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GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Nichols algebras of finite Gelfand-Kirillovdimension
Ivan Angiono
Universidad Nacional de Cordoba
Banff, Canada - September, 2015
Joint work with N. Andruskiewitsch and I. Heckenberger
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Plan of the talk
1 Gelfand-Kirillov dimension and pointed Hopf algebras.
2 Nichols algebras of diagonal type.
3 Nichols algebras of blocks.
4 Nichols algebras of decomposable modules.
5 Pre-Nichols algebras and liftings.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Plan of the talk
1 Gelfand-Kirillov dimension and pointed Hopf algebras.
2 Nichols algebras of diagonal type.
3 Nichols algebras of blocks.
4 Nichols algebras of decomposable modules.
5 Pre-Nichols algebras and liftings.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Plan of the talk
1 Gelfand-Kirillov dimension and pointed Hopf algebras.
2 Nichols algebras of diagonal type.
3 Nichols algebras of blocks.
4 Nichols algebras of decomposable modules.
5 Pre-Nichols algebras and liftings.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Plan of the talk
1 Gelfand-Kirillov dimension and pointed Hopf algebras.
2 Nichols algebras of diagonal type.
3 Nichols algebras of blocks.
4 Nichols algebras of decomposable modules.
5 Pre-Nichols algebras and liftings.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Plan of the talk
1 Gelfand-Kirillov dimension and pointed Hopf algebras.
2 Nichols algebras of diagonal type.
3 Nichols algebras of blocks.
4 Nichols algebras of decomposable modules.
5 Pre-Nichols algebras and liftings.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Definition
Let A be a finitely generated C-algebra. If V is a finite-dimensionalgenerating subspace of A and AV ,n =
∑0≤j≤n V
n, then
GKdimA := limn→∞ logn dimAV ,n;
it does not depend on the choice of V . If A is not fin. gen., then
GKdimA := supGKdimB|B ⊆ A, B finitely generated.
Example
A commutative ⇒ GKdimA ∈ N0 ∪∞; if A fin. gen.,
GKdimA = Krull dimA = dim SpecA.
(Krull dim = sup of the lengths of all chains of prime ideals).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Definition
Let A be a finitely generated C-algebra. If V is a finite-dimensionalgenerating subspace of A and AV ,n =
∑0≤j≤n V
n, then
GKdimA := limn→∞ logn dimAV ,n;
it does not depend on the choice of V . If A is not fin. gen., then
GKdimA := supGKdimB|B ⊆ A, B finitely generated.
Example
A commutative ⇒ GKdimA ∈ N0 ∪∞; if A fin. gen.,
GKdimA = Krull dimA = dim SpecA.
(Krull dim = sup of the lengths of all chains of prime ideals).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Problem
Classify Hopf algebras H with GKdimH <∞.
Let G = G (H) = x ∈ H − 0 : ∆(x) = x ⊗ x.
H is pointed if H0 :=∑
C simple subcoalgebra C = CG (H).
Example
Group algebras, enveloping algebras are pointed.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Problem
Classify Hopf algebras H with GKdimH <∞.
Let G = G (H) = x ∈ H − 0 : ∆(x) = x ⊗ x.
H is pointed if H0 :=∑
C simple subcoalgebra C = CG (H).
Example
Group algebras, enveloping algebras are pointed.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Remark
Let G be a finitely generated group. G is virtually nilpotent ornilpotent-by-finite if it has a normal nilpotent subgroup N such that G/Nis finite. Then
GKdimCG <∞ ⇐⇒ growth G <∞ ⇐⇒ G virtually nilpotent.
⇐ Wolf, Milnor; ⇒ Gromov.
Remark
Let A be an algebra with an ascending filtration. ThenGKdimA ≥ GKdim grA; = holds if grA is fg.
Thus, g Lie algebra =⇒ GKdimU(g) = dim g.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Remark
Let G be a finitely generated group. G is virtually nilpotent ornilpotent-by-finite if it has a normal nilpotent subgroup N such that G/Nis finite. Then
GKdimCG <∞ ⇐⇒ growth G <∞ ⇐⇒ G virtually nilpotent.
⇐ Wolf, Milnor; ⇒ Gromov.
Remark
Let A be an algebra with an ascending filtration. ThenGKdimA ≥ GKdim grA; = holds if grA is fg.
Thus, g Lie algebra =⇒ GKdimU(g) = dim g.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Problem
Classify pointed Hopf algebras H with GKdimH <∞.
H0 ⊂ H1 ⊂ · · · ⊂ H =⋃
n≥0 Hn,
grH associated graded (Hopf) algebra,
grH ' R#CG , R = ⊕n≥0Rn is a Hopf algebra in CG
CGYD.
R is connected and coradically graded.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Problem
Classify pointed Hopf algebras H with GKdimH <∞.
H0 ⊂ H1 ⊂ · · · ⊂ H =⋃
n≥0 Hn,
grH associated graded (Hopf) algebra,
grH ' R#CG , R = ⊕n≥0Rn is a Hopf algebra in CG
CGYD.
R is connected and coradically graded.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Problem
Classify pointed Hopf algebras H with GKdimH <∞.
H0 ⊂ H1 ⊂ · · · ⊂ H =⋃
n≥0 Hn,
grH associated graded (Hopf) algebra,
grH ' R#CG , R = ⊕n≥0Rn is a Hopf algebra in CG
CGYD.
R is connected and coradically graded.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Problem
Classify pointed Hopf algebras H with GKdimH <∞.
H0 ⊂ H1 ⊂ · · · ⊂ H =⋃
n≥0 Hn,
grH associated graded (Hopf) algebra,
grH ' R#CG , R = ⊕n≥0Rn is a Hopf algebra in CG
CGYD.
R is connected and coradically graded.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Problem
Classify pointed Hopf algebras H with GKdimH <∞.
H0 ⊂ H1 ⊂ · · · ⊂ H =⋃
n≥0 Hn,
grH associated graded (Hopf) algebra,
grH ' R#CG , R = ⊕n≥0Rn is a Hopf algebra in CG
CGYD.
R is connected and coradically graded.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
CGCGYD = category of Yetter-Drinfeld modules over CG ;
V = ⊕g∈GVg is a G -graded vector space;
V is a left G -module such that g · Vh = Vghg−1 (compatibility).
Now c ∈ GL(V ⊗ V ), c(v ⊗ w) = g · w ⊗ v , for v ∈ Vg , w ∈ V , satisfies
(c ⊗ id)(id⊗c)(c ⊗ id) = (id⊗c)(c ⊗ id)(id⊗c);
so (V , c) is a braided vector space and CGCGYD is a braided tensor
category. we may consider Hopf algebras in CG
CGYD.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Definition
Given V ∈ CGCGYD, the Nichols algebra of V is the graded Hopf algebra
B(V ) = ⊕n≥0Bn(V ) in CGCGYD such that
B0(V ) = C, B1(V ) = P(V ), B(V ) = C〈B1(V )〉.
Hence B(V ) ' T (V )/J (V ).
Remark
J (V ) maximal graded Hopf ideal generated by elements in ⊕n≥2Bn(V ).
Problem (very difficult)
Determine the ideal J (V ) (it depends crucially on the braiding c on V ).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Definition
Given V ∈ CGCGYD, the Nichols algebra of V is the graded Hopf algebra
B(V ) = ⊕n≥0Bn(V ) in CGCGYD such that
B0(V ) = C, B1(V ) = P(V ), B(V ) = C〈B1(V )〉.
Hence B(V ) ' T (V )/J (V ).
Remark
J (V ) maximal graded Hopf ideal generated by elements in ⊕n≥2Bn(V ).
Problem (very difficult)
Determine the ideal J (V ) (it depends crucially on the braiding c on V ).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Definition
Given V ∈ CGCGYD, the Nichols algebra of V is the graded Hopf algebra
B(V ) = ⊕n≥0Bn(V ) in CGCGYD such that
B0(V ) = C, B1(V ) = P(V ), B(V ) = C〈B1(V )〉.
Hence B(V ) ' T (V )/J (V ).
Remark
J (V ) maximal graded Hopf ideal generated by elements in ⊕n≥2Bn(V ).
Problem (very difficult)
Determine the ideal J (V ) (it depends crucially on the braiding c on V ).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Summarizing: H pointed Hopf algebra,
grH ' R#CG the graded Hopf alg. from its coradical filtration,
R = ⊕n≥0Rn ∈ CG
CGYD coradically gr. connected Hopf alg. Then
GKdimH <∞ (∗)=⇒ GKdim grH <∞ (∗∗)⇐⇒ GKdimR <∞,
GKdimCG <∞.
Here, converse of (*) true when grH fg; ⇐ of (**) needs anargument.
V = R1: B(V ) → R, so GKdimH <∞ =⇒ GKdimB(V ) <∞.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Summarizing: H pointed Hopf algebra,
grH ' R#CG the graded Hopf alg. from its coradical filtration,
R = ⊕n≥0Rn ∈ CG
CGYD coradically gr. connected Hopf alg. Then
GKdimH <∞ (∗)=⇒ GKdim grH <∞ (∗∗)⇐⇒ GKdimR <∞,
GKdimCG <∞.
Here, converse of (*) true when grH fg; ⇐ of (**) needs anargument.
V = R1: B(V ) → R, so GKdimH <∞ =⇒ GKdimB(V ) <∞.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Summarizing: H pointed Hopf algebra,
grH ' R#CG the graded Hopf alg. from its coradical filtration,
R = ⊕n≥0Rn ∈ CG
CGYD coradically gr. connected Hopf alg. Then
GKdimH <∞ (∗)=⇒ GKdim grH <∞ (∗∗)⇐⇒ GKdimR <∞,
GKdimCG <∞.
Here, converse of (*) true when grH fg; ⇐ of (**) needs anargument.
V = R1: B(V ) → R, so GKdimH <∞ =⇒ GKdimB(V ) <∞.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Lifting Method (Andruskiewitsch-Schneider)Let us fix a virtually nilpotent group G .
Question
Determine all V ∈ CGCGYD such that GKdimB(V ) <∞.
Question
If V ∈ CGCGYD has GKdimB(V ) <∞, then find all coradically graded
connected Hopf algebras R ∈ CGCGYD such that
R1 ' V and GKdimR <∞.
Question
Compute all liftings of R, i.e. all H such that grH ' R#G .
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Lifting Method (Andruskiewitsch-Schneider)Let us fix a virtually nilpotent group G .
Question
Determine all V ∈ CGCGYD such that GKdimB(V ) <∞.
Question
If V ∈ CGCGYD has GKdimB(V ) <∞, then find all coradically graded
connected Hopf algebras R ∈ CGCGYD such that
R1 ' V and GKdimR <∞.
Question
Compute all liftings of R, i.e. all H such that grH ' R#G .
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
GK dimensionPointed Hopf algebrasNichols algebras
Lifting Method (Andruskiewitsch-Schneider)Let us fix a virtually nilpotent group G .
Question
Determine all V ∈ CGCGYD such that GKdimB(V ) <∞.
Question
If V ∈ CGCGYD has GKdimB(V ) <∞, then find all coradically graded
connected Hopf algebras R ∈ CGCGYD such that
R1 ' V and GKdimR <∞.
Question
Compute all liftings of R, i.e. all H such that grH ' R#G .
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Γ abelian group, V ∈ CΓCΓYD V = ⊕g∈ΓVg , each Vg ∈ Γ−Mod.
Assume dimV <∞ =⇒ Vg is a direct sum of indecomposables.
Point of label q ∈ C×: braided v. sp. (V , c) of dim. 1, c = q id.Cχg ∈ CΓ
CΓYD of dimension 1, homogeneous of degree g , Γ acts by
χ ∈ Γ and χ(g).
Γ = 〈g〉 ' Z. V(ε, `) ∈ CZCZYD homogeneous of degree g, dimension
` > 1, the action of g given by a Jordan block of size ` andeigenvalue ε. These braided vector spaces are called blocks.
Every finite-dimensional indecomposable, not simple, in CΓCΓYD is a
block as braided vector space.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Γ abelian group, V ∈ CΓCΓYD V = ⊕g∈ΓVg , each Vg ∈ Γ−Mod.
Assume dimV <∞ =⇒ Vg is a direct sum of indecomposables.
Point of label q ∈ C×: braided v. sp. (V , c) of dim. 1, c = q id.Cχg ∈ CΓ
CΓYD of dimension 1, homogeneous of degree g , Γ acts by
χ ∈ Γ and χ(g).
Γ = 〈g〉 ' Z. V(ε, `) ∈ CZCZYD homogeneous of degree g, dimension
` > 1, the action of g given by a Jordan block of size ` andeigenvalue ε. These braided vector spaces are called blocks.
Every finite-dimensional indecomposable, not simple, in CΓCΓYD is a
block as braided vector space.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Γ abelian group, V ∈ CΓCΓYD V = ⊕g∈ΓVg , each Vg ∈ Γ−Mod.
Assume dimV <∞ =⇒ Vg is a direct sum of indecomposables.
Point of label q ∈ C×: braided v. sp. (V , c) of dim. 1, c = q id.Cχg ∈ CΓ
CΓYD of dimension 1, homogeneous of degree g , Γ acts by
χ ∈ Γ and χ(g).
Γ = 〈g〉 ' Z. V(ε, `) ∈ CZCZYD homogeneous of degree g, dimension
` > 1, the action of g given by a Jordan block of size ` andeigenvalue ε. These braided vector spaces are called blocks.
Every finite-dimensional indecomposable, not simple, in CΓCΓYD is a
block as braided vector space.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Γ abelian group, V ∈ CΓCΓYD V = ⊕g∈ΓVg , each Vg ∈ Γ−Mod.
Assume dimV <∞ =⇒ Vg is a direct sum of indecomposables.
Point of label q ∈ C×: braided v. sp. (V , c) of dim. 1, c = q id.Cχg ∈ CΓ
CΓYD of dimension 1, homogeneous of degree g , Γ acts by
χ ∈ Γ and χ(g).
Γ = 〈g〉 ' Z. V(ε, `) ∈ CZCZYD homogeneous of degree g, dimension
` > 1, the action of g given by a Jordan block of size ` andeigenvalue ε. These braided vector spaces are called blocks.
Every finite-dimensional indecomposable, not simple, in CΓCΓYD is a
block as braided vector space.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
(V , c) is of diagonal type: ∃ a basis (xi )i∈I` and a matrix q = (qij)i,j∈Iθ ,qii 6= 1 for all i , with connected diagram, such that
c(xi ⊗ xj) = qij xj ⊗ xi , i , j ∈ I`.
Question
Classify all braided vector spaces (V , c) of diagonal type with such thatGKdimB(V ) <∞.
q is
generic: qii , qijqji ∈ C ∪ 1 −G∞, ∀i 6= j ∈ Iθ;
torsion: qii , qijqji ∈ G∞, ∀i 6= j ∈ Iθ;
semigeneric: otherwise.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
q of Cartan type: ∃ a generalized Cartan matrix a = (aij)i,j∈Iθ such that
qijqji = qaijii , ∀i 6= j ∈ Iθ.
Remark
q of Cartan type is either generic or of torsion.
(Andruskiewitsch–A–Rosso) Assume q gen. ThenGKdimB(V ) <∞⇔ q of finite Cartan type.
Conjecture
q of Cartan type, GKdimB(V ) <∞⇒ finite type.
We know: if a is of affine type, then GKdimB(V ) =∞.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
q of Cartan type: ∃ a generalized Cartan matrix a = (aij)i,j∈Iθ such that
qijqji = qaijii , ∀i 6= j ∈ Iθ.
Remark
q of Cartan type is either generic or of torsion.
(Andruskiewitsch–A–Rosso) Assume q gen. ThenGKdimB(V ) <∞⇔ q of finite Cartan type.
Conjecture
q of Cartan type, GKdimB(V ) <∞⇒ finite type.
We know: if a is of affine type, then GKdimB(V ) =∞.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
q of Cartan type: ∃ a generalized Cartan matrix a = (aij)i,j∈Iθ such that
qijqji = qaijii , ∀i 6= j ∈ Iθ.
Remark
q of Cartan type is either generic or of torsion.
(Andruskiewitsch–A–Rosso) Assume q gen. ThenGKdimB(V ) <∞⇔ q of finite Cartan type.
Conjecture
q of Cartan type, GKdimB(V ) <∞⇒ finite type.
We know: if a is of affine type, then GKdimB(V ) =∞.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
q of Cartan type: ∃ a generalized Cartan matrix a = (aij)i,j∈Iθ such that
qijqji = qaijii , ∀i 6= j ∈ Iθ.
Remark
q of Cartan type is either generic or of torsion.
(Andruskiewitsch–A–Rosso) Assume q gen. ThenGKdimB(V ) <∞⇔ q of finite Cartan type.
Conjecture
q of Cartan type, GKdimB(V ) <∞⇒ finite type.
We know: if a is of affine type, then GKdimB(V ) =∞.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Conjecture
q torsion, GKdimB(V ) <∞⇒ dimB(V ) <∞.
More generally,
Conjecture
GKdimB(V ) <∞⇒ GRS is finite (it has a PBW basis with finite set ofgenerators) Heckenberger’s classification.
Theorem
Last conjecture is true when dimV = 2.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Conjecture
q torsion, GKdimB(V ) <∞⇒ dimB(V ) <∞.
More generally,
Conjecture
GKdimB(V ) <∞⇒ GRS is finite (it has a PBW basis with finite set ofgenerators) Heckenberger’s classification.
Theorem
Last conjecture is true when dimV = 2.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Conjecture
q torsion, GKdimB(V ) <∞⇒ dimB(V ) <∞.
More generally,
Conjecture
GKdimB(V ) <∞⇒ GRS is finite (it has a PBW basis with finite set ofgenerators) Heckenberger’s classification.
Theorem
Last conjecture is true when dimV = 2.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Block V(ε, `): ∃ a basis (xi )i∈I` such that
c(xi ⊗ xj) =
εx1 ⊗ xi , j = 1
(εxj + xj−1)⊗ xi , j ≥ 2,i ∈ I`.
Theorem
1 GKdimB(V(ε, `)) <∞ ⇐⇒ ` = 2 and ε ∈ ±1.
2 B(V(1, 2)) = C〈x1, x2|x2x1 − x1x2 + 12x
21 〉 Jordan plane.
GKdimB(V(1, 2)) = 2, with basis xa1xb2 : a, b ∈ N0.
3 B(V(−1, 2)) = C〈x1, x2|x21 , x2x12 − x12x2 − x1x12〉 super Jordan
plane. Here x12 = adc x2 x1 = x2x1 + x1x2.
GKdimB(V(−1, 2)) = 2, with basisxa1xb12x
c2 : a ∈ 0, 1, b, c ∈ N0.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Block V(ε, `): ∃ a basis (xi )i∈I` such that
c(xi ⊗ xj) =
εx1 ⊗ xi , j = 1
(εxj + xj−1)⊗ xi , j ≥ 2,i ∈ I`.
Theorem
1 GKdimB(V(ε, `)) <∞ ⇐⇒ ` = 2 and ε ∈ ±1.
2 B(V(1, 2)) = C〈x1, x2|x2x1 − x1x2 + 12x
21 〉 Jordan plane.
GKdimB(V(1, 2)) = 2, with basis xa1xb2 : a, b ∈ N0.
3 B(V(−1, 2)) = C〈x1, x2|x21 , x2x12 − x12x2 − x1x12〉 super Jordan
plane. Here x12 = adc x2 x1 = x2x1 + x1x2.
GKdimB(V(−1, 2)) = 2, with basisxa1xb12x
c2 : a ∈ 0, 1, b, c ∈ N0.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Block V(ε, `): ∃ a basis (xi )i∈I` such that
c(xi ⊗ xj) =
εx1 ⊗ xi , j = 1
(εxj + xj−1)⊗ xi , j ≥ 2,i ∈ I`.
Theorem
1 GKdimB(V(ε, `)) <∞ ⇐⇒ ` = 2 and ε ∈ ±1.
2 B(V(1, 2)) = C〈x1, x2|x2x1 − x1x2 + 12x
21 〉 Jordan plane.
GKdimB(V(1, 2)) = 2, with basis xa1xb2 : a, b ∈ N0.
3 B(V(−1, 2)) = C〈x1, x2|x21 , x2x12 − x12x2 − x1x12〉 super Jordan
plane. Here x12 = adc x2 x1 = x2x1 + x1x2.
GKdimB(V(−1, 2)) = 2, with basisxa1xb12x
c2 : a ∈ 0, 1, b, c ∈ N0.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Block V(ε, `): ∃ a basis (xi )i∈I` such that
c(xi ⊗ xj) =
εx1 ⊗ xi , j = 1
(εxj + xj−1)⊗ xi , j ≥ 2,i ∈ I`.
Theorem
1 GKdimB(V(ε, `)) <∞ ⇐⇒ ` = 2 and ε ∈ ±1.
2 B(V(1, 2)) = C〈x1, x2|x2x1 − x1x2 + 12x
21 〉 Jordan plane.
GKdimB(V(1, 2)) = 2, with basis xa1xb2 : a, b ∈ N0.
3 B(V(−1, 2)) = C〈x1, x2|x21 , x2x12 − x12x2 − x1x12〉 super Jordan
plane. Here x12 = adc x2 x1 = x2x1 + x1x2.
GKdimB(V(−1, 2)) = 2, with basisxa1xb12x
c2 : a ∈ 0, 1, b, c ∈ N0.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Γ abelian: V ∈ CΓCΓYD → V = V1 ⊕ · · · ⊕ Vt ⊕ Vt+1 ⊕ · · · ⊕ Vθ, where
V1, . . . ,Vt ∈ CΓCΓYD are blocks;
Vt+1, . . . ,Vθ ∈ CΓCΓYD are points.
V ∈ CΓCΓYD, dimV = 3, not of diagonal type
⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,
χ1, χ2 ∈ Γ and η : Γ→ C is a (χ1, χ1)-derivation.
V1 = Vg1 (χ1, η) ∈ CΓCΓYD is indecomposable with basis (xi )i∈I2 ;
V2 = Cχ2g2∈ CΓ
CΓYD is irreducible with base (x3);
η(g1) 6= 0 (V not diagonal type); may suppose η(g1) = 1.
Set qij = χj(gi ), i , j ∈ I2; ε = q11; a = q−121 η(g2).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Γ abelian: V ∈ CΓCΓYD → V = V1 ⊕ · · · ⊕ Vt ⊕ Vt+1 ⊕ · · · ⊕ Vθ, where
V1, . . . ,Vt ∈ CΓCΓYD are blocks;
Vt+1, . . . ,Vθ ∈ CΓCΓYD are points.
V ∈ CΓCΓYD, dimV = 3, not of diagonal type
⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,
χ1, χ2 ∈ Γ and η : Γ→ C is a (χ1, χ1)-derivation.
V1 = Vg1 (χ1, η) ∈ CΓCΓYD is indecomposable with basis (xi )i∈I2 ;
V2 = Cχ2g2∈ CΓ
CΓYD is irreducible with base (x3);
η(g1) 6= 0 (V not diagonal type); may suppose η(g1) = 1.
Set qij = χj(gi ), i , j ∈ I2; ε = q11; a = q−121 η(g2).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Γ abelian: V ∈ CΓCΓYD → V = V1 ⊕ · · · ⊕ Vt ⊕ Vt+1 ⊕ · · · ⊕ Vθ, where
V1, . . . ,Vt ∈ CΓCΓYD are blocks;
Vt+1, . . . ,Vθ ∈ CΓCΓYD are points.
V ∈ CΓCΓYD, dimV = 3, not of diagonal type
⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,
χ1, χ2 ∈ Γ and η : Γ→ C is a (χ1, χ1)-derivation.
V1 = Vg1 (χ1, η) ∈ CΓCΓYD is indecomposable with basis (xi )i∈I2 ;
V2 = Cχ2g2∈ CΓ
CΓYD is irreducible with base (x3);
η(g1) 6= 0 (V not diagonal type); may suppose η(g1) = 1.
Set qij = χj(gi ), i , j ∈ I2; ε = q11; a = q−121 η(g2).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Γ abelian: V ∈ CΓCΓYD → V = V1 ⊕ · · · ⊕ Vt ⊕ Vt+1 ⊕ · · · ⊕ Vθ, where
V1, . . . ,Vt ∈ CΓCΓYD are blocks;
Vt+1, . . . ,Vθ ∈ CΓCΓYD are points.
V ∈ CΓCΓYD, dimV = 3, not of diagonal type
⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,
χ1, χ2 ∈ Γ and η : Γ→ C is a (χ1, χ1)-derivation.
V1 = Vg1 (χ1, η) ∈ CΓCΓYD is indecomposable with basis (xi )i∈I2 ;
V2 = Cχ2g2∈ CΓ
CΓYD is irreducible with base (x3);
η(g1) 6= 0 (V not diagonal type); may suppose η(g1) = 1.
Set qij = χj(gi ), i , j ∈ I2; ε = q11; a = q−121 η(g2).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Γ abelian: V ∈ CΓCΓYD → V = V1 ⊕ · · · ⊕ Vt ⊕ Vt+1 ⊕ · · · ⊕ Vθ, where
V1, . . . ,Vt ∈ CΓCΓYD are blocks;
Vt+1, . . . ,Vθ ∈ CΓCΓYD are points.
V ∈ CΓCΓYD, dimV = 3, not of diagonal type
⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,
χ1, χ2 ∈ Γ and η : Γ→ C is a (χ1, χ1)-derivation.
V1 = Vg1 (χ1, η) ∈ CΓCΓYD is indecomposable with basis (xi )i∈I2 ;
V2 = Cχ2g2∈ CΓ
CΓYD is irreducible with base (x3);
η(g1) 6= 0 (V not diagonal type); may suppose η(g1) = 1.
Set qij = χj(gi ), i , j ∈ I2; ε = q11; a = q−121 η(g2).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Γ abelian: V ∈ CΓCΓYD → V = V1 ⊕ · · · ⊕ Vt ⊕ Vt+1 ⊕ · · · ⊕ Vθ, where
V1, . . . ,Vt ∈ CΓCΓYD are blocks;
Vt+1, . . . ,Vθ ∈ CΓCΓYD are points.
V ∈ CΓCΓYD, dimV = 3, not of diagonal type
⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,
χ1, χ2 ∈ Γ and η : Γ→ C is a (χ1, χ1)-derivation.
V1 = Vg1 (χ1, η) ∈ CΓCΓYD is indecomposable with basis (xi )i∈I2 ;
V2 = Cχ2g2∈ CΓ
CΓYD is irreducible with base (x3);
η(g1) 6= 0 (V not diagonal type); may suppose η(g1) = 1.
Set qij = χj(gi ), i , j ∈ I2; ε = q11; a = q−121 η(g2).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Γ abelian: V ∈ CΓCΓYD → V = V1 ⊕ · · · ⊕ Vt ⊕ Vt+1 ⊕ · · · ⊕ Vθ, where
V1, . . . ,Vt ∈ CΓCΓYD are blocks;
Vt+1, . . . ,Vθ ∈ CΓCΓYD are points.
V ∈ CΓCΓYD, dimV = 3, not of diagonal type
⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,
χ1, χ2 ∈ Γ and η : Γ→ C is a (χ1, χ1)-derivation.
V1 = Vg1 (χ1, η) ∈ CΓCΓYD is indecomposable with basis (xi )i∈I2 ;
V2 = Cχ2g2∈ CΓ
CΓYD is irreducible with base (x3);
η(g1) 6= 0 (V not diagonal type); may suppose η(g1) = 1.
Set qij = χj(gi ), i , j ∈ I2; ε = q11; a = q−121 η(g2).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
The braiding is given in the basis (xi )i∈I3 by
(c(xi ⊗ xj))i,j∈I3 =
εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1
εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2
q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3
.
We may assume ε2 = 1: c2|V1⊗V2
= id ⇐⇒ q12q21 = 1 and a = 0.
c2|V1⊗V2
is determined by q12q21 and the parameter a.
Definition
q12q21 = the interaction between the block and the point; it is
weak if q12q21 = 1, mild if q12q21 = −1, strong if q12q21 /∈ ±1.
The ghost: a normalized version of a,
G =
−2a, ε = 1,
a, ε = −1.
If G ∈ N, then we say that the ghost is discrete.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
The braiding is given in the basis (xi )i∈I3 by
(c(xi ⊗ xj))i,j∈I3 =
εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1
εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2
q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3
.
We may assume ε2 = 1: c2|V1⊗V2
= id ⇐⇒ q12q21 = 1 and a = 0.
c2|V1⊗V2
is determined by q12q21 and the parameter a.
Definition
q12q21 = the interaction between the block and the point; it is
weak if q12q21 = 1, mild if q12q21 = −1, strong if q12q21 /∈ ±1.
The ghost: a normalized version of a,
G =
−2a, ε = 1,
a, ε = −1.
If G ∈ N, then we say that the ghost is discrete.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
The braiding is given in the basis (xi )i∈I3 by
(c(xi ⊗ xj))i,j∈I3 =
εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1
εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2
q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3
.
We may assume ε2 = 1: c2|V1⊗V2
= id ⇐⇒ q12q21 = 1 and a = 0.
c2|V1⊗V2
is determined by q12q21 and the parameter a.
Definition
q12q21 = the interaction between the block and the point; it is
weak if q12q21 = 1, mild if q12q21 = −1, strong if q12q21 /∈ ±1.
The ghost: a normalized version of a,
G =
−2a, ε = 1,
a, ε = −1.
If G ∈ N, then we say that the ghost is discrete.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
The braiding is given in the basis (xi )i∈I3 by
(c(xi ⊗ xj))i,j∈I3 =
εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1
εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2
q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3
.
We may assume ε2 = 1: c2|V1⊗V2
= id ⇐⇒ q12q21 = 1 and a = 0.
c2|V1⊗V2
is determined by q12q21 and the parameter a.
Definition
q12q21 = the interaction between the block and the point; it is
weak if q12q21 = 1, mild if q12q21 = −1, strong if q12q21 /∈ ±1.
The ghost: a normalized version of a,
G =
−2a, ε = 1,
a, ε = −1.
If G ∈ N, then we say that the ghost is discrete.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
The braiding is given in the basis (xi )i∈I3 by
(c(xi ⊗ xj))i,j∈I3 =
εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1
εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2
q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3
.
We may assume ε2 = 1: c2|V1⊗V2
= id ⇐⇒ q12q21 = 1 and a = 0.
c2|V1⊗V2
is determined by q12q21 and the parameter a.
Definition
q12q21 = the interaction between the block and the point; it is
weak if q12q21 = 1, mild if q12q21 = −1, strong if q12q21 /∈ ±1.
The ghost: a normalized version of a,
G =
−2a, ε = 1,
a, ε = −1.
If G ∈ N, then we say that the ghost is discrete.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
The braiding is given in the basis (xi )i∈I3 by
(c(xi ⊗ xj))i,j∈I3 =
εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1
εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2
q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3
.
We may assume ε2 = 1: c2|V1⊗V2
= id ⇐⇒ q12q21 = 1 and a = 0.
c2|V1⊗V2
is determined by q12q21 and the parameter a.
Definition
q12q21 = the interaction between the block and the point; it is
weak if q12q21 = 1, mild if q12q21 = −1, strong if q12q21 /∈ ±1.
The ghost: a normalized version of a,
G =
−2a, ε = 1,
a, ε = −1.
If G ∈ N, then we say that the ghost is discrete.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Theorem
V b. v. s., braiding as above, GKdimB(V ) <∞⇒
interaction ε q22 G B(V ) GKdimweak ±1 1 or /∈ G∞ 0 B(V(ε, 1))⊗B(Cx3) 3
∈ G∞ − 1 21 1 discrete B(L(1, 1,G)) G + 3
−1 discrete B(L(1,−1,G)) 2∈ G′3 1 B(L(1, ω,G)) 2
−1 1 discrete B(L(−1, 1,G)) G + 3−1 discrete B(L(−1,−1,G)) G + 2
mild −1 −1 1 B(C) 2
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Theorem
V generators and relations
L(1, 1,G) C〈x1, x2, x3|x2x1 − x1x2 +12x2
1 , x1x3 − q12 x3x1,z1+G , ztzt+1 − q21q22 zt+1zt , 0 ≤ t < G〉
L(1,−1,G) C〈x1, x2, x3|x2x1 − x1x2 +12x2
1 ,x1x3 − q12 x3x1, z1+G , z
2t , 0 ≤ t ≤ G〉
L(−1, 1,G) C〈x1, x2, x3|x21 , x2x12 − x12x2 − x1x12,
x1x3 − q12 x3x1, x12x3 − q212x3x12, z1+2G ,
z22k+1, z2kz2k+1 − q21q22 z2k+1z2k , 0 ≤ k < G〉
L(−1,−1,G) C〈x1, x2, x3|x21 , x2x12 − x12x2 − x1x12, x
23 ,
x1x3 − q12 x3x1, x12x3 − q212x3x12, z1+2G ,
z22k , z2k−1z2k − q21q22 z2kz2k−1, 0 < k ≤ G〉
L(1, ω, 1) C〈x1, x2, x3|x2x1 − x1x2 +12x2
1 ,x1x3 − q12 x3x1, z2, x
33 , z
31 , z
31,0〉
C C〈x1, x2, x3|x21 , x2x12 − x12x2 − x1x12,
x23 , f
21 , z
21 , x12x3 − q2
12x3x12,x2z1 + q12z1x2 − q12 f0x2 − 1
2f1〉
Here, zt = (adc x2)tx3, ft = (adc x1)zt .
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Theorem
V b. v. s., braiding as above. Assume Conjecture on diagonal braidingsis true. GKdimB(V ) <∞⇒
interaction ε diagram of VJ (Gi ,Gj)
weak 1ωi
ω2 −1j
(0, 1)
(1, 0)−1i
−1 −1j
(1, 0)
(2, 0)−1i
ω −1j
(1, 0)
−1i
r−1 rj
(1, 0)
−1i
r−1 rj,
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
One blockOne block, one pointOne block, several point
Theorem
interaction ε diagram of VJ (Gi ,Gj)
mild, (−1, 1) −1 −1i
−1 −1j
(1, 0)
weak 1 (1,0,...,0)−1 −1 −1 −1 −1 −1
(1,0,...,0)−1 ω ω2
ω ω2
(1,0,...,0)−1 ω ω2
ω2 ω
mild, −1 (1,0,...,0)−1 −1 −1 −1 −1 −1
(−1, 1, . . . , 1)
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
R = ⊕n≥0Rn ∈ H
HYD graded connected Hopf algebra, V = R1
R is a Nichols algebra when
1 R is generated by V : R is a pre-Nichols algebra (Masuoka).
2 R is coradically graded ⇐⇒ P(R) = V : R is a post-Nicholsalgebra.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
R = ⊕n≥0Rn ∈ H
HYD graded connected Hopf algebra, V = R1
R is a Nichols algebra when
1 R is generated by V : R is a pre-Nichols algebra (Masuoka).
2 R is coradically graded ⇐⇒ P(R) = V : R is a post-Nicholsalgebra.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
T (V )
Ω=⊕n≥2Ωn
**//
""
B(V )
!!
// T c(V )
B
==
E
<<
T (V ) free algebra T c(V ) free coalgebra B(V ) Nichols alg.Ωn quantum symm. B pre-Nichols E post-Nichols
Definition
Pre(V ): poset of pre-Nichols, ≤ is ; min. T (V ), max. B(V ).Post(V ): poset of post-Nichols, ≤ is ⊆; min. B(V ), max. T c(V ).
Remark
dimV <∞⇒ Φ : Pre(V )→ Post(V ∗), Φ(R) = Rd , anti-isom.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
T (V )
Ω=⊕n≥2Ωn
**//
""
B(V )
!!
// T c(V )
B
==
E
<<
T (V ) free algebra T c(V ) free coalgebra B(V ) Nichols alg.Ωn quantum symm. B pre-Nichols E post-Nichols
Definition
Pre(V ): poset of pre-Nichols, ≤ is ; min. T (V ), max. B(V ).Post(V ): poset of post-Nichols, ≤ is ⊆; min. B(V ), max. T c(V ).
Remark
dimV <∞⇒ Φ : Pre(V )→ Post(V ∗), Φ(R) = Rd , anti-isom.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Let K be a Hopf algebra, V ∈ KKYD finite-dimensional.
Lemma
B a pre-Nichols algebra of a V , E = Bd . Then GKdim E ≤ GKdimB. If Eis fin. gen., then GKdim E = GKdimB.
Definition
V is pre-bounded if every chain
· · · < B[3] < B[2] < B[1] < B[0] = B(V ), (1)
of pre-Nichols algebras over V with finite GKdim, is finite.
Lemma
E ∈ KKYD post-Nichols algebra of V , GKdim E <∞. If V ∗ is
pre-bounded, then E is fin. gen. and GKdim E = GKdim Ed .In particular, if the only pre-Nichols algebra of V ∗ with finite GKdim isB(V ∗), then E = B(V ).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Let K be a Hopf algebra, V ∈ KKYD finite-dimensional.
Lemma
B a pre-Nichols algebra of a V , E = Bd . Then GKdim E ≤ GKdimB. If Eis fin. gen., then GKdim E = GKdimB.
Definition
V is pre-bounded if every chain
· · · < B[3] < B[2] < B[1] < B[0] = B(V ), (1)
of pre-Nichols algebras over V with finite GKdim, is finite.
Lemma
E ∈ KKYD post-Nichols algebra of V , GKdim E <∞. If V ∗ is
pre-bounded, then E is fin. gen. and GKdim E = GKdim Ed .In particular, if the only pre-Nichols algebra of V ∗ with finite GKdim isB(V ∗), then E = B(V ).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Let K be a Hopf algebra, V ∈ KKYD finite-dimensional.
Lemma
B a pre-Nichols algebra of a V , E = Bd . Then GKdim E ≤ GKdimB. If Eis fin. gen., then GKdim E = GKdimB.
Definition
V is pre-bounded if every chain
· · · < B[3] < B[2] < B[1] < B[0] = B(V ), (1)
of pre-Nichols algebras over V with finite GKdim, is finite.
Lemma
E ∈ KKYD post-Nichols algebra of V , GKdim E <∞. If V ∗ is
pre-bounded, then E is fin. gen. and GKdim E = GKdim Ed .In particular, if the only pre-Nichols algebra of V ∗ with finite GKdim isB(V ∗), then E = B(V ).
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Example
(V , c) θ := dimV <∞; c = the usual flip. Then
B(V ) = S(V ), T (V ) ' U(L(V )) (free Lie algebra on V ).
B a pre-Nichols algebra of V ⇒ U(P(B)); P(B) graded Lie algebragenerated by V .
L N-graded Lie algebra generated by L1 ' V ⇒ U(L) is apre-Nichols algebra of V . L finite-dimensional ⇒ GKdimU(L) <∞.Such Lie algebras are nilpotent, there are infinitely many.
Remark
The only pre-Nichols or post-Nichols algebra of V with finite GKdim isB(V ) when:
V is of finite generic Cartan type (Andruskiewitsch-Schneider),
V is of Jordan or super Jordan type.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Example
(V , c) θ := dimV <∞; c = the usual flip. Then
B(V ) = S(V ), T (V ) ' U(L(V )) (free Lie algebra on V ).
B a pre-Nichols algebra of V ⇒ U(P(B)); P(B) graded Lie algebragenerated by V .
L N-graded Lie algebra generated by L1 ' V ⇒ U(L) is apre-Nichols algebra of V . L finite-dimensional ⇒ GKdimU(L) <∞.Such Lie algebras are nilpotent, there are infinitely many.
Remark
The only pre-Nichols or post-Nichols algebra of V with finite GKdim isB(V ) when:
V is of finite generic Cartan type (Andruskiewitsch-Schneider),
V is of Jordan or super Jordan type.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Example
(V , c) θ := dimV <∞; c = the usual flip. Then
B(V ) = S(V ), T (V ) ' U(L(V )) (free Lie algebra on V ).
B a pre-Nichols algebra of V ⇒ U(P(B)); P(B) graded Lie algebragenerated by V .
L N-graded Lie algebra generated by L1 ' V ⇒ U(L) is apre-Nichols algebra of V . L finite-dimensional ⇒ GKdimU(L) <∞.Such Lie algebras are nilpotent, there are infinitely many.
Remark
The only pre-Nichols or post-Nichols algebra of V with finite GKdim isB(V ) when:
V is of finite generic Cartan type (Andruskiewitsch-Schneider),
V is of Jordan or super Jordan type.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Example
(V , c) θ := dimV <∞; c = the usual flip. Then
B(V ) = S(V ), T (V ) ' U(L(V )) (free Lie algebra on V ).
B a pre-Nichols algebra of V ⇒ U(P(B)); P(B) graded Lie algebragenerated by V .
L N-graded Lie algebra generated by L1 ' V ⇒ U(L) is apre-Nichols algebra of V . L finite-dimensional ⇒ GKdimU(L) <∞.Such Lie algebras are nilpotent, there are infinitely many.
Remark
The only pre-Nichols or post-Nichols algebra of V with finite GKdim isB(V ) when:
V is of finite generic Cartan type (Andruskiewitsch-Schneider),
V is of Jordan or super Jordan type.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Example
(V , c) θ := dimV <∞; c = the usual flip. Then
B(V ) = S(V ), T (V ) ' U(L(V )) (free Lie algebra on V ).
B a pre-Nichols algebra of V ⇒ U(P(B)); P(B) graded Lie algebragenerated by V .
L N-graded Lie algebra generated by L1 ' V ⇒ U(L) is apre-Nichols algebra of V . L finite-dimensional ⇒ GKdimU(L) <∞.Such Lie algebras are nilpotent, there are infinitely many.
Remark
The only pre-Nichols or post-Nichols algebra of V with finite GKdim isB(V ) when:
V is of finite generic Cartan type (Andruskiewitsch-Schneider),
V is of Jordan or super Jordan type.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Example
(V , c) θ := dimV <∞; c = the usual flip. Then
B(V ) = S(V ), T (V ) ' U(L(V )) (free Lie algebra on V ).
B a pre-Nichols algebra of V ⇒ U(P(B)); P(B) graded Lie algebragenerated by V .
L N-graded Lie algebra generated by L1 ' V ⇒ U(L) is apre-Nichols algebra of V . L finite-dimensional ⇒ GKdimU(L) <∞.Such Lie algebras are nilpotent, there are infinitely many.
Remark
The only pre-Nichols or post-Nichols algebra of V with finite GKdim isB(V ) when:
V is of finite generic Cartan type (Andruskiewitsch-Schneider),
V is of Jordan or super Jordan type.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Example
(V , c) θ := dimV <∞; c = the usual flip. Then
B(V ) = S(V ), T (V ) ' U(L(V )) (free Lie algebra on V ).
B a pre-Nichols algebra of V ⇒ U(P(B)); P(B) graded Lie algebragenerated by V .
L N-graded Lie algebra generated by L1 ' V ⇒ U(L) is apre-Nichols algebra of V . L finite-dimensional ⇒ GKdimU(L) <∞.Such Lie algebras are nilpotent, there are infinitely many.
Remark
The only pre-Nichols or post-Nichols algebra of V with finite GKdim isB(V ) when:
V is of finite generic Cartan type (Andruskiewitsch-Schneider),
V is of Jordan or super Jordan type.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Theorem
1 D = (g , χ, η) a Jordanian YD-triple for CG , V = Vg (χ, η). Letλ ∈ C be such that
λ = 0, if χ2 6= ε.
U = U(D, λ) = T (V )#CG/〈x2x1 − x1x2 + 12x
21 − λ(1− g2)〉 is a
lifting of a Jordan plane. Moreover, every lifting of a Jordan plane isU for some data.
2 D = (g , χ, η) a super Jordanian YD-triple for CG , V = Vg (χ, η).Let λ ∈ C be such that
λ = 0, if χ2 6= ε.
U = U(D, λ) =T (V )#CG/〈x2
1 − λ(1− g2), x2x12 − x12x2 − x1x12 + 2λx2 + λx1g2〉
is a lifting of a super Jordan plane. Moreover, every lifting of asuper Jordan plane is U for some data.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Theorem
1 D = (g , χ, η) a Jordanian YD-triple for CG , V = Vg (χ, η). Letλ ∈ C be such that
λ = 0, if χ2 6= ε.
U = U(D, λ) = T (V )#CG/〈x2x1 − x1x2 + 12x
21 − λ(1− g2)〉 is a
lifting of a Jordan plane. Moreover, every lifting of a Jordan plane isU for some data.
2 D = (g , χ, η) a super Jordanian YD-triple for CG , V = Vg (χ, η).Let λ ∈ C be such that
λ = 0, if χ2 6= ε.
U = U(D, λ) =T (V )#CG/〈x2
1 − λ(1− g2), x2x12 − x12x2 − x1x12 + 2λx2 + λx1g2〉
is a lifting of a super Jordan plane. Moreover, every lifting of asuper Jordan plane is U for some data.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
Theorem
1 D = (g , χ, η) a Jordanian YD-triple for CG , V = Vg (χ, η). Letλ ∈ C be such that
λ = 0, if χ2 6= ε.
U = U(D, λ) = T (V )#CG/〈x2x1 − x1x2 + 12x
21 − λ(1− g2)〉 is a
lifting of a Jordan plane. Moreover, every lifting of a Jordan plane isU for some data.
2 D = (g , χ, η) a super Jordanian YD-triple for CG , V = Vg (χ, η).Let λ ∈ C be such that
λ = 0, if χ2 6= ε.
U = U(D, λ) =T (V )#CG/〈x2
1 − λ(1− g2), x2x12 − x12x2 − x1x12 + 2λx2 + λx1g2〉
is a lifting of a super Jordan plane. Moreover, every lifting of asuper Jordan plane is U for some data.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
q = (qij)i,j∈Iθ V . Bq = B(V ) = T (V )/Jq.
cqij := −min n ∈ N0 : (n + 1)qii (1− qniiqijqji ) = 0 , i 6= j .
Assume dimB(V ) <∞.
i ∈ I is a Cartan vertex of q if qijqji(?)= q
cqijii , for all j 6= i .
α = sqi1si2 . . . sik (αi ) ∈ ∆q+ Cartan root of q if i ∈ I Cartan vertex of
ρik . . . ρi2ρi1 (q).
Definition
S = set of generators of Jq [A]. Let Jq ⊃ Iq := 〈S ∪S2 −S1〉
S1 = powers root vectors ENαα , α Cartan root;
S2 = q Serre rel.(adc Ei )1−cqij Ej , i 6= j such that 1− cqij = ord qii .
The distinguished pre-Nichols algebra of V is Bq = T (V )/Iq.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
q = (qij)i,j∈Iθ V . Bq = B(V ) = T (V )/Jq.
cqij := −min n ∈ N0 : (n + 1)qii (1− qniiqijqji ) = 0 , i 6= j .
Assume dimB(V ) <∞.
i ∈ I is a Cartan vertex of q if qijqji(?)= q
cqijii , for all j 6= i .
α = sqi1si2 . . . sik (αi ) ∈ ∆q+ Cartan root of q if i ∈ I Cartan vertex of
ρik . . . ρi2ρi1 (q).
Definition
S = set of generators of Jq [A]. Let Jq ⊃ Iq := 〈S ∪S2 −S1〉
S1 = powers root vectors ENαα , α Cartan root;
S2 = q Serre rel.(adc Ei )1−cqij Ej , i 6= j such that 1− cqij = ord qii .
The distinguished pre-Nichols algebra of V is Bq = T (V )/Iq.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
q = (qij)i,j∈Iθ V . Bq = B(V ) = T (V )/Jq.
cqij := −min n ∈ N0 : (n + 1)qii (1− qniiqijqji ) = 0 , i 6= j .
Assume dimB(V ) <∞.
i ∈ I is a Cartan vertex of q if qijqji(?)= q
cqijii , for all j 6= i .
α = sqi1si2 . . . sik (αi ) ∈ ∆q+ Cartan root of q if i ∈ I Cartan vertex of
ρik . . . ρi2ρi1 (q).
Definition
S = set of generators of Jq [A]. Let Jq ⊃ Iq := 〈S ∪S2 −S1〉
S1 = powers root vectors ENαα , α Cartan root;
S2 = q Serre rel.(adc Ei )1−cqij Ej , i 6= j such that 1− cqij = ord qii .
The distinguished pre-Nichols algebra of V is Bq = T (V )/Iq.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
The Lusztig algebra Lq of V is the graded dual of Bq.
BqBq→Lq.
Theorem
GKdim Bq = GKdimq Lq = |Cartan roots|.I. A., Distinguished pre-Nichols algebras, Transf. Gr., to appear.
N. Andruskiewitsch, I. A., F. Rossi Bertone. The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type.
arXiv:1501.04518
Example
q = (qij)i,j∈Iθ of Cartan type, i. e. qijqji = qcqijii , for all j 6= i . Assume
(ord qijqji , 210) = 1 for all i , j . Then
Bq = C〈x1, . . . , xθ| quantum Serre rel., power root vectors 〉;
Bq = C〈x1, . . . , xθ| quantum Serre rel. 〉
GKdim Bq = GKdimq Lq = |∆+|.
Ivan Angiono Nichols alg of finite GK dim
GK dim and pointed Hopf algsNichols algebras of diagonal type
Nichols algebras of a block + pointsPre-Nichols algebras and liftings
Definition, propertiesDistinguished pre-Nichols algebras
The Lusztig algebra Lq of V is the graded dual of Bq.
BqBq→Lq.
Theorem
GKdim Bq = GKdimq Lq = |Cartan roots|.I. A., Distinguished pre-Nichols algebras, Transf. Gr., to appear.
N. Andruskiewitsch, I. A., F. Rossi Bertone. The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type.
arXiv:1501.04518
Example
q = (qij)i,j∈Iθ of Cartan type, i. e. qijqji = qcqijii , for all j 6= i . Assume
(ord qijqji , 210) = 1 for all i , j . Then
Bq = C〈x1, . . . , xθ| quantum Serre rel., power root vectors 〉;
Bq = C〈x1, . . . , xθ| quantum Serre rel. 〉
GKdim Bq = GKdimq Lq = |∆+|.
Ivan Angiono Nichols alg of finite GK dim