# Classification of two-term tilting complexes over Brauer graph algebras

@article{Adachi2015ClassificationOT,
title={Classification of two-term tilting complexes over Brauer graph algebras},
author={Takahide Adachi and Takuma Aihara and Aaron Chan},
journal={Mathematische Zeitschrift},
year={2015},
volume={290},
pages={1-36}
}
• Published 19 April 2015
• Mathematics
• Mathematische Zeitschrift
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.

### On tilting complexes over blocks covering cyclic blocks

. Let p be a prime number, k an algebraically closed ﬁeld of characteristic p , ˜ G a ﬁnite group, and G a normal subgroup of ˜ G having a p -power index in ˜ G . Moreover let B be a block of kG with

### Brauer tree algebras have $\binom{2n}{n}$ $2$-tilting complexes

We show that any Brauer tree algebra has precisely (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

### On $\tau$-tilting finiteness of block algebras of direct products of finite groups

. We discuss ﬁniteness/inﬁniteness of τ -tilting modules over tensor products of two symmetric algebras. As an application, we discuss that over block algebras of direct products of ﬁnite groups.

### On induced modules of inertial-invariant support $\tau$-tilting modules over blocks of finite groups

In this article, we prove that induced modules of support τ -tilting modules over blocks of ﬁnite groups satisfying inertial invariant condition are also support τ -tilting modules.

### On $\tau$-tilting finiteness of symmetric algebras of polynomial growth, $0$-Hecke and $0$-Schur algebras

• Mathematics
• 2022
. In this paper, we report on the τ -tilting ﬁniteness of some classes of ﬁnite-dimensional algebras over an algebraically closed ﬁeld, including symmetric algebras of polynomial growth, 0-Hecke

### R T ] 1 3 A pr 2 02 2 ON τ-TILTING MODULES OVER TRIVIAL EXTENSIONS OF GENTLE TREE ALGEBRAS

• Mathematics
• 2022
. We show that trivial extensions of gentle tree algebras are exactly Brauer tree algebras without exceptional vertex. As a consequence, the number of support τ tilting modules over the trivial

### On $\tau$-tilting modules over trivial extensions of gentle tree algebras

• Mathematics
• 2022
. We show that trivial extensions of gentle tree algebras are exactly Brauer tree algebras without exceptional vertex. As a consequence, the number of support τ tilting modules over the trivial

### Fans and polytopes in tilting theory I: Foundations

• Mathematics
• 2022
For a finite dimensional algebra A over a field k, the 2-term silting complexes of A gives a simplicial complex ∆(A) called the g-simplicial complex. We give tilting theoretic interpretations of the

### Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

In the representation theory of finite-dimensional algebras A over a field, the classification of 2-term (pre)silting complexes is an important problem. One of the useful tool is the g-vector cones

## References

SHOWING 1-10 OF 31 REFERENCES

### The Grothendieck group of the stable category of symmetric special biserial algebras

Cartan matrices of symmetric special biserial algebras are described, and the order of the Grothendieck group for the stable category of these algebras is computed. In particular, a criterion for the

### Walking around the Brauer tree

LetG be a finite group, and k a field of finite characteristic p, such that the polynomial x¦G¦ –1 splits completely in k[x]. Let Β be a kG-block which has defect group D which is cylclic of order pd

### ON QUASI-BASS ORDERS

• Mathematics
• 1972
We study quasi-Bass orders over complete local Dedekind rings, i.e. orders of which every indecomposable representation module is a direct summand of an over-ring. We give a method allowing us to

### Auslander-reiten sequences with few middle terms and applications to string algebrass

• Mathematics
• 1987
(1987). Auslander-reiten sequences with few middle terms and applications to string algebrass. Communications in Algebra: Vol. 15, No. 1-2, pp. 145-179.

### Pointed Brauer Trees

• Mathematics
• 2001
Abstract In this paper we establish a one-to-one correspondence between Brauer trees with an additional structure that we call “pointing” and equivalence classes of tilting complexes of the