Category O and quasi-hereditary algebras

Daiva Pučinskaitė

University of Kiel

07.07.2011Seminar Bremen-Hamburg-Kiel

BGG category O(g),

BGG category O(g),

Bound quiver algebra,

BGG category O(g),

Bound quiver algebra,

Quasi-hereditary algebra,

BGG category O(g),

Bound quiver algebra,

Quasi-hereditary algebra,

1-quasi-hereditary algebra.

Highest weight modules in O(sl2(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] = 2x ,[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,

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,

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,

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, . . .

}

Highest weight modules in O(sl2(C))

Verma module V (3)= U(g).v = spanC{ v , f .v , f2.v , . . . , f n.v , . . .}

Highest weight modules in O(sl2(C))

Verma module V (3) = U(g).v = spanC{ v , f .v , f2.v , . . . , f n.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

v•3

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

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. . .

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

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

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

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

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

The block O3(sl2(C))

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) =