# nichols algebras of nite gelfand-kirillov dimension nichols algebras of diagonal type nichols...

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GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

Nichols algebras of finite Gelfand-Kirillov dimension

Iván Angiono

Universidad Nacional de Cordoba

Banff, Canada - September, 2015

Joint work with N. Andruskiewitsch and I. Heckenberger

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-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.

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-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.

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-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.

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-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.

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-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.

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Definition

Let A be a finitely generated C-algebra. If V is a finite-dimensional generating 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 := sup{GKdimB|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).

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Definition

Let A be a finitely generated C-algebra. If V is a finite-dimensional generating 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 := sup{GKdimB|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).

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Problem

Classify Hopf algebras H with GKdimH

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Problem

Classify Hopf algebras H with GKdimH

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Remark

Let G be a finitely generated group. G is virtually nilpotent or nilpotent-by-finite if it has a normal nilpotent subgroup N such that G/N is finite. Then

GKdimCG

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Remark

Let G be a finitely generated group. G is virtually nilpotent or nilpotent-by-finite if it has a normal nilpotent subgroup N such that G/N is finite. Then

GKdimCG

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Problem

Classify pointed Hopf algebras H with GKdimH

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Problem

Classify pointed Hopf algebras H with GKdimH

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Problem

Classify pointed Hopf algebras H with GKdimH

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Problem

Classify pointed Hopf algebras H with GKdimH

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

Problem

Classify pointed Hopf algebras H with GKdimH

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols algebras

CG CGYD = 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 CGCGYD.

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols 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 ).

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols 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 ).

Iván Angiono Nichols alg of finite GK dim

GK dim and pointed Hopf algs Nichols algebras of diagonal type

Nichols algebras of a block + points Pre-Nichols algebras and liftings

GK dimension Pointed Hopf algebras Nichols 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