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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

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