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 1 Hopf algebras of small GK-dimension 2 Five-minute primer on affine algebraic groups 3 Hopf algebra epimorphisms and crossed products 4 Towards a definition of a noncommutative solvable group 5 Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 2 / 19
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
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
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
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
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
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
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
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
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: Commutative: 1 Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19
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: Commutative: 1 H = k [ X ] , ∆( X ) = X ⊗ 1 + 1 ⊗ X or H = k [ X ± 1 ] , ∆( X ) = X ⊗ X Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19
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: Commutative: 1 H = k [ X ] , ∆( X ) = X ⊗ 1 + 1 ⊗ X or H = k [ X ± 1 ] , ∆( X ) = X ⊗ X Cocommutative: Above two, and 2 Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19
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: Commutative: 1 H = k [ X ] , ∆( X ) = X ⊗ 1 + 1 ⊗ X or H = k [ X ± 1 ] , ∆( X ) = X ⊗ X Cocommutative: Above two, and 2 H = kD , D infinite dihedral group � x ± 1 , g : g 2 = 1 , gxg = x − 1 � . case t = 0 of: Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 4 / 19
2. Hopf algebras of small GK-dimension Taft algebras 3 H = T ( n , t , ξ ) = k � x , g : g n = 1 , xg = ξ gx � ∆( g ) = g ⊗ g ∆( x ) = x ⊗ g t + 1 ⊗ x Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 5 / 19
2. Hopf algebras of small GK-dimension Taft algebras 3 H = T ( n , t , ξ ) = k � x , g : g n = 1 , xg = ξ gx � ∆( g ) = g ⊗ g ∆( x ) = x ⊗ g t + 1 ⊗ x Generalised Liu algebras: Special case is: for n , w coprime 4 integers, n > 1 , w ≥ 1 , H = L ( n , w , ξ ) = k � x ± 1 , y : yx = ξ xy , y n = 1 − x nw � x group-like, ∆( y ) = y ⊗ x w + 1 ⊗ y Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 5 / 19
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
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
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
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
2. Hopf algebras of small GK-dimension B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples: Γ = Z × Z or Z ⋊ Z Group algebras: k Γ, 1 Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19
2. Hopf algebras of small GK-dimension B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples: Γ = Z × Z or Z ⋊ Z Group algebras: k Γ, 1 Enveloping algebras: U ( g ), dim ( g ) = 2 2 Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19
2. Hopf algebras of small GK-dimension B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples: Γ = Z × Z or Z ⋊ Z Group algebras: k Γ, 1 Enveloping algebras: U ( g ), dim ( g ) = 2 2 Generalised q -Borels: For n ∈ Z , q ∈ k ∗ , 3 A ( n , q ) = k � x ± 1 , y : xy = qyx � x group-like, ∆( y ) = y ⊗ 1 + x n ⊗ y Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 7 / 19
2. Hopf algebras of small GK-dimension B: H a domain of GK-dimension two (Goodearl-Zhang(’10), Wang-Zhang-Zhuang(’11)) Examples: Γ = Z × Z or Z ⋊ Z Group algebras: k Γ, 1 Enveloping algebras: U ( g ), dim ( g ) = 2 2 Generalised q -Borels: For n ∈ Z , q ∈ k ∗ , 3 A ( n , q ) = k � x ± 1 , y : xy = qyx � x group-like, ∆( y ) = y ⊗ 1 + x n ⊗ y So A (1 , q 2 ) is positive Borel in U q ( 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
2. Hopf algebras of small GK-dimension Gorenstein singularities: Special case: Let n ∈ Z , n ≥ 1, and 4 q a primitive (6 n )th root of 1 in k . Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 8 / 19
2. Hopf algebras of small GK-dimension Gorenstein singularities: Special case: Let n ∈ Z , n ≥ 1, and 4 q a primitive (6 n )th root of 1 in k . Let σ ∈ Aut ( k [ y ]), σ ( y ) = qy , and form B := k [ y ][ x ± 1 ; σ ] . Set A = k � y 2 , y 3 � ⊂ k [ y ] . Ken Brown (University of Glasgow) Noetherian Hopf algebras Shanghai 13.9.2011 8 / 19
2. Hopf algebras of small GK-dimension Gorenstein singularities: Special case: Let n ∈ Z , n ≥ 1, and 4 q a primitive (6 n )th root of 1 in k . Let σ ∈ Aut ( k [ y ]), σ ( y ) = qy , and form B := k [ y ][ x ± 1 ; σ ] . Set A = k � y 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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