look at 2-d-determined algebras, as defined in [11]. If A is a 2-d-determined algebra, we prove that if the AGS resolution is minimal, then E(A) is generated in degrees 0,1, and 2. However, if the AGS resolution is not minimal, we prove that E(A) need not be K2.

Preprojective algebras of d-representation finite species with relations

In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We generalize this result to the species case.We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by the mapping cone of a chain map, which is homogeneous of degree 1 with respect to a certain grading. We also study a higher dimensional analogue of representation finite hereditary algebras called d-representation finite algebras. One source of d-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.

Elliptic R-matrices and Feigin and Odesskii’s elliptic algebras

The algebras $$Q_{n,k}(E,\tau )$$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $$n>k\ge 1$$ , a complex elliptic curve E, and a point $$\tau \in E$$ . The main result in this paper is that $$Q_{n,k}(E,\tau )$$ has the same Hilbert series as the polynomial ring on n variables when $$\tau $$ is not a torsion point. We also show that $$Q_{n,k}(E,\tau )$$ is a Koszul algebra, hence of global dimension n when $$\tau $$ is not a torsion point, and, for all but countably many $$\tau $$ , $$Q_{n,k}(E,\tau )$$ is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining $$Q_{n,k}(E,\tau )$$ is the image of an operator $$R_{\tau }(\tau )$$ that belongs to a family of operators $$R_{\tau }(z):{\mathbb {C}}^n\otimes {\mathbb {C}}^n\rightarrow {\mathbb {C}}^n\otimes {\mathbb {C}}^n$$ , $$z\in {\mathbb {C}}$$ , that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.

Koszul homology of quotients by edge ideals

We show that the Koszul homology algebra of a quotient of a polynomial ring by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients by edge ideals of trees.

Yoneda A∞-algebras and lattices of vector spaces

Generalizing some clever computations for monomial algebras by J. Chuang and A. King in an unpublished article, we provide a sufficient criterion that allows in some cases to explicitly compute the A∞-algebra structure of the Yoneda algebra ExtA•(K,K) of a nonnegatively graded connected algebra A=TV/I in the case where the A∞-algebra structure of the Yoneda algebra is given as the dual of (signed) inclusions appearing in the lattice of submodules of TV representing the Tor groups TorpA(K,K) for p∈N. This includes the case of monomial algebras A considered by Chuang and King, but also generalized Koszul algebras, and other nice algebras. The previous nice description of the A∞-(co)algebra structure is however far from universal. Indeed, we also give an example of a nonnegatively graded connected algebra A where the A∞-algebra structure of the Yoneda algebra can never be obtained by dualizing the (signed) inclusions of the Tor groups.

The structure of Koszul algebras defined by four quadrics

Avramov, Conca, and Iyengar ask whether βiS(R)≤(gi) for all i when R=S/I is a Koszul algebra minimally defined by g quadrics. In recent work, we give an affirmative answer to this question when g≤4 by completely classifying the possible Betti tables of Koszul algebras defined by height-two ideals of four quadrics. Continuing this work, the current paper proves a structure theorem for Koszul algebras defined by four quadrics. We show that all these Koszul algebras are LG-quadratic, proving that an example of Conca of a Koszul algebra that is not LG-quadratic is minimal in terms of number of defining equations. We then characterize precisely when these rings are absolutely Koszul, and establish the equivalence of the absolutely Koszul and Backelin–Roos properties up to field extensions for such rings (in characteristic zero). The combination of the above paper with the current one provides a fairly complete picture of all Koszul algebras defined by g≤4 quadrics.

G-quadratic, LG-quadratic, and Koszul quotients of exterior algebras

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.

A Class of Koszul Algebra and Some Homological Invariants through Circulant Matrices and Cycles

Recent advances in graph theory, linear algebra, and commutative algebra render us to tackle problems in one bough of mathematics with assistance and guidance from others. We will elaborate foremost and conceptually fathomless homological invariants inextricably linked with circulant matrices and cycles through various path lengths in this article, as well as a class of Koszul algebra, which portrays combinatorial correlation, in the end.

Koszul algebras and flow lattices

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph Γ with a spanning tree T, we associate a finite dimensional Koszul algebra AΓ,T. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated AΓ,T modules is isomorphic to the Euclidean lattice ZE(Γ), and we describe the sublattices of integer cuts and integer flows on Γ in terms of the representation theory of AΓ,T. The grading on AΓ,T gives rise to q-analogs of the lattices of integer cuts and flows; these q-lattices depend non-trivially on the choice of spanning tree. We give a q-analog of the matrix-tree theorem, and prove that the q-flow lattice of (Γ1,T1) is isomorphic to the q-flow lattice of (Γ2,T2) if and only if there is a cycle preserving bijection from the edges of Γ1 to the edges of Γ2 taking the spanning tree T1 to the spanning tree T2. This gives a q-analog of a classical theorem of Caporaso-Viviani and Su-Wagner.

Hochschild cohomology, finiteness conditions and a generalization of d-Koszul algebras

Given a finite-dimensional algebra [Formula: see text] and [Formula: see text], we construct a new algebra [Formula: see text], called the stretched algebra, and relate the homological properties of [Formula: see text] and [Formula: see text]. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that [Formula: see text] has (Fg) if and only if [Formula: see text] has (Fg). We also consider projective resolutions and apply our results in the case where [Formula: see text] is a [Formula: see text]-Koszul algebra for some [Formula: see text].

Koszul algebras and quadratic duals in Galois cohomology

We investigate the Galois cohomology of finitely generated maximal pro-p quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with the p-Zassenhaus filtration on the group. This paper also contains a survey of Koszul property in Galois cohomology and its relation with absolute Galois groups.

Quantizations [formula omitted] and [formula omitted] as quadratic-linear algebras

We prove that in q-Weyl generators the multi-parameter quantizations of type An+ and Bn+ are quadratic-linear Koszul algebras.

Right-angled Artin groups and enhanced Koszul properties

Abstract Let 𝔽 {\mathbb{F}} be a finite field. We prove that the cohomology algebra H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} with coefficients in 𝔽 {\mathbb{F}} of a right-angled Artin group G Γ {G_{\Gamma}} is a strongly Koszul algebra for every finite graph Γ. Moreover, H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} is a universally Koszul algebra if, and only if, the graph Γ associated to the group G Γ {G_{\Gamma}} has the diagonal property. From this, we obtain several new examples of pro-p groups, for a prime number p, whose continuous cochain cohomology algebra with coefficients in the field of p elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro-p Galois groups of fields formulated by J. Mináč et al.

Betti numbers of Koszul algebras defined by four quadrics

Let I be an ideal generated by quadrics in a standard graded polynomial ring S over a field. A question of Avramov, Conca, and Iyengar asks whether the Betti numbers of R=S/I over S can be bounded above by binomial coefficients on the minimal number of generators of I if R is Koszul. This question has been answered affirmatively for Koszul algebras defined by three quadrics and Koszul almost complete intersections with any number of generators. We give a strong affirmative answer to the above question in the case of four quadrics by completely determining the Betti tables of height two ideals of four quadrics defining Koszul algebras.

Quadratic algebras arising from Hopf operads generated by a single element

The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of $n$-ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.

Morita theory and singularity categories

We propose an analogue of the bounded derived category for an augmented ring spectrum, defined in terms of a notion of Noether normalization. In many cases we show this category is independent of the chosen normalization. Based on this, we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are Koszul dual in the style of the BGG correspondence. Examples of interest include Koszul algebras and Ginzburg DG-algebras, C⁎(BG) for finite groups (or for compact Lie groups with orientable adjoint representation), cochains in rational homotopy theory and various examples from chromatic homotopy theory.

Preprojective algebra structure on skew-group algebras

We give a class of finite subgroups G<SL(n,k) for which the skew-group algebra k[x1,…,xn]#G does not admit the grading structure of a higher preprojective algebra. Namely, we prove that if a finite group G<SL(n,k) is conjugate to a subgroup of SL(n1,k)×SL(n2,k), for some n1,n2≥1, then the skew-group algebra k[x1,…,xn]#G is not Morita equivalent to a higher preprojective algebra. This is related to the preprojective algebra structure on the tensor product of two Koszul bimodule Calabi–Yau algebras. We prove that such an algebra cannot be endowed with a grading structure as required for a higher preprojective algebra.Moreover, we construct explicitly the bound quiver of the higher preprojective algebra over a finite-dimensional Koszul algebra of finite global dimension. We show in addition that preprojective algebras over higher representation-infinite Koszul algebras are derivation-quotient algebras whose relations are given by a superpotential.

Universally Koszul and initially Koszul properties of Orlik–Solomon algebras

Let [Formula: see text] be a field with [Formula: see text] and [Formula: see text] an exterior algebra over [Formula: see text] with a standard grading [Formula: see text]. Let [Formula: see text] be a graded algebra, where [Formula: see text] is a graded ideal in [Formula: see text]. In this paper, we study universally Koszul and initially Koszul properties of [Formula: see text] and find classes of ideals [Formula: see text] which characterize such properties of [Formula: see text]. As applications, we classify arrangements whose Orlik–Solomon algebras are universally Koszul or initially Koszul. These results are related to a long-standing question of Shelton–Yuzvinsky [B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. 56 (1997) 477–490].