graded blocks of group algebras - university of leicester · 2014-09-12 · any two brauer tree...

Graded Blocks of Group Algebras

Dusko Bogdanic

Mathematical InstituteUniversity of Oxford

University of Leicester, 21 June 2010

Dusko Bogdanic (Oxford) Graded Blocks of Group Algebras Leicester, 21 June 2010 1 / 27

Motivation

Theorem (Rouquier, 2001)

Let Db(A-mod) ∼= Db(B-mod). If A is graded, then B is graded.

Problem

How do we effectively transfer gradings between derived equivalentalgebras A and B?

Motivation

Problem

Motivation

Problem

If the grading on A has some interesting properties, does the resultinggrading on B has the same properties?

Overview

1 Preliminaries

I Brauer tree algebras

I Graded algebras

2 The tilting complex

3 Example

4 The general case

5 Properties of the resulting grading

6 Classification of gradings

Brauer tree algebras

Γ finite connected tree with e edges.

Γ is a Brauer tree of type (m, e) if given:

I Cyclic ordering of the edges adjacent to a given vertex,

I The exceptional vertex v , to whom we assign the multiplicity m.

~~~~~~~

◦ S1 ◦ S2 •S3

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Γ finite connected tree with e edges.

Γ is a Brauer tree of type (m, e) if given:

I Cyclic ordering of the edges adjacent to a given vertex,

I The exceptional vertex v , to whom we assign the multiplicity m.

~~~~~~~

◦ S1 ◦ S2 •S3

@@@@@@@

Brauer tree algebrasA is a Brauer tree algebra associated with Γ if

I the isomorphism classes of simple A-modules are in one-to-onecorrespondence with the edges of Γ,

I Let a part of Γ be

◦ ◦

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U1 @@@@@@@

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The structure of PS is given by

I V V1,V2, . . . ,Vt ,I U U1,U2, . . . ,Ur ; S ,U1,U2, . . . ,Ur ; . . . ; S ,U1,U2, . . . ,Ur .

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U1 @@@@@@@

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U1 @@@@@@@

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I V V1,V2, . . . ,Vt ,

I U U1,U2, . . . ,Ur ; S ,U1,U2, . . . ,Ur ; . . . ; S ,U1,U2, . . . ,Ur .

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Vt��•

Ur��

U1 @@@@@@@

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Γ is of type (2, 4)

~~~~~~~

◦ S1 ◦ S2 •S3

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PS1 =S1

, PS2 =

, PS3 =

Γ is of type (2, 4)

~~~~~~~

◦ S1 ◦ S2 •S3

@@@@@@@

PS1 =S1

, PS2 =

, PS3 =

The Brauer star of type (m, e)

◦ ◦

◦ •

S2~~~~~~~ S1

@@@@@@@

S5~~~~~~~◦

◦ ◦

The projective indecomposable module Pj is uniserial.

The Brauer star of type (m, e)

◦ ◦

◦ •

S2~~~~~~~ S1

@@@@@@@

S5~~~~~~~◦

◦ ◦

The projective indecomposable module Pj is uniserial.

Any two Brauer tree algebras associated with the same Brauer tree Γand defined over the same field k are Morita equivalent.

A basic Brauer tree algebra corresponding to a Brauer tree Γ isisomorphic to the algebra kQ/I .

Q and I are easily constructed from Γ.

AΓ := kQ/I

Any two Brauer tree algebras associated with the same Brauer tree Γand defined over the same field k are Morita equivalent.

A basic Brauer tree algebra corresponding to a Brauer tree Γ isisomorphic to the algebra kQ/I .

Q and I are easily constructed from Γ.

AΓ := kQ/I

Graded algebras

Definition

A is a graded algebra if A is the direct sum of subspaces A =⊕

i∈Z Ai ,such that AiAj ⊂ Ai+j , i , j ∈ Z.

ai ∈ Ai deg(ai ) = i

If Ai = 0 for i < 0, then A is positively graded.

Definition

An A-module M is graded if M =⊕

i∈Z Mi , and AiMj ⊂ Mi+j .

N = M〈n〉 denotes the graded module given by Nj = Mn+j , j ∈ Z.

Definition

An A-module homomorphism f between two graded modules M and N isa homomorphism of graded modules if f (Mi ) ⊆ Ni , for all i ∈ Z.

Graded algebras

Definition

Graded algebras

Definition

Graded algebras

Definition

Graded algebrasX = (X i , d i )i∈Z is a graded complex if X i is a graded module, d i is ahomomorphism of graded A-modules, for all i .

f = {f i}i∈Z ∈ HomA(X ,Y ) is a homomorphism of graded complexesif f i is a homomorphism of graded modules.

X 〈j〉 denotes the complex (X 〈j〉)i := X i 〈j〉 and d iX 〈j〉 := d i .

Theorem

If X and Y are graded A-modules (or complexes), then

HomA(X ,Y ) ∼=⊕i∈Z

HomA−gr (X ,Y 〈i〉).

HomgrA(X ,Y ) :=⊕

i∈Z HomA−gr (X ,Y 〈i〉)The subspace Ht(X ,Y ) of zero homotopic maps is homogeneous.From this we get a grading on HomKb(PA)(X ,Y )

HomgrKb(PA)(X ,Y ) := HomgrA(X ,Y )/Ht(X ,Y )

Graded algebrasX = (X i , d i )i∈Z is a graded complex if X i is a graded module, d i is ahomomorphism of graded A-modules, for all i .f = {f i}i∈Z ∈ HomA(X ,Y ) is a homomorphism of graded complexesif f i is a homomorphism of graded modules.

Theorem

HomgrA(X ,Y ) :=⊕

Theorem

HomgrA(X ,Y ) :=⊕

Theorem

HomgrA(X ,Y ) :=⊕

Theorem

HomgrA(X ,Y ) :=⊕

i∈Z HomA−gr (X ,Y 〈i〉)

The subspace Ht(X ,Y ) of zero homotopic maps is homogeneous.From this we get a grading on HomKb(PA)(X ,Y )

Theorem

HomgrA(X ,Y ) :=⊕

i∈Z HomA−gr (X ,Y 〈i〉)The subspace Ht(X ,Y ) of zero homotopic maps is homogeneous.

From this we get a grading on HomKb(PA)(X ,Y )

Theorem

HomgrA(X ,Y ) :=⊕

The tilting complex

Theorem (Rickard 1989)

Up to derived equivalence, a Brauer tree algebra is determined by thenumber of edges and the multiplicity of the exceptional vertex.

Γ an arbitrary Brauer tree of type (m, e)

S the Brauer star of type (m, e)

Db(AS -mod) ∼= Db(AΓ-mod)

AS is tightly graded, i.e. AS∼=

⊕∞i=0(rad AS)i/(rad AS)i+1.

AΓ is graded.

Question

What is the corresponding grading on AΓ?

The tilting complex

AΓ is graded.

Question

The tilting complex

AΓ is graded.

Question

The tilting complex

AΓ is graded.

Question

The tilting complex

AΓ is graded.

Question

The tilting complex

There exist a tilting complex T such that EndKb(PAS)(T )op ∼= AΓ.

T can be constructed by taking Green’s walk around Γ.

There are many such tilting complexes (studied by Rickard, Schaps &Zakay-Illouz).

T is a direct sum of e indecomposable complexes with at most twonon-zero terms.

The summand Ti corresponds to the edge Si of Γ.

The tilting complex

Calculating EndK b(PAS)(T )op

If T is a graded complex, then EndgrKb(PAS)(T )op is a graded algebra.

When identifying EndKb(PAS)(T )op with AΓ, HomKb(PAS

)(Ti ,Tj)

corresponds to the space spanned by all paths starting at i and endingat j in the quiver of AΓ.

Theorem (B. 2007)

Let AΓ be a basic Brauer tree algebra corresponding to an arbitrary Brauertree Γ. Then there exists a positive grading on AΓ.

)(Ti ,Tj)

Theorem (B. 2007)

)(Ti ,Tj)

Theorem (B. 2007)

Example

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Example

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Example

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Example

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S2~~~~~~~

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Example

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S2~~~~~~~

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Example

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S2~~~~~~~

S5~~~~~~~

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Example

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S2~~~~~~~

S5~~~~~~~

S6~~~~~~~

Example

T is the direct sum of:

T1 : 0 → P1 → 0T2 : 0 → P2 → 0T3 : 0 → P3 → 0T4 : 0 → P3 → P4〈5〉 → 0T5 : 0 → P3 → P5〈4〉 → 0T6 : 0 → P5〈4〉 → P6〈9〉 → 0

The differentials are maps of maximal ranks.

This complex tilts from AS to AΓ: EndKb(PAS)(T ) ∼= Aop

HomgrKb(PAS)(Ti ,Tj)

i• α−→j•

Example

S4•0

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��

AA��S3•

AA��

]];;;;;;;;

The general case

Edges i and j of a Brauer tree Γ are adjacent to the same vertex.

i and j are at the same distance from the exceptional vertex:

Si~~~~~~~

Sj @@@@@@@

• Sl ◦

Si~~~~~~~

Sj @@@@@@@

The general case

Edges i and j of a Brauer tree Γ are adjacent to the same vertex.

i and j are at the same distance from the exceptional vertex:

Si~~~~~~~

Sj @@@@@@@ α��

• Sl ◦

Si~~~~~~~

Sj @@@@@@@ β��

If i > j , then deg(α) = i− j. If i < j , then deg(α) = e− (j− i).

deg(β) = 0.

The general case

The distance of one the edges, say i , is one less than the distance of j :

• Sl ◦ Si ◦

Sj~~~~~~~

• Sl ◦ Si ◦

@@@@@@@

The general case

The distance of one the edges, say i , is one less than the distance of j :

• Sl ◦ Si ◦

Sj~~~~~~~

• Sl ◦ Si ◦

@@@@@@@

deg(α) = me;

deg(β) = 0.

The general case

◦ S11 ◦ S10 ◦ S9 • S1 ◦S2

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S8~~~~~~~ S6 ◦ S7 ◦

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S5~~~~~~~

The general case

S8•0

��S11•

0 ** S10•11

)) S9•11

jj8 )) S1•3

11 )) S7•0

S2•11

��

S3•0

S4•0

The subalgebra A0

S8•0

��S11•

0 ** S10•0

)) S9• S1• S6•

S7•0

S2•0

��

S3•0

S4•0

The subalgebra A0

The quiver of A0 is a union of directed rooted trees.

A0 is a quasi-hereditary algebra.

The global dimension of A0 is bounded by the length (as a rootedtree) of the quiver of A0.

From A0 we can recover AΓ.

The subalgebra A0

The quiver of A0 is a union of directed rooted trees.

A0 is a quasi-hereditary algebra.

The global dimension of A0 is bounded by the length (as a rootedtree) of the quiver of A0.

From A0 we can recover AΓ.

The group OutK (AΓ)

Gradings are controlled by Out(A).

A =⊕

i∈Z Ai ! π : Gm → Out(A).

A ”good” case is when the maximal tori of Out(A) are isomorphic toGm.

In this case there is essentially unique grading on A.

A =⊕

Theorem (B. 2008)

Let Γ be an arbitrary Brauer tree of type (m, e). Then

OutK (AΓ) ∼= Aut(k[x ]/(xm+1)).

Corollary (B. 2008)

There is a unique grading on AΓ up to graded Morita equivalence andrescaling.

Theorem (B. 2008)

Let Γ be an arbitrary Brauer tree of type (m, e). Then

OutK (AΓ) ∼= Aut(k[x ]/(xm+1)).

Corollary (B. 2008)

There is a unique grading on AΓ up to graded Morita equivalence andrescaling.

graded blocks of group algebras - university of leicester · 2014-09-12 · any two brauer tree...

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