We give a complete derived equivalence classification of all symmetric algebras of domestic representation type over an algebraically closed field. This completes previous work by R. Bocian and the authors, where in this paper we solve the crucial problem of distinguishing standard and nonstandard algebras up to derived equivalence. Our main tool are generalized Reynolds ideals, introduced by B. K"ulshammer for symmetric algebras in positive characteristic, and recently shown by A. Zimmermann (math.RA/0509580) to be invariants under derived equivalences.
We give a complete derived equivalence classification of all nonstandard representation-infinite domestic selfinjective algebras over an algebraically closed field. As a consequence, also a complete stable equivalence classification of these algebras is obtained.
This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between weakly symmetric algebras o…
Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are integers at least two, satisfying $1/a+1/b+1/c>1$. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type $(a,b,c)$. Equivalently, in a representation-theoretic context, we can work in the mesh category of $\mathbb{Z}\tildeΔ$ over $k$, wher…
We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.
We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is ju…
We give a characterization of the sets of objects of the derived category of a block of a finite group algebra (or other symmetric algebra) that occur as the set of images of simple modules under an equivalence of derived categories. We give some applications to proving that certain blocks have equivalent derived categories.
We construct nontrivial auto-equivalences of stable module categories for elementary, local symmetric algebras over a field k. These auto-equivalences are modeled after the spherical twists of Seidel and Thomas and the $\mathbb{P}^n$-twists of Huybrechts and Thomas, which yield auto-equivalences of the derived category of coherent sheaves on a variety. For group algebras of p-groups in characteristic p we recover many of the auto-equivalences corresponding to endo-trivial modules. We also obtain analogous auto-equivalences for local algebras of dihedral and semi-dihedral type, which are not gr…
We use the dimension and the Lie algebra structure of the first Hochschild cohomology group to distinguish some algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type. In particular, we complete the classification of algebras of dihedral type that was mostly determined by Zhou and Zimmermann.
We define and study sl\_2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Broué's abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also con…
We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to be invariant under derived equivalence. We study transformations of Brauer complexes which preserve class of derived equivalence. Additionally, we establish a new invariant of derived equivalence of symmetric SB-algebras. As a consequence, symmetric SB-algebras with Brauer complex of genus 0 are classified. Keywords: Brauer tree algebras, special biserial al…
A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of Gorenstein properties of endomorphism rings.
We consider tilting mutations of a weakly symmetric algebra at a subset of simple modules, as recently introduced by T. Aihara. These mutations are defined as the endomorphism rings of certain tilting complexes of length 1. Starting from a weakly symmetric algebra A, presented by a quiver with relations, we give a detailed description of the quiver and relations of the algebra obtained by mutating at a single loopless vertex of the quiver of A. In this form the mutation procedure appears similar to, although significantly more complicated than, the mutation procedure of Derksen, Weyman and Zel…
We describe the structure and properties of the finite-dimensional symmetric algebras over an algebraically closed field $K$ which are socle equivalent to the general weighted surface algebras of triangulated surfaces, investigated in Erdmann and Skowroński (J. Algebra 544:170-227, 2020). In particular, we prove that all these algebras are tame periodic algebras of period 4. The main results of this paper form an essential step towards a classification of all symmetric tame periodic algebras of period 4.
We provide a combinatorial algorithm for constructing the stable Auslander-Reiten component containing a given indecomposable module of a symmetric special biserial algebra using only information from its underlying Brauer graph. We also show that the structure of the Auslander-Reiten quiver is closely related to the distinct Green walks around the Brauer graph and detail the relationship between the precise shape of the stable Auslander-Reiten components for domestic Brauer graph algebras and their underlying graph. Furthermore, we show that the specific component containing a given simple or…
In this paper the class of Brauer graph algebras is proved to be closed under derived equivalence. For that we use the rank of the maximal torus of the identity component $Out^0(A)$ of the group of outer automorphisms of a symmetric stably biserial algebra $A$.
We prove the existence of Morita equivalences between the spin blocks at the extremal points of strings in the block-reduced crystal graph. When the parities of the core partitions are not preserved, these equivalences require crossovers, with a block of the symmtric group Morita equivalent to a block of the alternating group and vice versa. The result permits in improvement of the bound for Donovan's Conjecture given by Kessar.
Let $A$ be a symmetric $k$-algebra over a perfect field $k$. Külshammer defined for any integer $n$ a mapping $ζ\_n$ on the degree 0 Hochschild cohomology and a mapping $κ\_n$ on the degree 0 Hochschild homology of $A$ as adjoint mappings of the respective $p$-power mappings with respect to the symmetrizing bilinear form. In an earlier paper it is shown that $ζ\_n$ is invariant under derived equivalences. In the present paper we generalize the definition of $κ\_n$ to higher Hochschild homology and show the invariance of $κ$ and its generalization under derived equivalences. This provides fine …
The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.
Generalized Reynolds ideals are ideals of the center of a symmetric algebra over a field of positive characteristic. They have been shown by the second author to be invariant under derived equivalences. In this paper we determine the generalized Reynolds ideals of algebras of dihedral and semidihedral type (as defined by Erdmann), in characteristic 2. In this way we solve some open problems about scalars occurring in the derived equivalence classification of these algebras.
We show that the $p$-power maps in the first Hochschild cohomology space of finite-dimensional selfinjective algebras over a field of prime characteristic $p$ commute with stable equivalences of Morita type on the subgroup of classes represented by integrable derivations. We show, by giving an example, that the $p$-power maps do not necessarily commute with arbitrary transfer maps in the Hochschild cohomology of symmetric algebras.
We generalise the notion of separable equivalence, originally presented by Linckelmann (2011), to an equivalence relation on additive categories. We use this generalisation to show that from an initial equivalence between two algebras we may build equivalences between many related categories. We also show that separable equivalence preserves the representation type of an algebra. This generalises Linckelmann's result, where he showed this in the case of symmetric algebras. We use these theorems to show that the group algebras of several small cyclic groups cannot be separably equivalent. This …
Using that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.
A well-known result of Scopes states that there are only finitely many Morita equivalence classes of $p$-blocks of symmetric groups with a given weight (or defect). In this note we investigate a lower bound on the number of those Morita equivalence classes.
Gendo-symmetric algebras were introduced by Fang and Koenig as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost $ν$-stable derived equivalences, introduced by Hu and Xi, between representation-finite gendo-sy…
On a derived equivalence classification of weakly symmetric algebras of Euclidean type
Shizuoka University（静冈大学）；Kyoto University（京都大学）；Osaka Central Advanced Mathematical Institute（大阪中央高等数学研究所）；Tokyo University of Science（东京理科大学）；Universidad de Murcia（穆尔西亚大学）
Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $τ$ we compare the notions of $τ$-representation equivalence with $τ$-isospectrality. We exhibit infinitely many $K$-types $τ$ so that, for arbitrary discrete subgroups $Γ$ and $Γ'$ of $G$, if the multiplicities of $λ$ in the spectra of the Laplace operators acting on sections of the induced $τ$-vector bundles over $Γ\backslash G/K$ and $Γ'\backslash G/K$ agree for all but finitely many $λ$, then $Γ$ and $Γ'$ are $τ$-representation equivalent in $G$ (i.e.\ $\dim \operatorname{Hom}_G(V_π, L^2…
We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety GL_2n/Sp_2n. We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of GL_2n/Sp_2n. We s…
We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the algebras of [Bocian-Holm-Skowroński, J. Pure Appl. Algebra 2004], where they study the weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type.
We classify the rational Cherednik algebras H_c(W) (and their spherical subalgebras) up to isomorphism and Morita equivalence in case when W is the symmetric group and `c' is a generic parameter value.
Approximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly $n$-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry,and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals…
We show that the bigroupoid of separable symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of finitely semi-simple Calabi-Yau categories are equivalent. To this end, we construct a trace on the category of finitely-generated representations of a symmetric, separable Frobenius algebra, given by the composite of the Frobenius form with the Hattori-Stallings trace. In order to put this result into perspective, we introduce the concept of the equivariantization of a 2-functor between bicategories endowed with a G-action. We prove that the equivalence between Froben…
We show that finitely generated cohomology is invariant under separable equivalences for all algebras. As a result, we obtain a proof of the finite generation of cohomology for finite symmetric tensor categories in characteristic zero, as conjectured by Etingof and Ostrik. Moreover, for such categories we also determine the representation dimension and the Rouquier dimension of the stable category. Finally, we recover a number of results on the cohomology of stably equivalent and singularly equivalent algebras.
