The Quiver of a Block of Categorymath.fau.edu/~dpucinskaite/talks/BocaRaton_2013.pdf · Boca Raton 12 April, 2013. Overview. Overview Finite dimensional C-algebras. Overview Finite

The Quiver of a Block of Category O

Daiva Pucinskaite

University of Kiel (Germany)

Algebra Logic Seminar + Mathematical Sciences

Colloquium

Boca Raton

12 April, 2013

Overview

Overview

Finite dimensional C-algebras

Overview

Finite dimensional C-algebras↓

Overview


A

Overview


A

finitely generatedleft A−modules

( category modA)

=

M is a C-spacewith an actionA×M → M

Overview


A


( category modA)

=


Overview


A


( category modA)

=


Overview


Overview


Overview


{associative algebras} {non-associative algebras}∪ ∪

{A ; A is a basic algebras}{

g ;g is a semi-simple

Lie algebra

}

Overview





Lie algebra

}

Overview


bound quiver algebras

A




Lie algebra

}

Overview


bound quiver algebras simple Lie algebras

A g




Lie algebra

}

Overview



A g




Lie algebra

}

Overview



A g




Lie algebra

}[

finite dimensionalA-modules

] [blocks of category

O(g)

]

Overview



A g




Lie algebra

}[



O(g)

]

C-algebra A given by a quiver Q and relations


◮ Q = (Qver,Qarr) is a quiver



Qver ={

1•, . . . ,

n•}

is the set of vertices



Qver ={

1•, . . . ,

n•}


Qarr is the set of arrows



Qver ={

1•, . . . ,

n•}



CQ = spanC{paths}

Example.



Qver ={

1•, . . . ,

n•}



CQ = spanC{paths}

Example. Q =(

1•

2•)

a

b



Qver ={

1•, . . . ,

n•}



CQ = spanC{paths}

Example. Q =Qver =

{1•,

2•}

(1•

2•)

a

b



Qver ={

1•, . . . ,

n•}



CQ = spanC{paths}

Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b



Qver ={

1•, . . . ,

n•}



◮ CQ = spanC {paths}

Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1•



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1,



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a,



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a, ab,



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a, ab, aba,



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2•



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}



◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths

Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

a→

2•)(

2•

b→

1•

a→

2•)

= aba

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

a→

2•)(

2•

b→

1•

a→

2•)

= aba

(2•

b→

1•

a→

2•)(

1•

a→

2•)

= 0

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t

r=1 cr ·pr is a relation, where pr = (i → · · · → j)

Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

a→

2•)(

2•

b→

1•

a→

2•)

= aba

(2•

b→

1•

a→

2•)(

1•

a→

2•)

= 0

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉

Example. Q =Qver =

{1•,

2•}

Qarr = {a, b}

(1•

a→

2•)(

2•

b→

1•

a→

2•)

= aba

(2•

b→

1•

a→

2•)(

1•

a→

2•)

= 0

(1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉

Example.(

1•

2•)

a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉

Example. The C-algebra [A = C(

1•

2•)

/ 〈ab〉]a

b

1• {e1, a, ab, aba, . . .}

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉


1•

2•)

/ 〈ab〉]a

b

1• {e1, a, ab, aba, . . .}|{z}

=0

|{z}=0

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉


1•

2•)

/ 〈ab〉]a

b

1• {e1, a, ab, aba, . . .} = {e1, a}|{z}

=0

|{z}=0

2• {e2, b, ba, bab, . . .}



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉


1•

2•)

/ 〈ab〉]a

b

1• {e1, a, ab, aba, . . .} = {e1, a}|{z}

=0

|{z}=0

2• {e2, b, ba, bab, . . .}|{z}

=0



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉


1•

2•)

/ 〈ab〉]a

b

1• {e1, a, ab, aba, . . .} = {e1, a}|{z}

=0

|{z}=0

2• {e2, b, ba, bab, . . .} = {e2, b, ba}|{z}

=0



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉

Example. The C-algebra dimC[A = C(

1•

2•)

/ 〈ab〉] = 5a

b

1• {e1, a, ab, aba, . . .} = {e1, a}|{z}

=0

|{z}=0

2• {e2, b, ba, bab, . . .} = {e2, b, ba}|{z}

=0



Qver ={

1•, . . . ,

n•}




◮ ρ =∑t


A = CQ/I where I = 〈ρ1, . . . , ρm〉

Example. The C-algebra dimC[A = C(

1•

2•)

/ 〈ab〉] = 5a

b

· e1 a e2 b ab

e1 e1 0 0 b 0a a 0 0 ab 0e2 0 a e2 0 ab

b 0 0 b 0 0ab 0 0 ab 0 0

1• {e1, a, ab, aba, . . .} = {e1, a}|{z}

=0

|{z}=0

2• {e2, b, ba, bab, . . .} = {e2, b, ba}|{z}

=0

Overview



A g




Lie algebra

}[



O(g)

]

Overview



A g




Lie algebra

}[



O(g)

]

Finite dimensional A = CQ/I-modules (modA)


Qver ={

1•, . . . ,

n•}


Qver ={

1•, . . . ,

n•} {simple A-modules S(i)}


Qver ={

1•, . . . ,


{projective indecomposable A-modules P(i)}


Qver ={

1•, . . . ,



P(i)= Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}


Qver ={

1•, . . . ,



P(i) = Aei= 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}


Qver ={

1•, . . . ,



P(i) = Aei = 〈ei 〉= spanC {p = (i → · · · ) is a path of A}


Qver ={

1•, . . . ,



P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}


Qver ={

1•, . . . ,




Example.


Qver ={

1•, . . . ,




Example. The C-algebra A = C(

1•

2•)

/ 〈ab〉a

b

P(1) {e1, a}

P(2) {e2, b, ba}

0 ⊂ 〈a〉 ⊂ 〈e1〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

0 ⊂ 〈ba〉 ⊂ 〈b〉 ⊂ 〈e2〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

︸︷︷︸

S(2)


Qver ={

1•, . . . ,





1•

2•)

/ 〈ab〉a

b

P(1) {e1, a}

P(2) {e2, b, ba}

0 ⊂ 〈a〉 ⊂ 〈e1〉 0 ⊂ 〈ba〉 ⊂ 〈b〉 ⊂ 〈e2〉


Qver ={

1•, . . . ,





1•

2•)

/ 〈ab〉a

b

P(1) {e1, a}

P(2) {e2, b, ba}

0 ⊂ 〈a〉 ⊂ 〈e1〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

0 ⊂ 〈ba〉 ⊂ 〈b〉 ⊂ 〈e2〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

︸︷︷︸

S(2)


Qver ={

1•, . . . ,





1•

2•)

/ 〈ab〉a

b

P(1) {e1, a}

P(2) {e2, b, ba}

0 ⊂ 〈a〉 ⊂ 〈e1〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

0 ⊂ 〈ba〉 ⊂ 〈b〉 ⊂ 〈e2〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

︸︷︷︸

S(2)

〈e1〉 = P(1)! v0

〈a〉 ! v4

0

〈e2〉 = P(2)! u4

〈b〉 ! v0

〈ba〉 ! v4

0


Qver ={

1•, . . . ,




M ∈ modA ! M is a factor module of⊕

i∈QverP(i)ri


1•

2•)

/ 〈ab〉a

b

P(1) {e1, a}

P(2) {e2, b, ba}

0 ⊂ 〈a〉 ⊂ 〈e1〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

0 ⊂ 〈ba〉 ⊂ 〈b〉 ⊂ 〈e2〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

︸︷︷︸

S(2)

〈e1〉 = P(1)! v0

〈a〉 ! v4

0

〈e2〉 = P(2)! u4

〈b〉 ! v0

〈ba〉 ! v4

0


Qver ={

1•, . . . ,




M ∈ modA ! M is a factor module of⊕

i∈QverP(i)ri


1•

2•)

/ 〈ab〉a

b

P(1) {e1, a}

P(2) {e2, b, ba}

0 ⊂ 〈a〉 ⊂ 〈e1〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

0 ⊂ 〈ba〉 ⊂ 〈b〉 ⊂ 〈e2〉︸︷︷︸

S(2)

︸︷︷︸

S(1)

︸︷︷︸

S(2)

〈e1〉 = P(1)! v0

〈a〉 ! v4

0

〈e2〉 = P(2)! u4

〈b〉 ! v0

〈ba〉 ! v4

0

Overview



A g




Lie algebra

}[



O(g)

]

Overview



A g




Lie algebra

}[



O(g)

]

Finite dimensional simple Lie algebras over C


Definition


Definition

A Lie algebra is a C-vector space g


Definition

A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:


Definition


[x , x ] = 0


Definition


[x , x ] = 0

[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g


Definition


[x , x ] = 0


Example.


Definition


[x , x ] = 0


Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0}


Definition


[x , x ] = 0


Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0} with[A,B ] = AB − BA is a simple Lie algebra.


Definition


[x , x ] = 0


A Lie algebra g is simple



Definition


[x , x ] = 0


A Lie algebra g is simple if it has no non-trivial ideals



Definition


[x , x ] = 0


A Lie algebra g is simple if it has no non-trivial ideals and is notabelian (i.e. if there exist x , y ∈ g with [x , y ] 6= 0)



Definition


[x , x ] = 0





Definition


[x , x ] = 0




Finite dimensional simple Lie algebras over C are classified

Simple Lie algebra sl2(C)


◮ Cartan decomposition



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+


◮ Cartan decomposition g = g− ⊕ h⊕ g+

sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+

◮ The universal enveloping algebra U(g) of g



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+


U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+



= spanC {fmhner ; m, n, r ∈ N0}



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+




◮ C-vector space V is a sl2-module



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+




◮ C-vector space V is a sl2-module via an action sl2×V → V



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+





(f , v) 7→ f .v (h, v) 7→ h.v (e, v) 7→ e.v



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+





(f , v) 7→ f .v (h, v) 7→ h.v (e, v) 7→ e.v

V is a U(sl2)-module



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+





(f , v) 7→ f .v (h, v) 7→ h.v (e, v) 7→ e.v

V is a U(sl2)-module via f mhner .v = f . . . f︸︷︷︸

m

. h . . . h︸︷︷︸

n

. e . . . e︸︷︷︸

r

.v



sl2(C) = C

(

0 01 0

)

⊕ C

(

1 00 −1

)

⊕ C

(

0 10 0

)

︸︷︷︸

f

︸︷︷︸

h

︸︷︷︸

e︸︷︷︸

(sl2)−

︸︷︷︸

h

︸︷︷︸

(sl2)+





(f , v) 7→ f .v (h, v) 7→ h.v (e, v) 7→ e.v

V is a U(sl2)-module via f mhner .v = f . . . f︸︷︷︸

m

. h . . . h︸︷︷︸

n

. e . . . e︸︷︷︸

r

.v

{g−modules} ←→ {U(g)−modules}



f .βm=βm+1, h.βm = (3− 2m)βm,


The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}

f .βm=βm+1, h.βm = (3− 2m)βm,



f .βm=βm+1, h.βm = (3− 2m)βm,

β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .



f .αn=αn+1,f .βm=βm+1, h.βm = (3− 2m)βm,

β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .



f .αn=αn+1,f .βm=βm+1, h.βm = (3− 2m)βm,

β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .



f .αn=αn+1, h.αn = (3− 2n)αn,f .βm=βm+1, h.βm = (3− 2m)βm,

β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .



f .αn=αn+1, h.αn = (3− 2n)αn, e.αn=n(4− n)αn−1,f .βm=βm+1, h.βm = (3− 2m)βm,

β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .



f .αn=αn+1, h.αn = (3− 2n)αn, e.αn=n(4− n)αn−1,f .βm=βm+1, h.βm = (3− 2m)βm,

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .



f .αn=αn+1, h.αn = (3− 2n)αn, e.αn=n(4− n)αn−1,f .βm=βm+1, h.βm = (3− 2m)βm, e.βm=m(4−m)βm−1 + αm−1

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

−21 −12 −5

α4α5α6α7••••. . .




−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

−21 −12 −5

α4α5α6α7••••. . .




−21 −12 −5

β4β5β6β7••••. . .

〈α4〉

0

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

〈α4〉

0

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

〈α4〉

0−21 −12 −5

α4α5α6α7••••. . .




−21 −12 −5

β4β5β6β7••••. . .

〈α0〉

〈α4〉

0

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

〈α0〉

〈α4〉

0

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

〈α4〉

0−21 −12 −5

α4α5α6α7••••. . .




−21 −12 −5

β4β5β6β7••••. . .

〈β4〉

〈α0〉

〈α4〉

0

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

〈α0〉

〈α4〉

0

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

〈α4〉

0−21 −12 −5

α4α5α6α7••••. . .

Overview



A g




Lie algebra

}[



O(g)

]

Overview



A g




Lie algebra

}[



O(g)

]

BGG Category O(g)

BGG Category O(g)

◮ Cartan-decomposition g = g− ⊕ h⊕ g+

BGG Category O(g)


◮ Universal enveloping algebra U(g),

BGG Category O(g)


◮ Universal enveloping algebra U(g), U(g+)

BGG Category O(g)



Definition (Bernstein, Gelfand, Gelfand)

BGG Category O(g)




The category O(g) is the full subcategory of g-modules M with

BGG Category O(g)





M is finitely generated,

BGG Category O(g)






M is h-semisimple,

BGG Category O(g)






M is h-semisimple,

dimCU(g+).m <∞ for all m ∈ M.

BGG Category O(g)






M is h-semisimple,


Example.

BGG Category O(g)






M is h-semisimple,


Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V

BGG Category O(g)






M is h-semisimple,



−21 −12 −5

β4β5β6β7••••. . .

−21 −12 −5 3 4 3

α0α1α2α3α4α5α6α7••••••••. . .

BGG Category O(g)






M is h-semisimple,



β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .

BGG Category O(g)






M is h-semisimple,



β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .

BGG Category O(g)






M is h-semisimple,



β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .

BGG Category O(g)






M is h-semisimple,



β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .

BGG Category O(g)






M is h-semisimple,



β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .

BGG Category O(g)






M is h-semisimple,


Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V is in O(sl2)

β4β5β6β7••••. . .

α0α1α2α3α4α5α6α7••••••••. . .

Overview



A g




Lie algebra

}[



O(g)

]

Overview



A g




Lie algebra

}[



O(g)

]

Blocks of category O(g)


O(g)


O(g)

◮ Important modules in O(g)


O(g)

◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}


O(g)


h∗λ ∈{ simple module S(λ) }

{projective indec. P(λ) }


O(g)




O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)


O(g)






O(g)

Oλ(g)






O(g)

Oλ(g)





◮ Important modules in Oλ(g) ! h∗(λ)


O(g)

Oλ(g)






h∗(λ) = {λ1, . . . , λn}


O(g)

Oλ(g)






h∗(λ) = {λ1, . . . , λn}

{simple modules} = {S(λ1), . . . ,S(λn)}


O(g)

Oλ(g)






h∗(λ) = {λ1, . . . , λn}

{simple modules} = {S(λ1), . . . ,S(λn)}{projective indec.} = {P(λ1), . . . ,P(λn)}


O(g)

Oλ(g)






h∗(λ) = {λ1, . . . , λn}


Aλ(g) := Endg (⊕n

i=1 P(λi ))


O(g)

Oλ(g)






h∗(λ) = {λ1, . . . , λn}



i=1 P(λi ))

◮ Aλ(g) is an associative finite dimensional bound quiver algebra


O(g)

Oλ(g)






h∗(λ) = {λ1, . . . , λn}



i=1 P(λi ))

◮ Aλ(g) is an associative finite dimensional bound quiver algebra

Overview



A g




Lie algebra

}[



O(g)

]

Overview



A g

mod Aλ(g) ∼ Oλ(g)

[finite dimensional

A-modules


O(g)

]

The algebra Aλ (sl2) for λ(h) ∈ N


Important modules in O(sl2)! h∗ = {λ : 〈h〉 → C ; λ is linear}



λ(h) ∈ N0 =⇒ h∗(λ) = {λ, −λ− 2}



λ(h) ∈ N0 =⇒ h∗(λ) = {λ, −λ− 2}

O3(sl2) ! h∗(3)



λ(h) ∈ N0 =⇒ h∗(λ) = {λ, −λ− 2}

O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}



λ(h) ∈ N0 =⇒ h∗(λ) = {λ, −λ− 2}

O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}

1−1←→ {P(3),P(−5)}



λ(h) ∈ N0 =⇒ h∗(λ) = {λ, −λ− 2}

O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}

1−1←→ {P(3),P(−5)}

P(3)α0α1α2α3α4α5α6α7••••••••. . .

β4β5β6β7••••. . .

P(−5)α0α1α2α3α4α5α6α7••••••••. . .



λ(h) ∈ N0 =⇒ h∗(λ) = {λ, −λ− 2}

O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}

1−1←→ {P(3),P(−5)}

P(3)α0α1α2α3α4α5α6α7••••••••. . .

β4β5β6β7••••. . .

P(−5)α0α1α2α3α4α5α6α7••••••••. . .



λ(h) ∈ N0 =⇒ h∗(λ) = {λ, −λ− 2}

O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}

1−1←→ {P(3),P(−5)}

P(3)α0α1α2α3α4α5α6α7••••••••. . .

β4β5β6β7••••. . .

P(−5)α0α1α2α3α4α5α6α7••••••••. . .

..Aλ(sl2) = Endsl2 (P(λ)⊕ P(−λ− 2)) ∼= C

(1•

2•)

/ 〈ab〉a

b..

modAλ (sl2) Oλ(sl2)


..Aλ(sl2) ∼= C

(1•

2•)

/ 〈ab〉a

b


..Aλ(sl2) ∼= C

(1•

2•)

/ 〈ab〉a

b

◮ projective indecomposable A3(sl2)-modules


..Aλ(sl2) ∼= C

(1•

2•)

/ 〈ab〉a

b


〈e1〉 = P(1)! α0

〈a〉 ! α4

0

〈e2〉 = P(2)! β4

〈b〉 ! α0

〈ba〉 ! α4

0


..Aλ(sl2) ∼= C

(1•

2•)

/ 〈ab〉a

b


〈e1〉 = P(1)! α0

〈a〉 ! α4

0

〈e2〉 = P(2)! β4

〈b〉 ! α0

〈ba〉 ! α4

0

◮ projective indecomposable modules in O3(sl2)


..Aλ(sl2) ∼= C

(1•

2•)

/ 〈ab〉a

b


〈e1〉 = P(1)! α0

〈a〉 ! α4

0

〈e2〉 = P(2)! β4

〈b〉 ! α0

〈ba〉 ! α4

0


P(3)α0α1α2α3α4α5α6α7••••••••. . .

β4β5β6β7••••. . .

P(−5)α0α1α2α3α4α5α6α7••••••••. . .


..Aλ(sl2) ∼= C

(1•

2•)

/ 〈ab〉a

b


〈e1〉 = P(1)! α0

〈a〉 ! α4

0

〈e2〉 = P(2)! β4

〈b〉 ! α0

〈ba〉 ! α4

0


P(3)α0α1α2α3α4α5α6α7••••••••. . .

β4β5β6β7••••. . .

P(−5)α0α1α2α3α4α5α6α7••••••••. . .


..Aλ(sl2) ∼= C

(1•

2•)

/ 〈ab〉a

b


〈e1〉 = P(1)! α0

〈a〉 ! α4

0

〈e2〉 = P(2)! β4

〈b〉 ! α0

〈ba〉 ! α4

0


P(3)α0α1α2α3α4α5α6α7••••••••. . .

β4β5β6β7••••. . .

P(−5)α0α1α2α3α4α5α6α7••••••••. . .

Overview



A g


[finite dimensional

A-modules


O(g)

]

Overview



A g


Overview



A g

mod A0(g) ∼ O0(g) principal block

Overview



A g


A0(g) = CQ/I

Overview



A g


A0(g) = CQ/I Qver1−1←→W := W (g) the Weyl group of g

Overview



A g



Overview



A g



Qarr ! Bruhat order 6 of W

Bruhat order on W (sln) ∼= Sym(n)


Sym(n) = {permutations of {1, . . . , n}}


Sym(n) = {permutations of {1, . . . , n}} = 〈(1, 2), . . . , (n−1, n)〉



l(σ) := min {m ; σ = τ1 · · · τm with τk ∈ {(1, 2), . . . , (n−1, n)}}




id2

(12)

– 0

– 1




id2

(12)

– 0

– 1

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3




id2

(12)

– 0

– 1

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6




id2

(12)

– 0

– 1

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6




σ ⊳ τ (σ is a small neighbor of τ)

id2

(12)

– 0

– 1

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6




σ ⊳ τ (σ is a small neighbor of τ) if

l(τ) = l(σ)− 1

id2

(12)

– 0

– 1

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

id2

(12)

– 0

– 1

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

id2

(12)

– 0

– 1

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

(Sym(2),6)

id2

(12)

– 0

– 1

(Sym(3),6)

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(Sym(4),6)

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

σ < τ if σ ⊳ σ1 ⊳ · · · ⊳ σt = τ

(Sym(2),6)

id2

(12)

– 0

– 1

(Sym(3),6)

id3

(12) (23)

(12)(23) (23)(12)

(12)(23)(12)

– 0

– 1

– 2

– 3

(Sym(4),6)

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6

Bruhat order on W (sl5) ∼= Sym(5)

(54321)•

• • • •

• • • • • • • • •

• • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • •

• • • • • • • • •

• • • •

•(12345)

Overview



A g



Qarr ! Bruhat order 6 of W

The quiver Q = (Qver, Qarr) of algebra A0(g)


1

ω0

um· · ·u1

w

vr· · ·v1


1

ω0

w ⊲ ui

um· · ·u1

w

vr· · ·v1


1

ω0

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


1

ω0

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


◮ Qver = W

1

ω0

Λ(w) := {v ∈ W ; w 6 v}Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


◮ Qver = W

◮ Qarr ! µ(w , v) := the number of arrows w → v

1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

µ(w , v) =

1 if w and v are neighbours (Bruhat order)0 if w and v are incomparable (Bruhat order)? else

1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

µ(w , v) =


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vvr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

µ(w , v) =


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vvr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

µ(w , v) =


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vvr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

µ(w , v) =


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vvr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

µ(w , v) =


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vvr· · ·v1


◮ Qver = W


µ(w , v) = µ(v , w)

µ(w , v) =


1

ω0

Λ(w) := {v ∈ W ; w 6 v}

w ⊲ ui

w ⊳ vi

um· · ·u1

w

vvr· · ·v1

Main Theorem

Main Theorem

Theorem

Main Theorem

Theorem

Let w ∈W . The following statements are equivalent

Main Theorem

Theorem


µ(w , v) = 0 for all v ∈ Λ(w) with w ⋪ v;

Main Theorem

Theorem



−0

− 1

− k − 1

− k

− m

1

ω0

aa

u1 ur

w

v

vr· · ·

· · ·

v1

Main Theorem

Theorem



−0

− 1

− k − 1

− k

− m

1

ω0

aa

u1 ur

w

v

vr· · ·

· · ·

v1

v

Main Theorem

Theorem


µ(w , v) = 0 for all v ∈ Λ(w) with w ⋪ v;∣∣{v ∈ Λ(w) ; l(v) = i

}∣∣ =

∣∣{v ∈ Λ(w) ; l(v) = l(w)− i

}∣∣ ∀i

−0

− 1

− k − 1

− k

− m

1

ω0

aa

u1 ur

w

v

vr· · ·

· · ·

v1

v

Main Theorem

Theorem



}∣∣ =

∣∣{v ∈ Λ(w) ; l(v) = l(w)− i

}∣∣ ∀i

−0

− 1

− k − 1

− k

− m

1

ω0

1

ω0

aa

u1 ur

w

vr· · ·

· · ·

v1

Main Theorem

Theorem



}∣∣ =

∣∣{v ∈ Λ(w) ; l(v) = l(w)− i

}∣∣ ∀i

−0

− 1

− k − 1

− k

− m

1

ω0

. · · · .

. · · · .

aa

u1 ur

w

vr· · ·

· · ·

v1

Main Theorem

Theorem



}∣∣ =

∣∣{v ∈ Λ(w) ; l(v) = l(w)− i

}∣∣ ∀i

−0

−

−

− i

− k

− k − i

− m

1

ω0

aa

w

· · ·

· · ·

· · ·

· · ·v1 · · · vr

u1 · · · ur





l(τ) = l(σ)− 1

τ = (i , j) · σ

σ < τ if σ ⊳ σ1 ⊳ · · · ⊳ σt = τ

(12)

(21)

– 0

– 1

(123)

(213) (132)

(231) (312)

(321)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

σ < τ if σ ⊳ σ1 ⊳ · · · ⊳ σt = τ

A0(sl2)

(12)

(21)

– 0

– 1

A0(sl3)

(123)

(213) (132)

(231) (312)

(321)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

σ < τ if σ ⊳ σ1 ⊳ · · · ⊳ σt = τ

A0(sl2)

(12)

(21)

– 0

– 1

A0(sl3)

(123)

(213) (132)

(231) (312)

(321)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

σ < τ if σ ⊳ σ1 ⊳ · · · ⊳ σt = τ

A0(sl2)

(12)

(21)

– 0

– 1

A0(sl3)

(123)

(213) (132)

(231) (312)

(321)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

σ < τ if σ ⊳ σ1 ⊳ · · · ⊳ σt = τ

A0(sl2)

(12)

(21)

– 0

– 1

A0(sl3)

(123)

(213) (132)

(231) (312)

(321)

– 0

– 1

– 2

– 3

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6





l(τ) = l(σ)− 1

τ = (i , j) · σ

σ < τ if σ ⊳ σ1 ⊳ · · · ⊳ σt = τ

A0(sl2)

(12)

(21)

– 0

– 1

A0(sl3)

(123)

(213) (132)

(231) (312)

(321)

– 0

– 1

– 2

– 3

A0(sl4)

(1234)

(2134) (1324) (1243)

(2314) (3124) (2143) (1342) (1423)

(3214) (2341) (2413) (3142) (4123) (1432)

(3241) (2431) (3412) (4213) (4132)

(3421) (4231) (4312)

(4321)

– 0

– 1

– 2

– 3

– 4

– 5

– 6


•

• • • •

• • • • • • • • •

• • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • •

• • • • • • • • •

• • • •

•


•

• • • •

• • • • • • • • •

• • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • •

• • • • • • • • •

• • • •

•

Main Theorem

Main Theorem

Theorem

Main Theorem

Theorem


−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

w

· · ·

· · ·

u1 ur

v1 vr

Main Theorem

Theorem


µ(w , v) = 0 for all v ∈ Λ(w) with w 6 ⊲v;

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

w

· · ·

· · ·

u1 ur

v1 vr

v

Main Theorem

Theorem


µ(w , v) = 0 for all v ∈ Λ(w) with w 6 ⊲v;∣∣{v ∈ Λ(w) | l(v) = l(w) + i

}∣∣ =

∣∣{v ∈ Λ(w) | l(v) = m− i

}∣∣

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

w

· · ·

· · ·

u1 ur

v1 vr

v

Main Theorem

Proposition

Main Theorem

Proposition

Let w ∈W and v ∈ Λ(w) with w 6 ⊲v.

Main Theorem

Proposition

Let w ∈W and v ∈ Λ(w) with w 6 ⊲v.

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

v

w

Main Theorem

Proposition

Let w ∈W and v ∈ Λ(w) with w 6 ⊲v. If µ(w , v) 6= 0

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

v

w

Main Theorem

Proposition

Let w ∈W and v ∈ Λ(w) with w 6 ⊲v. If µ(w , v) 6= 0

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

v

w

Main Theorem

Proposition

Let w ∈W and v ∈ Λ(w) with w 6 ⊲v. If µ(w , v) 6= 0 then there

exist i , j ∈ N with 0 ≤ i ≤ l(v) and l(w) ≤ j ≤ m such that

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

v

w

Main Theorem

Proposition



1

∣∣{u ∈ Λ(v) ; l(u) = l(v)− i

}∣∣ 6=

∣∣{u ∈ Λ(v) ; l(u) = i

}∣∣

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

v

w

Main Theorem

Proposition



1

∣∣{u ∈ Λ(v) ; l(u) = l(v)− i

}∣∣ 6=

∣∣{u ∈ Λ(v) ; l(u) = i

}∣∣

2

∣∣{u ∈ Λ(w) | l(u) = l(w) + j

}∣∣ 6=

∣∣{u ∈ Λ(w) | l(u) = m− j

}∣∣

−0

− k

− k + 1

− m − 1

− m

1

ω0

aa

v

w

Documents

The Quiver of a Block of Categorymath.fau.edu/~dpucinskaite/talks/BocaRaton_2013.pdf · Boca Raton 12 April, 2013. Overview. Overview Finite dimensional C-algebras. Overview Finite