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Noetherian Hopf algebras: speculations around noncommutative solvable groups Ken Brown University of Glasgow Shanghai 13.9.2011 Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 1 / 19 Plan Standing hypotheses

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Noetherian Hopf algebras: speculations around noncommutative solvable groups

Ken Brown

University of Glasgow

Shanghai 13.9.2011

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 1 / 19

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SLIDE 2

Plan

1

Standing hypotheses

2

Hopf algebras of small GK-dimension

3

Five-minute primer on affine algebraic groups

4

Hopf algebra epimorphisms and crossed products

5

Towards a definition of a noncommutative solvable group

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 2 / 19

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  • 1. Standing hypotheses

H affine noetherian Hopf k-algebra Coproduct ∆ : H − → H ⊗ H; counit ε : H − → k; antipode S : H − → H

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 3 / 19

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SLIDE 4
  • 1. Standing hypotheses

H affine noetherian Hopf k-algebra Coproduct ∆ : H − → H ⊗ H; counit ε : H − → k; antipode S : H − → H k = ¯ k, characteristic 0

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 3 / 19

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SLIDE 5
  • 1. Standing hypotheses

H affine noetherian Hopf k-algebra Coproduct ∆ : H − → H ⊗ H; counit ε : H − → k; antipode S : H − → H k = ¯ k, characteristic 0

Question

Does noetherian ⇒ affine?

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 3 / 19

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SLIDE 6
  • 1. Standing hypotheses

H affine noetherian Hopf k-algebra Coproduct ∆ : H − → H ⊗ H; counit ε : H − → k; antipode S : H − → H k = ¯ k, characteristic 0

Question

Does noetherian ⇒ affine? Yes in commutative case (Molnar, 1975) Yes in cocommutative case (exercise)

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 3 / 19

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SLIDE 7
  • 1. Standing hypotheses

H affine noetherian Hopf k-algebra Coproduct ∆ : H − → H ⊗ H; counit ε : H − → k; antipode S : H − → H k = ¯ k, characteristic 0

Question

Does noetherian ⇒ affine? Yes in commutative case (Molnar, 1975) Yes in cocommutative case (exercise)

Question

Does affine noetherian ⇒ S bijective? Yes if H semiprime or PI (Skryabin, 2006)

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 3 / 19

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SLIDE 8
  • 1. Standing hypotheses

H affine noetherian Hopf k-algebra Coproduct ∆ : H − → H ⊗ H; counit ε : H − → k; antipode S : H − → H k = ¯ k, characteristic 0

Question

Does noetherian ⇒ affine? Yes in commutative case (Molnar, 1975) Yes in cocommutative case (exercise)

Question

Does affine noetherian ⇒ S bijective? Yes if H semiprime or PI (Skryabin, 2006) We’ll always assume that S is bijective.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 3 / 19

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SLIDE 9
  • 2. Hopf algebras of small GK-dimension

Work of KAB, Goodearl, Liu, Wang, Zhuang, Zhang

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19

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SLIDE 10
  • 2. Hopf algebras of small GK-dimension

Work of KAB, Goodearl, Liu, Wang, Zhuang, Zhang A: H prime of GK-dimension one (KAB, Zhang (’10); Liu (’09))

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19

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SLIDE 11
  • 2. Hopf algebras of small GK-dimension

Work of KAB, Goodearl, Liu, Wang, Zhuang, Zhang A: H prime of GK-dimension one (KAB, Zhang (’10); Liu (’09)) Examples:

1

Commutative:

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19

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SLIDE 12
  • 2. Hopf algebras of small GK-dimension

Work of KAB, Goodearl, Liu, Wang, Zhuang, Zhang A: H prime of GK-dimension one (KAB, Zhang (’10); Liu (’09)) Examples:

1

Commutative: H = k[X], ∆(X) = X ⊗ 1 + 1 ⊗ X

  • r

H = k[X ±1], ∆(X) = X ⊗ X

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19

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SLIDE 13
  • 2. Hopf algebras of small GK-dimension

Work of KAB, Goodearl, Liu, Wang, Zhuang, Zhang A: H prime of GK-dimension one (KAB, Zhang (’10); Liu (’09)) Examples:

1

Commutative: H = k[X], ∆(X) = X ⊗ 1 + 1 ⊗ X

  • r

H = k[X ±1], ∆(X) = X ⊗ X

2

Cocommutative: Above two, and

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19

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SLIDE 14
  • 2. Hopf algebras of small GK-dimension

Work of KAB, Goodearl, Liu, Wang, Zhuang, Zhang A: H prime of GK-dimension one (KAB, Zhang (’10); Liu (’09)) Examples:

1

Commutative: H = k[X], ∆(X) = X ⊗ 1 + 1 ⊗ X

  • r

H = k[X ±1], ∆(X) = X ⊗ X

2

Cocommutative: Above two, and

H = kD, D infinite dihedral group x±1, g : g2 = 1, gxg = x−1. case t = 0 of:

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19

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  • 2. Hopf algebras of small GK-dimension

3

Taft algebras H = T(n, t, ξ) = kx, g : g n = 1, xg = ξgx ∆(g) = g ⊗ g ∆(x) = x ⊗ g t + 1 ⊗ x

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 5 / 19

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  • 2. Hopf algebras of small GK-dimension

3

Taft algebras H = T(n, t, ξ) = kx, g : g n = 1, xg = ξgx ∆(g) = g ⊗ g ∆(x) = x ⊗ g t + 1 ⊗ x

4

Generalised Liu algebras: Special case is: for n, w coprime integers, n > 1, w ≥ 1, H = L(n, w, ξ) = kx±1, y : yx = ξxy, y n = 1 − xnw x group-like, ∆(y) = y ⊗ xw + 1 ⊗ y

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 5 / 19

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  • 2. Hopf algebras of small GK-dimension

Question

Is every prime noetherian Hopf k-algebra H of GK-dimension one listed above?

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 6 / 19

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  • 2. Hopf algebras of small GK-dimension

Question

Is every prime noetherian Hopf k-algebra H of GK-dimension one listed above?

Theorem

(B - Zhang) Yes, if H has finite global dimension and prime PI-degree.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 6 / 19

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SLIDE 19
  • 2. Hopf algebras of small GK-dimension

Question

Is every prime noetherian Hopf k-algebra H of GK-dimension one listed above?

Theorem

(B - Zhang) Yes, if H has finite global dimension and prime PI-degree.

Question

Does every prime noetherian Hopf k-algebra H of GK-dimension one have finite global dimension?

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 6 / 19

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  • 2. Hopf algebras of small GK-dimension

B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11))

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19

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  • 2. Hopf algebras of small GK-dimension

B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples:

1

Group algebras: kΓ, Γ = Z × Z or Z ⋊ Z

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19

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  • 2. Hopf algebras of small GK-dimension

B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples:

1

Group algebras: kΓ, Γ = Z × Z or Z ⋊ Z

2

Enveloping algebras: U(g), dim(g) = 2

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19

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  • 2. Hopf algebras of small GK-dimension

B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples:

1

Group algebras: kΓ, Γ = Z × Z or Z ⋊ Z

2

Enveloping algebras: U(g), dim(g) = 2

3

Generalised q-Borels: For n ∈ Z, q ∈ k∗, A(n, q) = kx±1, y : xy = qyx x group-like, ∆(y) = y ⊗ 1 + xn ⊗ y

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19

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  • 2. Hopf algebras of small GK-dimension

B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples:

1

Group algebras: kΓ, Γ = Z × Z or Z ⋊ Z

2

Enveloping algebras: U(g), dim(g) = 2

3

Generalised q-Borels: For n ∈ Z, q ∈ k∗, A(n, q) = kx±1, y : xy = qyx x group-like, ∆(y) = y ⊗ 1 + xn ⊗ y So A(1, q2) is positive Borel in Uq(sl(2)); A(n, 1) = O(G), G 2-dim connected solvable algebraic group

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19

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  • 2. Hopf algebras of small GK-dimension

4

Gorenstein singularities: Special case: Let n ∈ Z, n ≥ 1, and q a primitive (6n)th root of 1 in k.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 8 / 19

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  • 2. Hopf algebras of small GK-dimension

4

Gorenstein singularities: Special case: Let n ∈ Z, n ≥ 1, and q a primitive (6n)th root of 1 in k. Let σ ∈ Aut(k[y]), σ(y) = qy, and form B := k[y][x±1; σ]. Set A = ky 2, y 3 ⊂ k[y].

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 8 / 19

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  • 2. Hopf algebras of small GK-dimension

4

Gorenstein singularities: Special case: Let n ∈ Z, n ≥ 1, and q a primitive (6n)th root of 1 in k. Let σ ∈ Aut(k[y]), σ(y) = qy, and form B := k[y][x±1; σ]. Set A = ky 2, y 3 ⊂ k[y]. Define H = A[x±1; σ] ⊂ B

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 8 / 19

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  • 2. Hopf algebras of small GK-dimension

4

Gorenstein singularities: Special case: Let n ∈ Z, n ≥ 1, and q a primitive (6n)th root of 1 in k. Let σ ∈ Aut(k[y]), σ(y) = qy, and form B := k[y][x±1; σ]. Set A = ky 2, y 3 ⊂ k[y]. Define H = A[x±1; σ] ⊂ B Then H is a Hopf algebra with ∆(x) = x ⊗ x, ∆(y 2) = y 2 ⊗ 1 + x2n ⊗ y 2, ∆(y 3) = y 3 ⊗ 1 + x3n ⊗ y 3

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 8 / 19

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  • 2. Hopf algebras of small GK-dimension

5

For n ≥ 2, define H = C(n) = k[y ±1][x; (y n − y)∂/∂y] ∆(y) = y ⊗ y, ∆(x) = x ⊗ y n−1 + 1 ⊗ x

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 9 / 19

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  • 2. Hopf algebras of small GK-dimension

5

For n ≥ 2, define H = C(n) = k[y ±1][x; (y n − y)∂/∂y] ∆(y) = y ⊗ y, ∆(x) = x ⊗ y n−1 + 1 ⊗ x Is the above the complete list (of noetherian affine Hopf domains of GK-dimension 2)?

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 9 / 19

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  • 2. Hopf algebras of small GK-dimension

Remark on classical subgroups: For any Hopf algebra H, [H, H] := H{xy − yx : x, y ∈ H}H is a Hopf ideal;

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 10 / 19

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  • 2. Hopf algebras of small GK-dimension

Remark on classical subgroups: For any Hopf algebra H, [H, H] := H{xy − yx : x, y ∈ H}H is a Hopf ideal; so H/[H, H] is a Hopf algebra.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 10 / 19

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  • 2. Hopf algebras of small GK-dimension

Remark on classical subgroups: For any Hopf algebra H, [H, H] := H{xy − yx : x, y ∈ H}H is a Hopf ideal; so H/[H, H] is a Hopf algebra. It’s commutative, so

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 10 / 19

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  • 2. Hopf algebras of small GK-dimension

Remark on classical subgroups: For any Hopf algebra H, [H, H] := H{xy − yx : x, y ∈ H}H is a Hopf ideal; so H/[H, H] is a Hopf algebra. It’s commutative, so H/[H, H] ∼ = O(G), G an affine algebraic group over k.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 10 / 19

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  • 2. Hopf algebras of small GK-dimension

Remark on classical subgroups: For any Hopf algebra H, [H, H] := H{xy − yx : x, y ∈ H}H is a Hopf ideal; so H/[H, H] is a Hopf algebra. It’s commutative, so H/[H, H] ∼ = O(G), G an affine algebraic group over k. “G is the biggest classical group contained in the quantum group H.”

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 10 / 19

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  • 2. Hopf algebras of small GK-dimension

Theorem

Goodearl, Zhang, 2010) Let the Hopf k-algebra H be an affine noetherian domain of GK-dimension 2. (*) Suppose H contains a classical subgroup of dimension at least 1. Then H is one of the algebras of types 1-5 listed above.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 11 / 19

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  • 2. Hopf algebras of small GK-dimension

Theorem

Goodearl, Zhang, 2010) Let the Hopf k-algebra H be an affine noetherian domain of GK-dimension 2. (*) Suppose H contains a classical subgroup of dimension at least 1. Then H is one of the algebras of types 1-5 listed above.

Theorem

(Wang, Zhang, Zhuang, 2011) There is an infinite family of affine noetherian Hopf k-algebra domains of GK-dimension 2 for which (*) fails.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 11 / 19

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  • 2. Hopf algebras of small GK-dimension

Theorem

Goodearl, Zhang, 2010) Let the Hopf k-algebra H be an affine noetherian domain of GK-dimension 2. (*) Suppose H contains a classical subgroup of dimension at least 1. Then H is one of the algebras of types 1-5 listed above.

Theorem

(Wang, Zhang, Zhuang, 2011) There is an infinite family of affine noetherian Hopf k-algebra domains of GK-dimension 2 for which (*) fails.

Question

Is every pointed noetherian Hopf k-algebra domain of GK-dimension 2 accounted for by the above 2 theorems?

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 11 / 19

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SLIDE 39
  • 3. Five-minute primer on algebraic groups

Recall that k is algebraically closed of characteristic 0. Let G be a connected affine algebraic group over k.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 12 / 19

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  • 3. Five-minute primer on algebraic groups

Recall that k is algebraically closed of characteristic 0. Let G be a connected affine algebraic group over k.

Definition

The unipotent radical of G, denoted U(G), is the largest connected normal unipotent subgroup of G.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 12 / 19

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SLIDE 41
  • 3. Five-minute primer on algebraic groups

Recall that k is algebraically closed of characteristic 0. Let G be a connected affine algebraic group over k.

Definition

The unipotent radical of G, denoted U(G), is the largest connected normal unipotent subgroup of G. U(G) is nilpotent, and there is a chain 1 = N0 ⊂ N1 ⊂ · · · ⊂ Nd = U(G), d = dim(U(G)), Ni ✁ G, Ni/Ni−1 ∼ = (k, +) for all i.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 12 / 19

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SLIDE 42
  • 3. Five-minute primer on algebraic groups

Recall that k is algebraically closed of characteristic 0. Let G be a connected affine algebraic group over k.

Definition

The unipotent radical of G, denoted U(G), is the largest connected normal unipotent subgroup of G. U(G) is nilpotent, and there is a chain 1 = N0 ⊂ N1 ⊂ · · · ⊂ Nd = U(G), d = dim(U(G)), Ni ✁ G, Ni/Ni−1 ∼ = (k, +) for all i. G/U(G) is reductive,

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 12 / 19

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SLIDE 43
  • 3. Five-minute primer on algebraic groups

Recall that k is algebraically closed of characteristic 0. Let G be a connected affine algebraic group over k.

Definition

The unipotent radical of G, denoted U(G), is the largest connected normal unipotent subgroup of G. U(G) is nilpotent, and there is a chain 1 = N0 ⊂ N1 ⊂ · · · ⊂ Nd = U(G), d = dim(U(G)), Ni ✁ G, Ni/Ni−1 ∼ = (k, +) for all i. G/U(G) is reductive, and G ∼ = U(G) ⋊ L(G).

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 12 / 19

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SLIDE 44
  • 3. Five-minute primer on algebraic groups

Recall that k is algebraically closed of characteristic 0. Let G be a connected affine algebraic group over k.

Definition

The unipotent radical of G, denoted U(G), is the largest connected normal unipotent subgroup of G. U(G) is nilpotent, and there is a chain 1 = N0 ⊂ N1 ⊂ · · · ⊂ Nd = U(G), d = dim(U(G)), Ni ✁ G, Ni/Ni−1 ∼ = (k, +) for all i. G/U(G) is reductive, and G ∼ = U(G) ⋊ L(G). G is solvable iff L(G) is a torus, (so isom. to (k∗)m).

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 12 / 19

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SLIDE 45
  • 4. Hopf epimorphisms and crossed products

(Joint with Steven O’Hagan (PhD student))

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 13 / 19

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  • 4. Hopf epimorphisms and crossed products

(Joint with Steven O’Hagan (PhD student)) Suppose H and ¯ H are Hopf k-algebras, and π : H ։ ¯ H is an epimorphism of Hopf algebras.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 13 / 19

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  • 4. Hopf epimorphisms and crossed products

(Joint with Steven O’Hagan (PhD student)) Suppose H and ¯ H are Hopf k-algebras, and π : H ։ ¯ H is an epimorphism of Hopf algebras. Then H is a right ¯ H-comodule algebra: ρ := (1 ⊗ π) ◦ ∆ : H − → H ⊗ ¯ H. The right ¯ H-coinvariants is the subalgebra of H Hcoπ = Hco¯

H := {a ∈ H : ρ(a) = a ⊗ 1}.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 13 / 19

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SLIDE 48
  • 4. Hopf epimorphisms and crossed products

(Joint with Steven O’Hagan (PhD student)) Suppose H and ¯ H are Hopf k-algebras, and π : H ։ ¯ H is an epimorphism of Hopf algebras. Then H is a right ¯ H-comodule algebra: ρ := (1 ⊗ π) ◦ ∆ : H − → H ⊗ ¯ H. The right ¯ H-coinvariants is the subalgebra of H Hcoπ = Hco¯

H := {a ∈ H : ρ(a) = a ⊗ 1}.

In “very nice cases”, we then find that H “decomposes” as a crossed product H ∼ = Hco¯

H ∗σ T,

T ∼ = ¯ H

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 13 / 19

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SLIDE 49
  • 4. Hopf epimorphisms and crossed products

Lemma

(Schneider, 1993) Given π : H ։ ¯ H, Hco ¯

H is a Hopf subalgebra of H

⇐ ⇒

co¯ HH = Hco¯ H, and this happens provided π is conormal.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 14 / 19

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SLIDE 50
  • 4. Hopf epimorphisms and crossed products

Lemma

(Schneider, 1993) Given π : H ։ ¯ H, Hco ¯

H is a Hopf subalgebra of H

⇐ ⇒

co¯ HH = Hco¯ H, and this happens provided π is conormal.

Definition

π is conormal ⇐ ⇒ for all x ∈ I := kerπ, x1S(x3) ⊗ x2 ∈ H ⊗ I and x2 ⊗ S(x1)x3 ∈ I ⊗ H.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 14 / 19

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SLIDE 51
  • 4. Hopf epimorphisms and crossed products

Lemma

(Schneider, 1993) Given π : H ։ ¯ H, Hco ¯

H is a Hopf subalgebra of H

⇐ ⇒

co¯ HH = Hco¯ H, and this happens provided π is conormal.

Definition

π is conormal ⇐ ⇒ for all x ∈ I := kerπ, x1S(x3) ⊗ x2 ∈ H ⊗ I and x2 ⊗ S(x1)x3 ∈ I ⊗ H. π : H = O(G) ։ O(N), then π is conormal ⇐ ⇒ N ✁ G.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 14 / 19

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  • 4. Hopf epimorphisms and crossed products

Examples:

1

h ✁ g, π : H := U(g) ։ U(g/h) =: ¯ H.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 15 / 19

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SLIDE 53
  • 4. Hopf epimorphisms and crossed products

Examples:

1

h ✁ g, π : H := U(g) ։ U(g/h) =: ¯

  • H. Then

U(g) ∼ = U(h) ∗σ U(g/h)

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 15 / 19

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SLIDE 54
  • 4. Hopf epimorphisms and crossed products

Examples:

1

h ✁ g, π : H := U(g) ։ U(g/h) =: ¯

  • H. Then

U(g) ∼ = U(h) ∗σ U(g/h)

2

H = O(SL(2)) ∼ = k[a, b, c, d]/ad − bc − 1, and π : H ։ ¯ H := k[¯ a, ¯ d : ¯ a¯ d = ¯ 1]

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 15 / 19

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SLIDE 55
  • 4. Hopf epimorphisms and crossed products

Examples:

1

h ✁ g, π : H := U(g) ։ U(g/h) =: ¯

  • H. Then

U(g) ∼ = U(h) ∗σ U(g/h)

2

H = O(SL(2)) ∼ = k[a, b, c, d]/ad − bc − 1, and π : H ։ ¯ H := k[¯ a, ¯ d : ¯ a¯ d = ¯ 1] The right coinvariants are Hco¯

H = k[cd, ba], not a Hopf

subalgebra, (equivalently, Cartan subgroup SL(2, k);

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 15 / 19

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SLIDE 56
  • 4. Hopf epimorphisms and crossed products

Examples:

1

h ✁ g, π : H := U(g) ։ U(g/h) =: ¯

  • H. Then

U(g) ∼ = U(h) ∗σ U(g/h)

2

H = O(SL(2)) ∼ = k[a, b, c, d]/ad − bc − 1, and π : H ։ ¯ H := k[¯ a, ¯ d : ¯ a¯ d = ¯ 1] The right coinvariants are Hco¯

H = k[cd, ba], not a Hopf

subalgebra, (equivalently, Cartan subgroup SL(2, k); and there is no crossed product decomposition (since no non-trivial units in O(SL(2)).

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 15 / 19

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  • 4. Hopf epimorphisms and crossed products

3

G connected affine algebraic group, π : H = O(G) ։ O(N), N ⊳ G, N unipotent, dim(N) = d.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 16 / 19

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  • 4. Hopf epimorphisms and crossed products

3

G connected affine algebraic group, π : H = O(G) ։ O(N), N ⊳ G, N unipotent, dim(N) = d. H ∼ = Hcoπ ⊗k O(N) = O(G/N)[x1, . . . , xd] as k-algebras.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 16 / 19

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  • 4. Hopf epimorphisms and crossed products

3

G connected affine algebraic group, π : H = O(G) ։ O(N), N ⊳ G, N unipotent, dim(N) = d. H ∼ = Hcoπ ⊗k O(N) = O(G/N)[x1, . . . , xd] as k-algebras.

Question

Let H be an affine noetherian Hopf k-algebra, and let N be a connected unipotent algebraic group with dim(N) = d. Suppose (*) π : H ։ ¯ H = O(N).

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 16 / 19

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SLIDE 60
  • 4. Hopf epimorphisms and crossed products

3

G connected affine algebraic group, π : H = O(G) ։ O(N), N ⊳ G, N unipotent, dim(N) = d. H ∼ = Hcoπ ⊗k O(N) = O(G/N)[x1, . . . , xd] as k-algebras.

Question

Let H be an affine noetherian Hopf k-algebra, and let N be a connected unipotent algebraic group with dim(N) = d. Suppose (*) π : H ։ ¯ H = O(N).

1

Does π factor through ˆ π : H ։ O(ˆ N), ˆ N unipotent, ˆ π conormal?

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 16 / 19

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SLIDE 61
  • 4. Hopf epimorphisms and crossed products

3

G connected affine algebraic group, π : H = O(G) ։ O(N), N ⊳ G, N unipotent, dim(N) = d. H ∼ = Hcoπ ⊗k O(N) = O(G/N)[x1, . . . , xd] as k-algebras.

Question

Let H be an affine noetherian Hopf k-algebra, and let N be a connected unipotent algebraic group with dim(N) = d. Suppose (*) π : H ։ ¯ H = O(N).

1

Does π factor through ˆ π : H ։ O(ˆ N), ˆ N unipotent, ˆ π conormal?

2

Is there a crossed product decomposition H ∼ = Hcoπ ∗σ ¯ H ?

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 16 / 19

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  • 4. Hopf epimorphisms and crossed products

Theorem

Assume set-up of above question.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 17 / 19

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  • 4. Hopf epimorphisms and crossed products

Theorem

Assume set-up of above question.

1

If H commutative then “yes” to both questions. (Example 3)

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 17 / 19

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  • 4. Hopf epimorphisms and crossed products

Theorem

Assume set-up of above question.

1

If H commutative then “yes” to both questions. (Example 3)

2

(Goodearl-Zhang, 2010) If d = 1 and GK − dim(H) = 2, then “yes” to both questions.

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 17 / 19

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  • 4. Hopf epimorphisms and crossed products

Theorem

Assume set-up of above question.

1

If H commutative then “yes” to both questions. (Example 3)

2

(Goodearl-Zhang, 2010) If d = 1 and GK − dim(H) = 2, then “yes” to both questions.

3

(Masuoka, 1991) If H is pointed, then “yes” to part (2).

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 17 / 19

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SLIDE 66

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 18 / 19

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  • 4. Hopf epimorphisms and crossed products

Consequences:

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 19 / 19

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  • 4. Hopf epimorphisms and crossed products

Consequences: Suppose π : H − → ¯ H = O(ˆ N), ˆ π conormal, dim(ˆ N) = d, and there exists a crossed product decomposition. Then

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 19 / 19

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  • 4. Hopf epimorphisms and crossed products

Consequences: Suppose π : H − → ¯ H = O(ˆ N), ˆ π conormal, dim(ˆ N) = d, and there exists a crossed product decomposition. Then Hco¯

H is a Hopf subalgebra of H, and

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 19 / 19

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  • 4. Hopf epimorphisms and crossed products

Consequences: Suppose π : H − → ¯ H = O(ˆ N), ˆ π conormal, dim(ˆ N) = d, and there exists a crossed product decomposition. Then Hco¯

H is a Hopf subalgebra of H, and

H ∼ = Hco¯

H[x1, δ1] . . . [xd, δd].

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 19 / 19

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SLIDE 71
  • 4. Hopf epimorphisms and crossed products

Consequences: Suppose π : H − → ¯ H = O(ˆ N), ˆ π conormal, dim(ˆ N) = d, and there exists a crossed product decomposition. Then Hco¯

H is a Hopf subalgebra of H, and

H ∼ = Hco¯

H[x1, δ1] . . . [xd, δd].

Hence we can (for example) get properties of H from those of Hco¯

H;

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 19 / 19

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SLIDE 72
  • 4. Hopf epimorphisms and crossed products

Consequences: Suppose π : H − → ¯ H = O(ˆ N), ˆ π conormal, dim(ˆ N) = d, and there exists a crossed product decomposition. Then Hco¯

H is a Hopf subalgebra of H, and

H ∼ = Hco¯

H[x1, δ1] . . . [xd, δd].

Hence we can (for example) get properties of H from those of Hco¯

H;

hope to classify ”almost classically unipotent” H - (i.e. when dimH = d + 1).

Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 19 / 19