Brauer Tree Algebras Research Articles

Broué's abelian defect group conjecture for blocks with cyclic hyperfocal subgroups

AbstractIn this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups.

The Ext-algebra of the Brauer tree algebra associated to a line

The Ext-algebra of the Brauer tree algebra associated to a line

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

The Singularity and Cosingularity Categories of C∗BG for Groups with Cyclic Sylow p-subgroups

We construct a differential graded algebra (DGA) modelling certain A_{infty } algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and H_{*}{Omega } BG{^{^{wedge }}_p}. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander–Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.

Blocks of profinite groups with cyclic defect group

We demonstrate that the blocks of a profinite group whose defect groups are cyclic have a Brauer tree algebra structure analogous to the case of finite groups. We show further that the Brauer tree of a block with defect group Z p $\mathbb {Z}_{p}$ is of star type.

The differential graded stable category of a self-injective algebra

Let A be a finite-dimensional, self-injective algebra, graded in non-positive degree. We define A -dgstab, the differential graded stable category of A , to be the Verdier quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We express A -dgstab as the triangulated hull of the orbit category A -grstab / Ω ( 1 ) , reducing computations in the dg-stable category to those in the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A -dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. We also provide a detailed description of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges.

Mutations and Pointing for Brauer Tree Algebras

Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between an ordinary Brauer tree algebra and the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [13], which is obtained from a sequence of mutations. In [1], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara's complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives.

Simplicial complexes and tilting theory for Brauer tree algebras

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the lattice polytope corresponding to the simplicial complex of 2-term tilting complexes. Via a geometric interpretation of derived equivalences, we show that the f-vector of the simplicial complexes of Brauer tree algebras only depends on 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 enumeration problem of Coxeter-biCatalan combinatorics.

Spherical objects of the multiplicity free Brauer tree algebra with two edges

We explicitly describe all spherical objects, as well as tilting objects, of the multiplicity free Brauer tree algebra with two edges.

FOLDINGS AND TWO-SIDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS

In this note, for a Brauer tree algebra $A$ and a star-shaped Brauer tree algebra $B$ which is derived equivalent to $A$, we give operations on the two-sided tilting complex $D_T$ of $A\otimes B^{op}$-modules constructed in [3] which is isomorphic to the Rickard tree-to-star complex $T$ constructed in [5] in $D^b(A)$, and we show that the operations on $D_T$ correspond to operations called $foldings$ on the Rickard tree-to-star complex $T$ given in [7].

Two-sided tilting complexes for Brauer tree algebras

In this note, we construct two-sided tilting complexes corresponding to one-sided tilting complexes for Brauer tree algebras.

Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring R(Λ,V) and we describe R(Λ,V) explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.

On Representation-Finite Gendo-Symmetric Biserial Algebras

Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) 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 in Hu and Xi (I. Nagoya Math. J., 200:107–152, 2010), between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.

Representation type for block algebras of Hecke algebras of classical type

We find representation type of the cyclotomic quiver Hecke algebras RΛ(β) of level two in affine type A. In particular, we have determined representation type for all the block algebras of Hecke algebras of classical type (except for characteristic two in type D), which has not been known for a long time. As an application of this result, we prove that block algebras of finite representation type are Brauer tree algebras whose Brauer trees are straight lines without exceptional vertex if the Hecke algebras are of classical type in good characteristic. We conjecture that this statement should hold for Hecke algebras of exceptional type.

Derived Picard groups of selfinjective Nakayama algebras

In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

Two-Term Tilting Complexes over Brauer Tree Algebras

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.

Two-Term Partial Tilting Complexes Over Brauer Tree Algebras

In this paper, all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 are described, using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application, all two-term tilting complexes over a Brauer star algebra are described and their endomorphism rings are computed.

Cuspidal [formula omitted]-modules and deformations of certain Brauer tree algebras

We show that the algebras describing blocks of the category of cuspidal weight (resp. generalized weight) sl n -modules are one-parameter (resp. multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley–Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.

Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra A Γ corresponding to an arbitrary Brauer tree Γ of type ( m , e ) . We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A S , whose tree is a star with the exceptional vertex in the middle, to A Γ . The grading on A S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras. We also compute Out K ( A Γ ) , the group of outer automorphisms that fix the isomorphism classes of simple A Γ -modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A Γ up to graded Morita equivalence and rescaling.