We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by different authors using geometric methods. We also obtain several new results about the parabolic category O.
In this paper all two-term tilting complexes over a Brauer tree algebra with multiplicity one are described using a classification of indecomposable two-term partial tilting complexes obtained earlier in a joint paper with M. Antipov. The endomorphism rings of such complexes are computed.
In this paper we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings.
We give a proof, based on the rigidity of tilting complexes, that the class of self-injective finite-dimensional algebras over an algebraically closed field is closed under derived equivalence.
In this paper, we compute the number of two-term tilting complexes for an arbitrary symmetric algebra with radical cube zero over an algebraically closed field. Firstly, we give a complete list of symmetric algebras with radical cube zero having only finitely many isomorphism classes of two-term tilting complexes in terms of their associated graphs. Secondly, we enumerate the number of two-term tilting complexes for each case in the list.
Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.
We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras.
We show that any Brauer tree algebra has precisely $\binom{2n}{n}$ $2$-tilting complexes, where $n$ is the number of edges of the associated Brauer tree. More explicitly, for an external edge $e$ and an integer $j\neq0$, we show that the number of $2$-tilting complexes $T$ with $g_e(T)=j$ is $\binom{2n-|j|-1}{n-1}$, where $g_e(T)$ denotes the $e$-th of the $g$-vector of $T$. To prove this, we use a geometric model of Brauer graph algebras on the closed oriented marked surfaces and a classification of $2$-tilting complexes due to Adachi-Aihara-Chan.
We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.
We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection.
We construct derived equivalences between group graded self-injective algebras, starting from equivalences between their 1-components, obtained via a construction of J. Rickard and S. Al-Nofayee.
In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding Coxeter group. Moreover, we provide the complete classification in the case of affine type by showing that any two-term silting complex belongs one of them. For this purpose, we also discuss the Krull-Schmidt property for the homotopy category of finitely generated projective modules over a complete ring.
Let $Λ$ be an Artin algebra and $K^b(proj(Λ))$ be the triangulated category of bounded co-chain complexes in $proj(Λ).$ It is well known that two-terms silting complexes in $K^b(proj(Λ))$ are described by the $τ$-tilting theory. The aim of this paper is to give a characterization of certain $n$-term silting complexes in $K^b(proj(Λ))$ which are induced by $Λ$-modules. In order to do that, we introduce the notions of $τ_n$-rigid, $τ_n$-tilting and $τ_{n,m}$-tilting $Λ$-modules. The latter is both a generalization of $τ$-tilting and tilting in $mod(Λ).$ It is also stated and proved some variant,…
We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular…
If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.
Given a set of 'simple-minded' objects in a derived category, Rickard constructed a complex, which over a symmetric algebra provides a derived equivalence sending the 'simple-minded' objects to simple ones. We characterise in terms of t-structures, when this complex is a tilting complex, show that there is an associated natural $t$-structure and we provide an alternative construction of this complex in terms of A-infinity-structures. Our approach is similar to that of Keller--Nicolás.
In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we …
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $τ^2 T\cong T$, where $τ$ is the Auslander-Reiten translation in $D^b(\operatorname{coh}\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we…
The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.
In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $Mod(C)$, from a skeletally small preadditive category $C$ to the category of abelian groups. We introduced the notion of a a generalized tilting category $T$, and extended Happel's theorem to $Mod(C)$. We proved that there is an equivalence of triangulated categories $D^b (Mod(C))\cong Db (Mod(T))$. In the case of dualizing varieties, we proved a version of Happel's theorem for the categories of finitely presented functors. We also proved in this paper, that there exists a relation b…
In this note we give an overview of rigidity properties and Loewy lengths of tilting modules in the BGG category O associated to a reductive Lie algebra. These results are well-known by several specialists, but seem difficult to find in the existing literature.
We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support $τ$-tilting modules, is a derived invariant. Moreover, we apply our result to the …
We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GL_n in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.
We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$ of this category associated with each reduced expression $\bf{w}$ of $w$ and give a sufficient condition on $\bf{w}$ such that $M(\bf{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\bf{w})$. Moreover, we compare it with a triangle equivalence given by Amiot-Reiten-Todorov for a cluster category.
Let $p$ be a prime number, $k$ an algebraically closed field of characteristic $p$, $\tilde{G}$ a finite group, and $G$ a normal subgroup of $\tilde{G}$ having a $p$-power index in $\tilde{G}$. Moreover let $B$ be a block of $kG$ with a cyclic defect group and $\tilde{B}$ be the unique block of $k\tilde{G}$ covering $B$. We study tilting complexes over the block $\tilde{B}$ and show that the block $\tilde{B}$ is a tilting-discrete algebra. Moreover we show that the set of all tilting complexes over $\tilde{B}$ is isomorphic to that over $B$ as partially ordered sets.
In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in \cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(n\geq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.
Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.
We construct a tilting object for the stable category of vector bundles on a weighted projective line X of type (2,2,2,2;λ), consisting of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical algebra associated with X of type (2,2,2,2).
We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.
$n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher dimensional Auslander-Reiten theory. We give a complete classification of $n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama algebras. In particular, we show that there are three kinds of Nakayama algebras that admit $n\mathbb{Z}$-cluster tilting subcategories: finite global dimension, selfinjective and non-Iwanaga-Gorenstein. Only the selfinjective ones can admit more than one $n\mathbb{Z}$-cluster tilting subcategory. It has been shown by the second author, that each such $n\mathbb{Z}$-clust…
This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 <= \ell <= d-1, we show a weakly d-cluster tilting subcategory T_{\ell} which has an indecomposable object with precisely \ell…
This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type $(2, 2, n)$, and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.
In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the $τ$-tilting theo…
In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.
For any good tilting module $T$ over a ring $A$, there exists an $n$-symmetric subcategory $\mathscr{E}$ of a module category such that the derived category of the endomorphism ring of $T$ is a recollement of the derived categories of $\mathscr{E}$ and $A$ in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of $\mathscr{E}$.
We study the t-structure induced by an n-tilting module T in the derived category D(R) of a ring R. Our main objective is to determine when the heart of the t-structure is a Grothendieck category. We obtain characterizations in terms of properties of the module category over the endomorphism ring of T and as a main result we prove that the heart is a Grothendieck category if and only if T is a pure projective $R$-module.
Motivated by $τ$-tilting theory developed by Adachi, Iyama and Reiten, for a finite-dimensional algebra $Λ$ with action by a finite group $G$, we introduce the notion of $G$-stable support $τ$-tilting modules. Then we establish bijections among $G$-stable support $τ$-tilting modules over $Λ$, $G$-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective $Λ$-modules, and $G$-stable functorially finite torsion classes in the category of finitely generated left $Λ$-modules. In the case when $Λ$ is the endomorphism of a $G$-stable cluster-tilt…
We introduce Wakamatsu-silting complexes (resp., Wakamatsu-tilting complexes) as a common generalization of both silting complexes (resp., tilting complexes) and Wakamatsu-tilting modules. Characterizations of Wakamatsu-silting complexes are given. In particular, we show that a complex $T$ is Wakamatsu-silting if and only if its dual $DT$ is Wakamatsu-silting. It is conjectured that all compact Wakamatsu-silting complexes are just silting complexes. We prove that the conjecture lies under the finitistic dimension conjecture.
We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category. We also obtain a classification of projective-injective modules in the full BGG category $\mathcal O$ for all simple classical Lie superalgebras. We then classify and give an explicit combinatorial description of parabolic subalgebras of the periplectic Lie superalgebras and …
Let $\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\rm mod}\mbox{-}Λ$, where $Λ$ is an artin algebra. Let $\mathcal{S}(\mathcal{M})$ denotes the full subcategory of $\mathcal{S}(Λ)$, the submodule category of $Λ$, consisting of all monomorphisms in $\mathcal{M}$. We construct two functors from $\mathcal{S}(\mathcal{M})$ to ${\rm mod}\mbox{-}\underline{\mathcal{M}}$, the category of finitely presented (coherent) additive contravariant functors on the stable category of $\mathcal{M}$. We show that these functors are full, dense and objective. So they induce equivalences from the quotie…
Let T be a locally finite triangulated category with an autoequivalence F such that the orbit category T/F is triangulated. We show that if X is an m-cluster tilting subcategory, then the image of X in T/F is an m-cluster tilting subcategory if and only if X is F-perodic. We show that for path-algebras of Dynking quivers δone may study the periodic properties of n-cluster tilting objects in the n-cluster category Cn(kδ) to obtain information on periodicity of the preimage as n-cluster tilting subcategories of Db(kδ). Finally we classify the periodic properties of all 2-cluster tilting objects …
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and each indecomposable tilting vector bundle $T$ on $C$ a certain complex $S(C,T)$ of $G \times \mathbb{G}_m$-equivariant coherent sheaves on $\mathcal{N}$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-$t$-structure. We then show that if $p$ is larger than the Coxeter number, then the hypercoho…
Support $τ$-tilting modules correspond to some classes of categorical objects bijectively, such as two-term tilting complexes for any finite dimensional symmetric algebra. This fact motivates us to classify support $τ$-tilting modules over blocks of finite groups. Therefore we classify support $τ$-tilting modules over particular blocks of finite groups by using the modular representation theoretical approaches including Clifford theory, Green's indecomposability theorem and so on.
Given a tilting object of the derived category of an abelian category of finite global dimension, we give (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring.
We present a new way to construct $n$-cluster tilting subcategories of abelian categories. Our method takes as input a direct system of abelian categories $\mathcal{A}_i$ with certain subcategories and, under reasonable conditions, outputs an $n$-cluster tilting subcategory of an admissible target $\mathcal{A}$ of the direct system. We apply this general method to a direct system of module categories $\text{mod}Λ_i$ of representation-directed algebras $Λ_i$ and obtain an $n$-cluster tilting subcategory $\mathcal{M}$ of a module category $\text{mod}\mathcal{C}$ of a locally bounded Krull-Schmid…
In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and …
In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke categor…
Let $k$ be a field. In this short note we give an example of a $2$-abelian $k$-category, realized as a $2$-cluster-tilting subcategory of the category $\operatorname{mod}\,A$ of finite dimensional (right) $A$-modules over a finite dimensional $k$-algebra $A$, for which the naive idea for constructing its "bounded derived category" as $2$-cluster-tilting subcategory of the bounded derived category of $\operatorname{mod}\,A$ cannot work.
For a triangulated category T, if C is a cluster-tilting subcategory of T, then the quotient category T\C is an abelian category. Under certain conditions, the converse also holds. This is an very important result of cluster-tilting theory, due to Koenig-Zhu and Beligiannis. Now let B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory C of B, which is a generalization of cluster tilting subcategory. We show that C is cluster tilting if and only if B/C is abelia…
Let B be an extriangulated category with enough projectives and enough injectives. Let C be a fully rigid subcategory of B which admits a twin cotorsion pair ((C,K),(K,D)). The quotient category B/K is abelian, we assume that it is hereditary and has finite length. In this article, we study the relation between support tilting subcategories of B/K and maximal relative rigid subcategories of B. More precisely, we show that the image of any cluster tilting subcategory of B is support tilting in B/K and any support tilting subcategory in B/K can be lifted to a unique relative maximal rigid subcat…
