Nichols algebras of nite Gelfand-Kirillov dimension · Nichols algebras of diagonal type Nichols algebras of a block + points Pre-Nichols algebras and liftings Nichols algebras of

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



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.




Plan of the talk









Plan of the talk









Plan of the talk









Plan of the talk









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).





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).





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.





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.





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.





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.





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.





Problem


H0 ⊂ H1 ⊂ · · · ⊂ H =⋃

n≥0 Hn,



CGYD.






Problem


H0 ⊂ H1 ⊂ · · · ⊂ H =⋃

n≥0 Hn,



CGYD.






Problem


H0 ⊂ H1 ⊂ · · · ⊂ H =⋃

n≥0 Hn,



CGYD.






Problem


H0 ⊂ H1 ⊂ · · · ⊂ H =⋃

n≥0 Hn,



CGYD.






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.





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 ).





Definition



B0(V ) = C, B1(V ) = P(V ), B(V ) = C〈B1(V )〉.

Hence B(V ) ' T (V )/J (V ).

Remark








Definition



B0(V ) = C, B1(V ) = P(V ), B(V ) = C〈B1(V )〉.

Hence B(V ) ' T (V )/J (V ).

Remark








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 ) <∞.










GKdimCG <∞.












GKdimCG <∞.







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 .






Question


Question




Question







Question


Question




Question





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.








































(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.




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 ) =∞.






Remark



Conjecture








Remark



Conjecture








Remark



Conjecture






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.




Conjecture


More generally,

Conjecture


Theorem





Conjecture


More generally,

Conjecture


Theorem





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.






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






c2 : a ∈ 0, 1, b, c ∈ N0.






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






c2 : a ∈ 0, 1, b, c ∈ N0.






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






c2 : a ∈ 0, 1, b, c ∈ N0.





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).









⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,



V2 = Cχ2g2∈ CΓ












⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,



V2 = Cχ2g2∈ CΓ












⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,



V2 = Cχ2g2∈ CΓ












⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,



V2 = Cχ2g2∈ CΓ












⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,



V2 = Cχ2g2∈ CΓ












⇒ V = Vg1 (χ1, η)⊕ Cχ2g2, g1, g2 ∈ Γ,



V2 = Cχ2g2∈ CΓ








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.







εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1

εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2

q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3

.


= id ⇐⇒ q12q21 = 1 and a = 0.

c2|V1⊗V2


Definition




G =

−2a, ε = 1,

a, ε = −1.








εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1

εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2

q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3

.


= id ⇐⇒ q12q21 = 1 and a = 0.

c2|V1⊗V2


Definition




G =

−2a, ε = 1,

a, ε = −1.








εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1

εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2

q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3

.


= id ⇐⇒ q12q21 = 1 and a = 0.

c2|V1⊗V2


Definition




G =

−2a, ε = 1,

a, ε = −1.








εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1

εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2

q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3

.


= id ⇐⇒ q12q21 = 1 and a = 0.

c2|V1⊗V2


Definition




G =

−2a, ε = 1,

a, ε = −1.








εx1 ⊗ x1 (εx2 + x1)⊗ x1 q12x3 ⊗ x1

εx1 ⊗ x2 (εx2 + x1)⊗ x2 q12x3 ⊗ x2

q21x1 ⊗ x3 q21(x2 + ax1)⊗ x3 q22x3 ⊗ x3

.


= id ⇐⇒ q12q21 = 1 and a = 0.

c2|V1⊗V2


Definition




G =

−2a, ε = 1,

a, ε = −1.






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





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 .





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,





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)




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.





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.





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.





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.





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 ).






Lemma


Definition


· · · < B[3] < B[2] < B[1] < B[0] = B(V ), (1)


Lemma








Lemma


Definition


· · · < B[3] < B[2] < B[1] < B[0] = B(V ), (1)


Lemma







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.





Example





Remark








Example





Remark








Example





Remark








Example





Remark








Example





Remark








Example





Remark








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.





Theorem


λ = 0, if χ2 6= ε.

U = U(D, λ) = T (V )#CG/〈x2x1 − x1x2 + 12x

21 − λ(1− g2)〉 is a



λ = 0, if χ2 6= ε.

U = U(D, λ) =T (V )#CG/〈x2

1 − λ(1− g2), x2x12 − x12x2 − x1x12 + 2λx2 + λx1g2〉






Theorem


λ = 0, if χ2 6= ε.

U = U(D, λ) = T (V )#CG/〈x2x1 − x1x2 + 12x

21 − λ(1− g2)〉 is a



λ = 0, if χ2 6= ε.

U = U(D, λ) =T (V )#CG/〈x2

1 − λ(1− g2), x2x12 − x12x2 − x1x12 + 2λx2 + λx1g2〉






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.











ρik . . . ρi2ρi1 (q).

Definition















ρik . . . ρi2ρi1 (q).

Definition









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 = |∆+|.





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 = |∆+|.


Nichols algebras of nite Gelfand-Kirillov dimension · Nichols algebras of diagonal type Nichols algebras of a block + points Pre-Nichols algebras and liftings Nichols algebras of

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