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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
Finite dimensional C-algebras↓
A
Overview
Finite dimensional C-algebras↓
A
finitely generatedleft A−modules
( category modA)
=
M is a C-spacewith an actionA×M → M
Overview
Finite dimensional C-algebras↓
A
finitely generatedleft A−modules
( category modA)
=
M is a C-spacewith an actionA×M → M
Overview
Finite dimensional C-algebras↓
A
finitely generatedleft A−modules
( category modA)
=
M is a C-spacewith an actionA×M → M
Overview
Finite dimensional C-algebras↓
Overview
Finite dimensional C-algebras↓
Overview
Finite dimensional C-algebras↓
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}
Overview
Finite dimensional C-algebras↓
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}
Overview
Finite dimensional C-algebras↓
bound quiver algebras
A
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
C-algebra A given by a quiver Q and relations
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
CQ = spanC{paths}
Example.
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
CQ = spanC{paths}
Example. Q =(
1•
2•)
a
b
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
CQ = spanC{paths}
Example. Q =Qver =
{1•,
2•}
(1•
2•)
a
b
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
CQ = spanC{paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1•
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1,
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1, a,
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1, a, ab,
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1, a, ab, aba,
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1, a, ab, aba, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1, a, ab, aba, . . .}
2•
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1, a, ab, aba, . . .}
2• {e2, b, ba, bab, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}
Example. Q =Qver =
{1•,
2•}
Qarr = {a, b}
(1•
2•)
a
b
1• {e1, a, ab, aba, . . .}
2• {e2, b, ba, bab, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ 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, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
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, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
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, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ 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, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
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, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
A = CQ/I where I = 〈ρ1, . . . , ρm〉
Example.(
1•
2•)
a
b
1• {e1, a, ab, aba, . . .}
2• {e2, b, ba, bab, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
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, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
A = CQ/I where I = 〈ρ1, . . . , ρm〉
Example. The C-algebra [A = C(
1•
2•)
/ 〈ab〉]a
b
1• {e1, a, ab, aba, . . .}|{z}
=0
|{z}=0
2• {e2, b, ba, bab, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
A = CQ/I where I = 〈ρ1, . . . , ρm〉
Example. The C-algebra [A = C(
1•
2•)
/ 〈ab〉]a
b
1• {e1, a, ab, aba, . . .} = {e1, a}|{z}
=0
|{z}=0
2• {e2, b, ba, bab, . . .}
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
A = CQ/I where I = 〈ρ1, . . . , ρm〉
Example. The C-algebra [A = C(
1•
2•)
/ 〈ab〉]a
b
1• {e1, a, ab, aba, . . .} = {e1, a}|{z}
=0
|{z}=0
2• {e2, b, ba, bab, . . .}|{z}
=0
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
A = CQ/I where I = 〈ρ1, . . . , ρm〉
Example. The C-algebra [A = C(
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
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
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
C-algebra A given by a quiver Q and relations
◮ Q = (Qver,Qarr) is a quiver
Qver ={
1•, . . . ,
n•}
is the set of vertices
Qarr is the set of arrows
◮ CQ = spanC {paths}CQ × CQ → CQ is given by the concatenation of paths
◮ ρ =∑t
r=1 cr ·pr is a relation, where pr = (i → · · · → j)
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
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Finite dimensional A = CQ/I-modules (modA)
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•}
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i)= Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei= 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉= spanC {p = (i → · · · ) is a path of A}
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
Example.
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
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)
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
Example. The C-algebra A = C(
1•
2•)
/ 〈ab〉a
b
P(1) {e1, a}
P(2) {e2, b, ba}
0 ⊂ 〈a〉 ⊂ 〈e1〉 0 ⊂ 〈ba〉 ⊂ 〈b〉 ⊂ 〈e2〉
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
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)
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
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)
〈e1〉 = P(1)! v0
〈a〉 ! v4
0
〈e2〉 = P(2)! u4
〈b〉 ! v0
〈ba〉 ! v4
0
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
M ∈ modA ! M is a factor module of⊕
i∈QverP(i)ri
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)
〈e1〉 = P(1)! v0
〈a〉 ! v4
0
〈e2〉 = P(2)! u4
〈b〉 ! v0
〈ba〉 ! v4
0
Finite dimensional A = CQ/I-modules (modA)
Qver ={
1•, . . . ,
n•} {simple A-modules S(i)}
{projective indecomposable A-modules P(i)}
P(i) = Aei = 〈ei 〉 = spanC {p = (i → · · · ) is a path of A}
M ∈ modA ! M is a factor module of⊕
i∈QverP(i)ri
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)
〈e1〉 = P(1)! v0
〈a〉 ! v4
0
〈e2〉 = P(2)! u4
〈b〉 ! v0
〈ba〉 ! v4
0
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Finite dimensional simple Lie algebras over C
Finite dimensional simple Lie algebras over C
Definition
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
Example.
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0}
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0} with[A,B ] = AB − BA is a simple Lie algebra.
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
A Lie algebra g is simple
Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0} with[A,B ] = AB − BA is a simple Lie algebra.
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
A Lie algebra g is simple if it has no non-trivial ideals
Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0} with[A,B ] = AB − BA is a simple Lie algebra.
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
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)
Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0} with[A,B ] = AB − BA is a simple Lie algebra.
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
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)
Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0} with[A,B ] = AB − BA is a simple Lie algebra.
Finite dimensional simple Lie algebras over C
Definition
A Lie algebra is a C-vector space g together with a bilinear map[−,−] : g× g→ g satisfying the following axioms:
[x , x ] = 0
[x , [y , z ]] − [[x , y ], z ] = [[z , x ], y ] for all x , y , z ∈ g
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)
Example. The space sln(C) := {A ∈ Matn(C) ; tr(A) = 0} with[A,B ] = AB − BA is a simple Lie algebra.
Finite dimensional simple Lie algebras over C are classified
Simple Lie algebra sl2(C)
Simple Lie algebra sl2(C)
◮ Cartan decomposition
Simple Lie algebra sl2(C)
◮ Cartan decomposition
sl2(C) = C
(
0 01 0
)
⊕ C
(
1 00 −1
)
⊕ C
(
0 10 0
)
Simple Lie algebra sl2(C)
◮ Cartan decomposition
sl2(C) = C
(
0 01 0
)
⊕ C
(
1 00 −1
)
⊕ C
(
0 10 0
)
︸ ︷︷ ︸
f
︸ ︷︷ ︸
h
︸ ︷︷ ︸
e
Simple Lie algebra sl2(C)
◮ Cartan decomposition
sl2(C) = C
(
0 01 0
)
⊕ C
(
1 00 −1
)
⊕ C
(
0 10 0
)
︸ ︷︷ ︸
f
︸ ︷︷ ︸
h
︸ ︷︷ ︸
e︸ ︷︷ ︸
(sl2)−
︸ ︷︷ ︸
h
︸ ︷︷ ︸
(sl2)+
Simple Lie algebra sl2(C)
◮ 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)+
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
= spanC {fmhner ; m, n, r ∈ N0}
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
= spanC {fmhner ; m, n, r ∈ N0}
◮ C-vector space V is a sl2-module
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
= spanC {fmhner ; m, n, r ∈ N0}
◮ C-vector space V is a sl2-module via an action sl2×V → V
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
= spanC {fmhner ; m, n, r ∈ N0}
◮ C-vector space V is a sl2-module via an action sl2×V → V
(f , v) 7→ f .v (h, v) 7→ h.v (e, v) 7→ e.v
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
= spanC {fmhner ; m, n, r ∈ N0}
◮ C-vector space V is a sl2-module via an action sl2×V → V
(f , v) 7→ f .v (h, v) 7→ h.v (e, v) 7→ e.v
V is a U(sl2)-module
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
= spanC {fmhner ; m, n, r ∈ N0}
◮ C-vector space V is a sl2-module via an action sl2×V → V
(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
Simple Lie algebra sl2(C)
◮ 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)+
◮ The universal enveloping algebra U(g) of g
U(sl2) ∼= C 〈f , h, e〉 / 〈[x , y ]− xy + yx ; x , y ∈ {f , h, e}〉
= spanC {fmhner ; m, n, r ∈ N0}
◮ C-vector space V is a sl2-module via an action sl2×V → V
(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}
Simple Lie algebra sl2(C)
Simple Lie algebra sl2(C)
f .βm=βm+1, h.βm = (3− 2m)βm,
Simple Lie algebra sl2(C)
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,
Simple Lie algebra sl2(C)
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,
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
f .αn=αn+1,f .βm=βm+1, h.βm = (3− 2m)βm,
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
f .αn=αn+1,f .βm=βm+1, h.βm = (3− 2m)βm,
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
f .αn=αn+1, h.αn = (3− 2n)αn,f .βm=βm+1, h.βm = (3− 2m)βm,
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N4}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N4}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
−21 −12 −5
α4α5α6α7••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N4}
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
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
−21 −12 −5
α4α5α6α7••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−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••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−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••••. . .
Simple Lie algebra sl2(C)
The action of sl2(C) = 〈f , h, e〉 on V = spanC {αn, β4+n ; n ∈ N0}
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
−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
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
BGG Category O(g)
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g),
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example.
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V
−21 −12 −5
β4β5β6β7••••. . .
−21 −12 −5 3 4 3
α0α1α2α3α4α5α6α7••••••••. . .
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
BGG Category O(g)
◮ Cartan-decomposition g = g− ⊕ h⊕ g+
◮ Universal enveloping algebra U(g), U(g+)
Definition (Bernstein, Gelfand, Gelfand)
The category O(g) is the full subcategory of g-modules M with
M is finitely generated,
M is h-semisimple,
dimCU(g+).m <∞ for all m ∈ M.
Example. The sl2(C) = 〈f 〉− ⊕ 〈h〉 ⊕ 〈e〉+-module V is in O(sl2)
β4β5β6β7••••. . .
α0α1α2α3α4α5α6α7••••••••. . .
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Blocks of category O(g)
Blocks of category O(g)
O(g)
Blocks of category O(g)
O(g)
◮ Important modules in O(g)
Blocks of category O(g)
O(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
Blocks of category O(g)
O(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
Blocks of category O(g)
O(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
Blocks of category O(g)
O(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
◮ Important modules in Oλ(g) ! h∗(λ)
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
◮ Important modules in Oλ(g) ! h∗(λ)
h∗(λ) = {λ1, . . . , λn}
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
◮ Important modules in Oλ(g) ! h∗(λ)
h∗(λ) = {λ1, . . . , λn}
{simple modules} = {S(λ1), . . . ,S(λn)}
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
◮ Important modules in Oλ(g) ! h∗(λ)
h∗(λ) = {λ1, . . . , λn}
{simple modules} = {S(λ1), . . . ,S(λn)}{projective indec.} = {P(λ1), . . . ,P(λn)}
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
◮ Important modules in Oλ(g) ! h∗(λ)
h∗(λ) = {λ1, . . . , λn}
{simple modules} = {S(λ1), . . . ,S(λn)}{projective indec.} = {P(λ1), . . . ,P(λn)}
Aλ(g) := Endg (⊕n
i=1 P(λi ))
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
◮ Important modules in Oλ(g) ! h∗(λ)
h∗(λ) = {λ1, . . . , λn}
{simple modules} = {S(λ1), . . . ,S(λn)}{projective indec.} = {P(λ1), . . . ,P(λn)}
Aλ(g) := Endg (⊕n
i=1 P(λi ))
◮ Aλ(g) is an associative finite dimensional bound quiver algebra
Blocks of category O(g)
O(g)
Oλ(g)
◮ Important modules in O(g)! h∗ = {λ : h→ C ; λ is linear}
h∗λ ∈{ simple module S(λ) }
{projective indec. P(λ) }
O(g) decomposes into a direct sum of(infinitely many) blocks Oλ(g)
◮ Important modules in Oλ(g) ! h∗(λ)
h∗(λ) = {λ1, . . . , λn}
{simple modules} = {S(λ1), . . . ,S(λn)}{projective indec.} = {P(λ1), . . . ,P(λn)}
Aλ(g) := Endg (⊕n
i=1 P(λi ))
◮ Aλ(g) is an associative finite dimensional bound quiver algebra
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
{associative algebras} {non-associative algebras}∪ ∪
{A ; A is a basic algebras}{
g ;g is a semi-simple
Lie algebra
}[
finite dimensionalA-modules
] [blocks of category
O(g)
]
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod Aλ(g) ∼ Oλ(g)
[finite dimensional
A-modules
] [blocks of category
O(g)
]
The algebra Aλ (sl2) for λ(h) ∈ N
The algebra Aλ (sl2) for λ(h) ∈ N
Important modules in O(sl2)! h∗ = {λ : 〈h〉 → C ; λ is linear}
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}
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}
O3(sl2) ! h∗(3)
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}
O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}
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}
O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}
1−1←→ {P(3),P(−5)}
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}
O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}
1−1←→ {P(3),P(−5)}
P(3)α0α1α2α3α4α5α6α7••••••••. . .
β4β5β6β7••••. . .
P(−5)α0α1α2α3α4α5α6α7••••••••. . .
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}
O3(sl2) ! h∗(3)1−1←→ {S(3),S(−5)}
1−1←→ {P(3),P(−5)}
P(3)α0α1α2α3α4α5α6α7••••••••. . .
β4β5β6β7••••. . .
P(−5)α0α1α2α3α4α5α6α7••••••••. . .
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}
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)
modAλ (sl2) Oλ(sl2)
..Aλ(sl2) ∼= C
(1•
2•)
/ 〈ab〉a
b
modAλ (sl2) Oλ(sl2)
..Aλ(sl2) ∼= C
(1•
2•)
/ 〈ab〉a
b
◮ projective indecomposable A3(sl2)-modules
modAλ (sl2) Oλ(sl2)
..Aλ(sl2) ∼= C
(1•
2•)
/ 〈ab〉a
b
◮ projective indecomposable A3(sl2)-modules
〈e1〉 = P(1)! α0
〈a〉 ! α4
0
〈e2〉 = P(2)! β4
〈b〉 ! α0
〈ba〉 ! α4
0
modAλ (sl2) Oλ(sl2)
..Aλ(sl2) ∼= C
(1•
2•)
/ 〈ab〉a
b
◮ projective indecomposable A3(sl2)-modules
〈e1〉 = P(1)! α0
〈a〉 ! α4
0
〈e2〉 = P(2)! β4
〈b〉 ! α0
〈ba〉 ! α4
0
◮ projective indecomposable modules in O3(sl2)
modAλ (sl2) Oλ(sl2)
..Aλ(sl2) ∼= C
(1•
2•)
/ 〈ab〉a
b
◮ projective indecomposable A3(sl2)-modules
〈e1〉 = P(1)! α0
〈a〉 ! α4
0
〈e2〉 = P(2)! β4
〈b〉 ! α0
〈ba〉 ! α4
0
◮ projective indecomposable modules in O3(sl2)
P(3)α0α1α2α3α4α5α6α7••••••••. . .
β4β5β6β7••••. . .
P(−5)α0α1α2α3α4α5α6α7••••••••. . .
modAλ (sl2) Oλ(sl2)
..Aλ(sl2) ∼= C
(1•
2•)
/ 〈ab〉a
b
◮ projective indecomposable A3(sl2)-modules
〈e1〉 = P(1)! α0
〈a〉 ! α4
0
〈e2〉 = P(2)! β4
〈b〉 ! α0
〈ba〉 ! α4
0
◮ projective indecomposable modules in O3(sl2)
P(3)α0α1α2α3α4α5α6α7••••••••. . .
β4β5β6β7••••. . .
P(−5)α0α1α2α3α4α5α6α7••••••••. . .
modAλ (sl2) Oλ(sl2)
..Aλ(sl2) ∼= C
(1•
2•)
/ 〈ab〉a
b
◮ projective indecomposable A3(sl2)-modules
〈e1〉 = P(1)! α0
〈a〉 ! α4
0
〈e2〉 = P(2)! β4
〈b〉 ! α0
〈ba〉 ! α4
0
◮ projective indecomposable modules in O3(sl2)
P(3)α0α1α2α3α4α5α6α7••••••••. . .
β4β5β6β7••••. . .
P(−5)α0α1α2α3α4α5α6α7••••••••. . .
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod Aλ(g) ∼ Oλ(g)
[finite dimensional
A-modules
] [blocks of category
O(g)
]
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod Aλ(g) ∼ Oλ(g)
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod A0(g) ∼ O0(g) principal block
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod A0(g) ∼ O0(g) principal block
A0(g) = CQ/I
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod A0(g) ∼ O0(g) principal block
A0(g) = CQ/I Qver1−1←→W := W (g) the Weyl group of g
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod A0(g) ∼ O0(g) principal block
A0(g) = CQ/I Qver1−1←→W := W (g) the Weyl group of g
Overview
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod A0(g) ∼ O0(g) principal block
A0(g) = CQ/I Qver1−1←→W := W (g) the Weyl group of g
Qarr ! Bruhat order 6 of W
Bruhat order on W (sln) ∼= Sym(n)
Bruhat order on W (sln) ∼= Sym(n)
Sym(n) = {permutations of {1, . . . , n}}
Bruhat order on W (sln) ∼= Sym(n)
Sym(n) = {permutations of {1, . . . , n}} = 〈(1, 2), . . . , (n−1, n)〉
Bruhat order on W (sln) ∼= Sym(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)}}
Bruhat order on W (sln) ∼= Sym(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
Bruhat order on W (sln) ∼= Sym(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
id3
(12) (23)
(12)(23) (23)(12)
(12)(23)(12)
– 0
– 1
– 2
– 3
Bruhat order on W (sln) ∼= Sym(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
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
Bruhat order on W (sln) ∼= Sym(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
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
Bruhat order on W (sln) ∼= Sym(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)}}
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
Bruhat order on W (sln) ∼= Sym(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)}}
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Finite dimensional C-algebras↓
bound quiver algebras simple Lie algebras
A g
mod A0(g) ∼ O0(g) principal block
A0(g) = CQ/I Qver1−1←→W := W (g) the Weyl group of g
Qarr ! Bruhat order 6 of W
The quiver Q = (Qver, Qarr) of algebra A0(g)
The quiver Q = (Qver, Qarr) of algebra A0(g)
1
ω0
um· · ·u1
w
vr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
1
ω0
w ⊲ ui
um· · ·u1
w
vr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
1
ω0
w ⊲ ui
w ⊳ vi
um· · ·u1
w
vr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
1
ω0
w ⊲ ui
w ⊳ vi
um· · ·u1
w
vr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
1
ω0
Λ(w) := {v ∈ W ; w 6 v}
w ⊲ ui
w ⊳ vi
um· · ·u1
w
vr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
1
ω0
Λ(w) := {v ∈ W ; w 6 v}
w ⊲ ui
w ⊳ vi
um· · ·u1
w
vr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ 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
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ 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
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
µ(w , v) = µ(v , w)
1
ω0
Λ(w) := {v ∈ W ; w 6 v}
w ⊲ ui
w ⊳ vi
um· · ·u1
w
vr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
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
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
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
vvr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
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
vvr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
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
vvr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
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
vvr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
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
vvr· · ·v1
The quiver Q = (Qver, Qarr) of algebra A0(g)
◮ Qver = W
◮ Qarr ! µ(w , v) := the number of arrows w → v
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
vvr· · ·v1
Main Theorem
Main Theorem
Theorem
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
µ(w , v) = 0 for all v ∈ Λ(w) with w ⋪ v;
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
µ(w , v) = 0 for all v ∈ Λ(w) with w ⋪ v;
−0
− 1
− k − 1
− k
− m
1
ω0
aa
u1 ur
w
v
vr· · ·
· · ·
v1
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
µ(w , v) = 0 for all v ∈ Λ(w) with w ⋪ v;
−0
− 1
− k − 1
− k
− m
1
ω0
aa
u1 ur
w
v
vr· · ·
· · ·
v1
v
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
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
Let w ∈W . The following statements are equivalent
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
1
ω0
aa
u1 ur
w
vr· · ·
· · ·
v1
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
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
vr· · ·
· · ·
v1
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
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
−
−
− i
− k
− k − i
− m
1
ω0
aa
w
· · ·
· · ·
· · ·
· · ·v1 · · · vr
u1 · · · ur
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sln) ∼= Sym(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)}}
σ ⊳ τ (σ is a small neighbor of τ) if
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
Bruhat order on W (sl5) ∼= Sym(5)
•
• • • •
• • • • • • • • •
• • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • •
• • • • • • • • •
• • • •
•
Bruhat order on W (sl5) ∼= Sym(5)
•
• • • •
• • • • • • • • •
• • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • • • • •
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• • • • • • • • •
• • • •
•
Main Theorem
Main Theorem
Theorem
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
−0
− k
− k + 1
− m − 1
− m
1
ω0
aa
w
· · ·
· · ·
u1 ur
v1 vr
Main Theorem
Theorem
Let w ∈W . The following statements are equivalent
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
Let w ∈W . The following statements are equivalent
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
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
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
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
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
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