Koszul Algebras Research Articles

The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦2 algebra would be another 𝒦2 algebra. We show that this is not necessarily the case by constructing a monomial 𝒦2 algebra for which the corresponding Yoneda algebra is not 𝒦2.

Koszul differential graded algebras and BGG correspondence II

The concept of Koszul differential graded (DG for short) algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module A k is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] (see also Proposition 2.2), A k is not compact if H(A) is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.

Selfinjective Koszul algebras of finite complexity

In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Open image in new window of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ l ≤ m there exist a family of modules of complexity l parameterized by G(l,m), the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL(2, C) over the exterior algebra of a 2-dimensional vector space.

Koszul duality for monoids and the operad of enriched rooted trees

We introduce here the notion of Koszul duality for monoids in the monoidal category of species with respect to the ordinary product. To each Koszul monoid we associate a class of Koszul algebras in the sense of Priddy, by taking the corresponding analytic functor. The operad A M of rooted trees enriched with a monoid M was introduced by the author. One special case of that is the operad of ordinary rooted trees, called in the recent literature the permutative non-associative operad. We prove here that A M is Koszul if and only if the corresponding monoid M is Koszul. In this way we obtain a wide family of Koszul operads, extending a recent result of Chapoton and Livernet, and providing an interesting link between Koszul duality for associative algebras and Koszul duality for operads.

Quivers with relations arising from Koszul algebras of [formula omitted]-invariants

Let g be a complex simple Lie algebra and let Ψ be an extremal set of positive roots. After Chari and Greenstein (2009) [9], one associates with Ψ an infinite dimensional Koszul algebra S Ψ g which is a graded subalgebra of the locally finite part of ( ( End V ) op ⊗ S ( g ) ) g , where V is the direct sum of all simple finite dimensional g -modules. We describe the structure of the algebra S Ψ g explicitly in terms of an infinite quiver with relations for g of types A and C. We also describe several infinite families of quivers and finite dimensional associative algebras arising from this construction.

Quadratic algebras with Ext algebras generated in two degrees

Green and Marcos (2005) [2] call a graded k -algebra δ -Koszul if the corresponding Yoneda algebra Ext ( k , k ) is finitely generated and there exists a function δ : N → N such that Ext i , j ( k , k ) is zero if j ≠ δ ( i ) . For any integer m ≥ 3 we exhibit a noncommutative quadratic δ -Koszul algebra for which the Yoneda algebra is generated in degrees ( 1 , 1 ) and ( m , m + 1 ) . These examples answer a question of Green and Marcos. These algebras are not Koszul but m -Koszul (in the sense of Backelin).

The Face Semigroup Algebra of a Hyperplane Arrangement

Abstract.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 the intersection lattice of the hyperplane arrangement. A complete systemof primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposablemodules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

Koszul algebras and finite Galois coverings

The relationships between Koszulity and finite Galois coverings are obtained, which provide a construction of Koszul algebras by finite Galois coverings.

Gerasimov's theorem and N ‐Koszul algebras

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.

Homological Algebra and Divergent Series

We study some features of infinite resolutions of Koszul algebras motivated by the developments in the string theory initiated by Berkovits.

Graded and Koszul Categories

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries.

Growth function for a class of monoids

In this article we study a class of monoids that includes Garside monoids, and give a simple combinatorial proof of a formula for the formal sum of all elements of the monoid. This leads to a formula for the growth function of the monoid in the homogeneous case, and can also be lifted to a resolution of the monoid algebra. These results are then applied to known monoids related to Coxeter systems: we give the growth function of the Artin-Tits monoids, and do the same for the dual braid monoids. In this last case we show that the monoid algebras of the dual braid monoids of type A and B are Koszul algebras. Nous étudions une classe de monoïdes incluant les monoïdes de Garside, et donnons une preuve combinatoire simple d'une formule pour la somme formelle de leurs éléments. Cela mène à une formule pour la fonction de croissance du monoïde dans le cas homogène, et peut être aussi relevé en une résolution de l'algèbre de monoïdes. Ces résultats sont ensuite appliqués aux monoïdes liés aux systèmes de Coxeter : nous donnons la fonction de croissance des monoïdes d'Artin-Tits ainsi que des monoïdes duaux ; pour ces derniers nous montrons que leur algèbre de monoïde en types A et B est une algèbre de Koszul.

A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Let g be a finite-dimensional simple Lie algebra and let S g be the locally finite part of the algebra of invariants ( End C V ⊗ S ( g ) ) g where V is the direct sum of all simple finite-dimensional modules for g and S ( g ) is the symmetric algebra of g . Given an integral weight ξ, let Ψ = Ψ ( ξ ) be the subset of roots which have maximal scalar product with ξ. Given a dominant integral weight λ and ξ such that Ψ is a subset of the positive roots we construct a finite-dimensional subalgebra S Ψ g ( ⩽ Ψ λ ) of S g and prove that the algebra is Koszul of global dimension at most the cardinality of Ψ. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Ψ. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.

Co-Point Modules Over Frobenius Koszul Algebras

Let A be a Frobenius Koszul algebra such that its Koszul dual A ! is a quantum polynomial algebra. Co-point modules over A were defined as dual notion of point modules over A ! with respect to the Koszul duality. In this article, we will see that various important functors between module categories over A used in representation theory of finite dimensional algebras send co-point modules to co-point modules. As a consequence, we will show that if (E, σ) is a geometric pair associated to A !, then the map σ:E → E is an automorphism of the point scheme E of A !, so that there is a bijection between isomorphism classes of left point modules over A ! and those of right point modules over A !.

COMBINATORICS AND N-KOSZUL ALGEBRAS

The numerical Hilbert series combinatorics for quadratic Koszul algebras was extended to N-Koszul algebras by Dubois-Violette and Popov [9]. In this paper, we give a striking application of this extension when the relations of the algebra are all the antisymmetric tensors of degree N over given variables. Furthermore, we present a new type of Hilbert series combinatorics, called comodule Hilbert series combinatorics, and due to Hai, Kriegk and Lorenz [15]. When the relations are all the antisymmetric tensors, a natural generalization of the MacMahon Master Theorem (MMT) is obtained from the comodule level, the original MMT corresponding to N = 2 and to polynomial algebras.

Koszul differential graded algebras and BGG correspondence

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 consisting of all compact DG modules modulo the full triangulated subcategory consisting of all the right DG modules with finite dimensional cohomology. The classical BGG correspondence can be derived from the DG version.

Category 𝒪 for the Virasoro algebra: cohomology and Koszulity

We investigate blocks of the Category O for the Virasoro algebra over C. We demonstrate that the blocks have Kazhdan‐Lusztig theories and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.

Noetherianity and Ext

Let R = R 0 ⊕ R 1 ⊕ R 2 ⊕ ⋯ be a graded algebra over a field K such that R 0 is a finite product of copies of K and each R i is finite dimensional over K . Set J = R 1 ⊕ R 2 ⊕ ⋯ and S = ⊕ n ≥ 0 Ext R n ( R / J , R / J ) . We study the properties of the categories of graded R -modules and S -modules that relate to the noetherianity of R . We pay particular attention to the case when R is a Koszul algebra and S is the Koszul dual to R .

MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Let $\Lambda=kQ/I$ 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 $\Lambda$ 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 $\Lambda$ over the enveloping algebra $\Lambda^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $\L$ 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 $\L$ 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 $\Lambda$ is shown to be surjective onto the graded centre of the Koszul dual.

Piecewise-Koszul algebras

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 ∞-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.