Category O and quasi-hereditary algebrasmath.fau.edu/~dpucinskaite/talks/Hamburg_2011.pdf · 2017....
Transcript of Category O and quasi-hereditary algebrasmath.fau.edu/~dpucinskaite/talks/Hamburg_2011.pdf · 2017....
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Category O and quasi-hereditary algebras
Daiva Pučinskaitė
University of Kiel
07.07.2011Seminar Bremen-Hamburg-Kiel
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BGG category O(g),
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BGG category O(g),
Bound quiver algebra,
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BGG category O(g),
Bound quiver algebra,
Quasi-hereditary algebra,
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BGG category O(g),
Bound quiver algebra,
Quasi-hereditary algebra,
1-quasi-hereditary algebra.
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Highest weight modules in O(sl2(C))
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Highest weight modules in O(sl2(C))
g = sl2(C)= C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
) [h, e] = 2x ,[h, y ] = −2[h, h] = 0,
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Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
) [h, e] = 2e[h, y ] = −2[h, h] = 0,
-
Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
) [h, e] = 2e[h, y ] = −2[h, h] = 0,. .︸ ︷︷ ︸
e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f
-
Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
) [h, e] = 2e[h, y ] = −2[h, h] = 0,. .︸ ︷︷ ︸
e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f
h = Ch is a Cartan-subalgebra, h∗ = {λ : h → C | λ is linear} ∼= C
-
Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
) [h, e] = 2e[h, y ] = −2[h, h] = 0,. .︸ ︷︷ ︸
e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f
h = Ch is a Cartan-subalgebra, h∗ = {λ : h → C | λ is linear} ∼= C
-
Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
)
. .︸ ︷︷ ︸e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f
h = Ch is a Cartan-subalgebra, h∗ = {λ : h → C | λ is linear} ∼= C
[h, e] = 2e,
[h, f ] = −2f ,
[h, h] = 0,
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Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
)
Φ = Φ+︸︷︷︸||
{2}
∪ Φ−︸︷︷︸||
{−2}
. .︸ ︷︷ ︸e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f
h = Ch is a Cartan-subalgebra, h∗ = {λ : h → C | λ is linear} ∼= C
[h, e] = 2e,
[h, f ] = −2f ,
[h, h] = 0,
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Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
)
Φ = Φ+︸︷︷︸||
{2}
∪ Φ−︸︷︷︸||
{−2}
. .︸ ︷︷ ︸e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f. .︸ ︷︷ ︸
g2=n+
. .︸ ︷︷ ︸g0
. .︸ ︷︷ ︸g−2=n−
h = Ch is a Cartan-subalgebra, h∗ = {λ : h → C | λ is linear} ∼= C
[h, e] = 2e,
[h, f ] = −2f ,
[h, h] = 0,
-
Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
)
Φ = Φ+︸︷︷︸||
{2}
∪ Φ−︸︷︷︸||
{−2}
. .︸ ︷︷ ︸e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f. .︸ ︷︷ ︸
g2=n+
. .︸ ︷︷ ︸g0
. .︸ ︷︷ ︸g−2=n−. .︸ ︷︷ ︸
b=Borel subalgebra
h = Ch is a Cartan-subalgebra, h∗ = {λ : h → C | λ is linear} ∼= C
[h, e] = 2e,
[h, f ] = −2f ,
[h, h] = 0,
-
Highest weight modules in O(sl2(C))
g = sl2(C) = C
(0 10 0
)⊕ C
(1 00 −1
)⊕ C
(0 01 0
)
Φ = Φ+︸︷︷︸||
{2}
∪ Φ−︸︷︷︸||
{−2}
. .︸ ︷︷ ︸e
. .︸ ︷︷ ︸h
. .︸ ︷︷ ︸f. .︸ ︷︷ ︸
g2=n+
. .︸ ︷︷ ︸g0
. .︸ ︷︷ ︸g−2=n−. .︸ ︷︷ ︸
b=Borel subalgebra
h = Ch is a Cartan-subalgebra, h∗ = {λ : h → C | λ is linear} ∼= C
[h, e] = 2e,
[h, f ] = −2f ,
[h, h] = 0,
U(n−) = spanC{1, f , f 2, f 3, . . . , f n, . . .
}
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Highest weight modules in O(sl2(C))
Verma module V (3)= U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
v•3
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .v••31
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
34
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
−21 −12 −5 3 4 3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
−21 −12 −5 3 4 3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
−21 −12 −5 3 4 3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
−21 −12 −5 3 4 3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
−21 −12 −5
f 4.vf 5.vf 6.vf 7.vV (−5)••••
-5-7-9-11. . .
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
−21 −12 −5 3 4 3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
−21 −12 −5
f 4.vf 5.vf 6.vf 7.vV (−5) = L(−5)••••
-5-7-9-11. . .
-
Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
−21 −12 −5 3 4 3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
−21 −12 −5
f 4.vf 5.vf 6.vf 7.vV (−5) = L(−5)••••
-5-7-9-11. . .
3 4 3
vf .vf 2.vf 3.vV (3)/V (−5)••••
31-1-3
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Highest weight modules in O(sl2(C))
Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}︸︷︷︸
3︸︷︷︸
1︸︷︷︸−1
︸︷︷︸3−2n
−21 −12 −5 3 4 3
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.vV (3)••••••••
31-1-3-5-7-9-11. . .
f .f n.v = f n+1.v , h.f n.v = (3 − 2n)f n.v , e.f n.v = n(4 − n)f n−1.v
−21 −12 −5
f 4.vf 5.vf 6.vf 7.vV (−5) = L(−5)••••
-5-7-9-11. . .
3 4 3
vf .vf 2.vf 3.vL(3)dimCL(3) = 4 ••••
31-1-3
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The block O3(sl2(C))
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The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ δ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
f 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
f 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
f 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3}= {3,−5}
f 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
f 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
−21 −12 −5 3 4 3
V (3)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
−21 −12 −5 3 4 3
V (3)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
−21 −12 −5
V (−5) = L(−5)f 4.vf 5.vf 6.vf 7.v
••••••••-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
−21 −12 −5 3 4 3
V (3)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
−21 −12 −5
V (−5) = L(−5)
L(3)vf .vf 2.vf 3.v
3 4 3
••••31-1-3
f 4.vf 5.vf 6.vf 7.v••••-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
−21 −12 −5 3 4 3
V (3) = P(3)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
−21 −12 −5
V (−5) = L(−5)
L(3)vf .vf 2.vf 3.v
3 4 3
••••31-1-3
f 4.vf 5.vf 6.vf 7.v••••-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
−21 −12 −5 3 4 3
V (3) = P(3)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
−21 −12 −5
V (−5) = L(−5)
L(3)vf .vf 2.vf 3.v
3 4 3
••••31-1-3
f 4.vf 5.vf 6.vf 7.v••••-5-7-9-11
. . .
−21 −12 −5
wf .wf 2.wf 3.w••••-5-7-9-11
. . .
−21 −12 −5 3 4 3
P(−5)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
-
The block O3(sl2(C))
The Weyl group of sl2(C) is W = {id, σ} with σ(λ) = −λ for all λ ∈ h∗ ∼= C
Φ+ = {2} ⇒ ρ = 2 12 = 1 ⇒ W · 3 = {id · 3, σ · 3} = {3,−5}
−21 −12 −5 3 4 3
V (3) = P(3)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
−21 −12 −5
V (−5) = L(−5)
L(3)vf .vf 2.vf 3.v
3 4 3
••••31-1-3
f 4.vf 5.vf 6.vf 7.v••••-5-7-9-11
. . .
−21 −12 −5
wf .wf 2.wf 3.w••••-5-7-9-11
. . .
−21 −12 −5 3 4 3
P(−5)vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . .
−21 −12 −5 3 4 3
P(−5)/V (−5)vf .vf 2.vf 3.vwf .wf 2.wf 3.w••••••••31-1-3-5-7-9-11
. . .
-
O3(sl2(C)) ∼ mod-A3
-
O3(sl2(C)) ∼ mod-A3
A3 ∼= EndO (P(3) ⊕ P(−5))∼=
a c d
b e
a
| a, b, c , d , e ∈ C
-
O3(sl2(C)) ∼ mod-A3
A3 ∼= EndO (P(3) ⊕ P(−5))∼=
a c d
b e
a
| a, b, c , d , e ∈ C
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . . P(3)
wf .wf 2.wf 3.w••••-5-7-9-11
. . .
P(−5)ṽf .ṽf 2.ṽf 3.ṽf 4.ṽf 5.ṽf 6.ṽf 7.ṽ••••••••31-1-3-5-7-9-11
. . .
1
1
v 7→ 0w 7→ w
-
O3(sl2(C)) ∼ mod-A3
A3 ∼= EndO (P(3) ⊕ P(−5)) ∼=
a c d
b e
a
| a, b, c , d , e ∈ C
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . . P(3)
wf .wf 2.wf 3.w••••-5-7-9-11
. . .
P(−5)ṽf .ṽf 2.ṽf 3.ṽf 4.ṽf 5.ṽf 6.ṽf 7.ṽ••••••••31-1-3-5-7-9-11
. . .
-
O3(sl2(C)) ∼ mod-A3
A3 ∼= EndO (P(3) ⊕ P(−5)) ∼=
a c d
b e
a
| a, b, c , d , e ∈ C
vf .vf 2.vf 3.vf 4.vf 5.vf 6.vf 7.v••••••••31-1-3-5-7-9-11
. . . P(3)
wf .wf 2.wf 3.w••••-5-7-9-11
. . .
P(−5)ṽf .ṽf 2.ṽf 3.ṽf 4.ṽf 5.ṽf 6.ṽf 7.ṽ••••••••31-1-3-5-7-9-11
. . .
v 7→ ṽw 7→ 0
!
0 1 00 0 00 0 0
-
Quiver algebra (example)
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k
k k
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα}
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k
k k
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα}
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k
k k
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k
k k
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β}
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k
k k
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k
k k
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k
k k
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
dimkA = dimkk(
1•
α−→
2•
β−→
3•)
= 6 is isomorphic to
k
k k
k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1
e2
e3 βα
α
β βα
βα 0 0 βα 0 0 0
dimkA = dimkk(
1•
α−→
2•
β−→
3•)
= 6 is isomorphic to
k
k k
k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1 e1 0 0 0 0 0
e2
e3 βα
α
β βα
βα 0 0 βα 0 0 0
dimkA = dimkk(
1•
α−→
2•
β−→
3•)
= 6 is isomorphic to
k
k k
k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1 e1 0 0 0 0 0
e2 0 e2 0 α 0 0
e3 βα
α
β βα
βα 0 0 βα 0 0 0
dimkA = dimkk(
1•
α−→
2•
β−→
3•)
= 6 is isomorphic to
k
k k
k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1 e1 0 0 0 0 0
e2 0 e2 0 α 0 0
e3 0 0 e3 0 β βα
α
β βα
βα 0 0 βα 0 0 0
dimkA = dimkk(
1•
α−→
2•
β−→
3•)
= 6 is isomorphic to
k
k k
k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1 e1 0 0 0 0 0
e2 0 e2 0 α 0 0
e3 0 0 e3 0 β βα
α α 0 0 0 0 0
β 0 β 0 βα 0 0
βα 0 0 βα 0 0 0
dimkA = dimkk(
1•
α−→
2•
β−→
3•)
= 6 is isomorphic to
k
k k
k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1 e1 0 0 0 0 0
e2 0 e2 0 α 0 0
e3 0 0 e3 0 β βα
α α 0 0 0 0 0
β 0 β 0 βα 0 0
βα 0 0 βα 0 0 0
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k 0 0
k k 0
k k k
-
Quiver algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1 e1 0 0 0 0 0
e2 0 e2 0 α 0 0
e3 0 0 e3 0 β βα
α α 0 0 0 0 0
β 0 β 0 βα 0 0
βα 0 0 βα 0 0 0
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k 0 0
k k 0
k k k
e1 7→
1 0 00 0 00 0 0
, e2 7→
0 0 00 1 00 0 0
, e3 7→
0 0 00 0 00 0 1
α 7→
0 0 01 0 00 0 0
, β 7→
0 0 00 0 00 1 0
, βα 7→
0 0 00 0 01 0 0
-
Quiver algebra (example)
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .}= {e1, α}
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .}= {e1, α}
2 {e2, β, αβ, βαβ, . . .}= {e2, β, αβ}
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)/ 〈βα〉 ] = 5
α
β
1 {e1, α, βα, αβα, . . .}= {e1, α}
2 {e2, β, αβ, βαβ, . . .}= {e2, β, αβ}
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)/ 〈βα〉 ] = 5
α
β
1 {e1, α, βα, αβα, . . .}= {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .}= {e2, β, αβ}
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)/ 〈βα〉 ] = 5
α
β
1 {e1, α, βα, αβα, . . .}= {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .}= {e2, β, αβ}︸︷︷︸=0
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)/ 〈βα〉 ] = 5
α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .}= {e2, β, αβ}︸︷︷︸=0
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)/ 〈βα〉 ] = 5
α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
A −→
a c e
b d
a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
A∼
−→
a c e
0 b d0 0 a
| a, b, c , d , e ∈ k
-
Quiver algebra (example)
The k-algebra dimk [A = k(
1•
2•)
/ 〈βα〉] = 5α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
A∼
−→
a c e
0 b d0 0 a
| a, b, c , d , e ∈ k
e1 7→
0 0 00 1 00 0 0
, e2 7→
1 0 00 0 00 0 1
α 7→
0 1 00 0 00 0 0
, β 7→
0 0 00 0 10 0 0
, αβ 7→
0 0 10 0 00 0 0
.
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
· e1 e2 e3 α β βα
e1 e1 0 0 0 0 0
e2 0 e2 0 α 0 0
e3 0 0 e3 0 β βα
α α 0 0 0 0 0
β 0 β 0 βα 0 0
βα 0 0 βα 0 0 0
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k 0 0
k k 0
k k k
e1 7→
1 0 00 0 00 0 0
, e2 7→
0 0 00 1 00 0 0
, e3 7→
0 0 00 0 00 0 1
α 7→
0 0 01 0 00 0 0
, β 7→
0 0 00 0 00 1 0
, βα 7→
0 0 00 0 01 0 0
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
A = k(
1•
α−→
2•
β−→
3•)
is isomorphic to
k 0 0
k k 0
k k k
e1 7→
1 0 00 0 00 0 0
, e2 7→
0 0 00 1 00 0 0
, e3 7→
0 0 00 0 00 0 1
α 7→
0 0 01 0 00 0 0
, β 7→
0 0 00 0 00 1 0
, βα 7→
0 0 00 0 01 0 0
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
〈e2)
〈β)
0
P(2)
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
〈e2)
〈β)
0
P(2)
〈e3)
0
P(3)
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
〈e2)
〈β)
0
P(2)
〈e3)
0
P(3)
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
〈e2)
〈β)
0
P(2)
〈e3)
0
P(3)
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
〈e2)
〈β)
0
P(2)
〈e3)
0
P(3)
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
〈e2)
〈β)
0
P(2)
∆(2) =
{
〈e3)
0
P(3)
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
〈e2)
〈β)
0
P(2)
∆(2) =
{
〈e3)
0
P(3)
∆(3) =
{
-
Quasi-hereditary algebra (example)
Let A be a k-algebra given by
the quiver1•
α−→
2•
β−→
3•
1 {e1, α, βα} ,
2 {e2, β} ,
3 {e3} ,
2 < 1 < 3
〈e1)
〈α)
〈βα)
0
P(1)
∆(1) =
∆(3) =
{
〈e2)
〈β)
0
P(2)
∆(2) =
{
∆(3) =
{ 〈e3)
0
P(3)
∆(3) =
{
-
Quasi-hereditary algebra (example)
-
Quasi-hereditary algebra (example)
The k-algebra A = k(
1•
2•)
/ 〈βα〉 2 < 1α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
A∼
−→
a c e
0 b d0 0 a
| a, b, c , d , e ∈ k
e1 7→
0 0 00 1 00 0 0
, e2 7→
1 0 00 0 00 0 1
α 7→
0 1 00 0 00 0 0
, β 7→
0 0 00 0 10 0 0
, αβ 7→
0 0 10 0 00 0 0
.
-
Quasi-hereditary algebra (example)
The k-algebra A = k(
1•
2•)
/ 〈βα〉 2 < 1α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
-
Quasi-hereditary algebra (example)
The k-algebra A = k(
1•
2•)
/ 〈βα〉 2 < 1α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
〈e1)
〈α)
0
∆(1) =
-
Quasi-hereditary algebra (example)
The k-algebra A = k(
1•
2•)
/ 〈βα〉 2 < 1α
β
1 {e1, α, βα, αβα, . . .} = {e1, α}︸︷︷︸=0
︸︷︷︸=0
2 {e2, β, αβ, βαβ, . . .} = {e2, β, αβ}︸︷︷︸=0
〈e1)
〈α)
0
〈e2)
〈β)
〈αβ)
0
∆(1) =
∆(2) ={
∆(1) =