Results on the Gelfand-Kirillov dimension

4.4. Results on the Gelfand-Kirillov dimension 97

Then the space M_kG⊕V has basis {m₁, . . . , m_r} ∪ {Fˆ_α|α ∈ Π}. If c^f is of exponential type, the braiding onM_kG⊕V is given by

c( ˆF_α⊗Fˆ_β) = v^−d(β,α)Fˆ_β⊗Fˆ_α, c( ˆF_α⊗m_j) = v^d(λj,α)m_j ⊗Fˆ_α,

c(m_i⊗Fˆ_β) = v^d(β,λi)Fˆ_β⊗m_i and

c(m_i⊗m_j) = f(λ_j, λ_i)m_j ⊗m_i =v^{−dϕ(λi,λj)}m_j ⊗m_i.

Let P := Π ˙∪{1, . . . , r}. If c^f is of strong exponential type there is always a matrix (bij)i,j∈P ∈Q^P×P such that the following conditions are satisfied:

∀α, β ∈Π : 2(α, β) = (α, α)b_αβ (1)

∀α ∈Π,1≤i≤r : 2(α, λi) = −(α, α)bαi (2)

∀α ∈Π,1≤i≤r : 2(α, λ_i) =−ϕ(λ_i, λ_i)b_iα (3)

∀1≤i, j ≤r : ϕ(λ_i, λ_j) +ϕ(λ_j, λ_i) = ϕ(λ_i, λ_i)b_ij (4)

∀1≤i≤r, i6=j ∈P, α∈Π : ϕ(λ_i, λ_i) = 0 ⇒b_ii= 2, b_ij = 0, b_iα = 0 (5) The matrix (b_ij)_i,j∈P will be called the extended Cartan matrix of M. Theorem 4.4.4. Letk be an algebraically closed field of characteristic zero, q ∈ k not a root of unity. Assume that the braiding c^f on the finite-dimensional integrable U_q(g)-module M is of exponential type with a sym-metric functionϕ (i.e. ϕ(λ, λ⁰) =ϕ(λ⁰, λ) for all λ, λ⁰ ∈Λ).

Then the Nichols algebra B(M, c^f) has finite Gelfand-Kirillov dimension if and only if c^f is of strong exponential type (with function ϕ) and the ex-tended Cartan matrix (b_ij) is a Cartan matrix of finite type.

Proof. Denote the basis ofM_kG⊕V byx_i, i∈P wherex_α := ˆF_αandx_i :=m_i for α∈Π and 1≤i≤r. The braiding of M_kG⊕V is of the form

c(x_i⊗x_j) =q_ijx_j⊗x_i ∀i, j ∈P, where the q_ij can be read off the formulas given above:

q_αβ =v^−d(β,α), q_αi =v^d(α,λi), q_iα =v^d(α,i)andq_ij =v^{−dϕ(λi,λj)} for all α, β ∈Π and 1≤i, j ≤r.

The if-part: By the definition of (b_ij) for all i, j ∈P q_ijq_ji =q_ii^bij.

For all α∈Π and for all 1≤i≤r with ϕ(λ_i, λ_i)6= 0 define d_α := d(α, α)

2 and d_i := dϕ(λ_i, λ_i) 2

4.4. Results on the Gelfand-Kirillov dimension 99 and for 1 ≤ i ≤ r with ϕ(λ_i, λ_i) = 0 define d_i := 1. These (d_i) are positive integers satisfying

d_ib_ij =d_jb_ji ∀i, j ∈P.

Because ϕ is symmetric one gets for all i, j ∈P q_ij =v^−dibij.

This means that the braiding onMkG⊕V is of Frobenius-Lusztig type with generalized Cartan matrix (b_ij). As (b_ij) is a finite Cartan matrix,B(M_kG⊕ V) has finite Gelfand-Kirillov dimension by [4, Theorem 2.10.]. Thus the subalgebra B(M) has finite Gelfand-Kirillov dimension [22, Lemma 3.1.].

The only-if-part: By 2.2.4B(M) has a PBW basis and because the Gelfand-Kirillov dimension is finite the set of PBW generators S_M must be finite.

Similarly B(V) has a PBW basis and because it has finite Gelfand-Kirillov dimension (see [4, Theorem 2.10.]) its set of PBW generators S_V is also finite. So the finite set

S:={s#1|s∈S_M}∪{˙ 1#s⁰|s⁰ ∈S_V}

forms a set of PBW generators forB(M)#B(V)∼=B(M_kG⊕V). This implies thatB(MkG⊕V) has finite Gelfand-Kirillov dimension. Now [38, Lemma 14 and 20] allows us to find integers c_ij ≤0, i, j ∈P such that

qijqji =q^c_ii^ij ∀i, j ∈P.

Using the definition of theq_ij one obtains that thec_ij must satisfy the equa-tions (1) − (4) from the definition of (b_ij) with b_ij, i, j ∈ P replaced by c_ij, i, j ∈ P. Because of relations (3) and (4), ϕ must satisfy the condi-tion from the definicondi-tion of strong exponential braidings. Furthermore one may assume c_ii = 2 for all i ∈ P and c_ij = 0 for all 1 ≤ i ≤ r with ϕ(λ_i, λ_i) = 0, i 6= j ∈ P. This means that b_ij = c_ij for all i, j ∈ P. Now observe that b_ij is a generalized Cartan matrix. Exactly as in the “only-if”

part of the proof the braiding inM_kG⊕V is of Frobenius-Lusztig type with generalized Cartan matrix (b_ij). By [4, Theorem 2.10.] (b_ij) is a finite Cartan matrix because B(M_kG⊕V) has finite Gelfand-Kirillov dimension.

Explicit calculations for simple U

_q

(g)-modules

Now the results above are used to determine for each finite-dimensional sim-ple comsim-plex Lie algebragall pairs (λ, ϕ) such that the Nichols algebra of the Uq(g)-module of highest weight λ together with the braiding defined by the functionϕ has finite Gelfand-Kirillov dimension. First observe that (as only

modules of highest weight are considered) one may assume that the function ϕ is of the form

ϕ(µ, ν) = (µ, ν) +x for µ, ν ∈Λ

for some x ∈ Q. This is true because the braiding c^f depends only on the values ϕ(λ⁰, λ⁰⁰) for those weights λ⁰, λ⁰⁰ ∈ Λ such that M_λ0 6= 0, M_λ00 6= 0.

They are all in the same coset of ZΦ in Λ. Thus one can choose x:=ϕ(λ, λ)−(λ, λ)

for any weight λ∈Λ with M_λ 6= 0.

Theorem 4.4.5. Assume thatk is an algebraically closed field of character-istic zero and that q∈k is not a root of unity. Let g be a finite-dimensional simple complex Lie algebra with weight lattice Λ. Fix a U_q(g)-module M of highest weight λ ∈ Λ and a value x ∈ Q. Let d⁰ be the least common multiple of the denominator of x and the determinant of the Cartan matrix of g. Let d:= 2d⁰ and fix v ∈k with v^d =q. Define a function

f : Λ×Λ→k^×, (λ, λ⁰)7→v^{d((λ,λ0)+x)}.

Then the Nichols algebra B(M, c^f_M,M) has finite Gelfand-Kirillov dimension if and only if the tuple g, λ, x occurs in Table 4.1.

Note that the braidingc^f_M,M may depend on the choice of v, but the Gelfand-Kirillov dimension ofB(M, c^f_M,M) does not.

In Table 4.1 also the type of the extended Cartan matrix (b_ij) and the value ϕ(λ, λ) are given. The weight λ_α always denotes the fundamental weight dual to the root α. The numbering of the roots is as in [13] and in Table 1.1.

Proof. Assume that the tuple g, λ, x leads to a Nichols algebra of finite Gelfand-Kirillov dimension and let (a_αβ)_α,β∈Π be the Cartan matrix for g.

Define for all α∈Π the integer d_α := ^(α,α)₂ . By Theorem 4.4.4 the extended Cartan matrix (bij)i,j∈P is a finite Cartan matrix. FurthermoreP = Π ˙∪{λ} andb_αβ =a_αβ forα, β ∈Π. First assume that (b_ij) is not a connected Cartan matrix. As (a_ij) is a connected Cartan matrix observe

b_λα = 0 =b_αλ ∀α ∈Π.

By the definition of (b_ij) this impliesλ= 0. This is the first line in the table.

Now assume that (b_ij) is a connected finite Cartan matrix and thus its Cox-eter graph contains no cycles. As (a_αβ) is also a connected finite Cartan matrix there is a unique root α∈Π such that

b_αλ, b_λα<0 and for allβ ∈Π\ {α}:b_βλ = 0 =b_λβ.

4.4. Results on the Gelfand-Kirillov dimension 101

Type ofg λ x Type of (b_ij) ϕ(λ, λ) D

any 0 any D∪A₀ any no relations

A_n, n≥1 λ_α1 or λ_αn ⁿ⁺²_n+1 A_n+1 2 2 A_n, n≥1 λ_α1 or λ_{αn n+1}¹ B_n+1 1 2 A_n, n≥1 2λ_α1 or 2λ_{αn n+1}⁴ C_n+1 (resp. B₂) 4 3 A_n, n≥3 λ_αn−1 orλ_{α2 n+1}⁴ D_n+1 2 2

A₁ λ_α1 ¹₆ G₂ ³₂ 4

A₁ 3λ_α1 ³₂ G₂ 6 2

A5 λα3

1

2 E6 2 2

A6 λα3 orλα4

2

7 E7 2 2

A₇ λ_α3 orλ_α5 ¹₈ E₈ 2 2

B_n, n ≥2 λ_α1 2 B_n+1 4 2

B₂ λ_α2 1 C₃ 2 2

B₃ λ_α3 1 F₄ 2 2

C_n, n≥3 λ_α1 1 C_n+1 2 2

C₃ λ_α3 1 F₄ 4 2

D_n, n≥5 λ_α1 1 D_n+1 2 2

D₄ λ_α1, λ_α3 or λ_α4 1 D₅ 2 2

D5 λα4 orλα5

3

4 E6 2 2

D6 λα5 orλα6

1

2 E7 2 2

D₇ λ_α6 orλ_α7 ¹₄ E₈ 2 2

E₆ λ_α1 orλ_α6 ²₃ E₇ 2 2

E₇ λ_α7 ¹₂ E₈ 2 2

Table 4.1: Highest weights with Nichols algebras of finite Gelfand-Kirillov dimension; D is the degree of the relations calculated with Theorem 4.5.5

This implies that

l := (α, λ)>0 and for allβ ∈Π\ {α}: (β, λ) = 0.

Observe, using the definition of (b_ij) andϕ, that l =−b_αλ(α, α)

2 =−b_αλd_α, ϕ(λ, λ) = b_αλ

b_λα(α, α) = 2b_αλ

b_λαd_α, and x=ϕ(λ, λ)−(λ, λ).

Furthermore we conclude λ = −b_αλλ_α, where λ_α is the weight dual to the root α, i.e.

(λ_α, β) = δ_β,αd_α.

In a case-by-case analysis we will now consider all finite connected Car-tan matrices (a_αβ)_α,β∈Π and all possible finite connected Cartan matrices (bij)_{i,j∈Π ˙∪{λ}} having (aαβ) as a submatrix. In each case we compute the values for l, ϕ(λ, λ) and x and decide if there is a tuple g, λ, x leading to the matrix (b_ij). For every case also the Dynkin diagram of (b_ij) with la-beled vertices is given. The vertices 1, . . . , n correspond to the simple roots α₁, . . . , α_n∈Π, the vertex ? corresponds to λ∈P.

Note that to calculatexwe must calculate the values (λ_α, λ_α) for someα∈Π.

To do this we use Table 1 from [13, 11.4] and the explicit construction of the root systems there. However in the case of the root system B_n we have to multiply the scalar product by 2 to obtain the normalization described in Subsection 1.2.3.

A_n →A_n+1, n ≥1:

•1 •· · · ·^{2 n−1}• ⁿ• •^? or •^? •¹ •· · · ·^{2 n−1}• •ⁿ

We have eitherα =α_n orα =α₁. In any case d_α = 1, b_αλ=b_λα=−1 and ϕ(λ, λ) = 2, λ =λ_α,(λ, λ) = n

n+ 1, x= n+ 2 n+ 1.

So λ = λ_α1 or λ = λ_αn together with x = ⁿ⁺²_n+1 extend the matrix of A_n to A_n+1.

A_n →B_n+1, n ≥1:

•1 •· · · ·^{2 n−1}• •ⁿ==⇒•^? or • ⇐^? ==•¹ •· · · ·^{2 n−1}• •ⁿ

4.4. Results on the Gelfand-Kirillov dimension 103 Here either α=α₁ orα =α_n. In any case d_α = 1, b_αλ=−1, b_λα=−2 and

ϕ(λ, λ) = 1, λ=λ_α,(λ, λ) = n

n+ 1, x= 1 n+ 1.

So λ = λ_α1 or λ = λ_αn together with x = _n+1¹ extend the matrix of A_n to B_n+1.

A_n→C_n+1, n ≥1:

•1 •· · · ·^{2 n−1}• ⁿ• ⇐==•^? or •^? ==⇒•¹ •· · · ·^{2 n−1}• ⁿ• Either α=α₁ orα=α_n. In any case d_α = 1, b_αλ =−2, b_λα =−1 and

ϕ(λ, λ) = 4, λ= 2λ_α,(λ, λ) = 4n

n+ 1, x= 4 n+ 1.

Soλ = 2λα1 or λ= 2λαn together withx = _n+1⁴ extend the matrix of An to C_n+1 (resp. B₂).

A_n→D_n+1, n ≥3:

•1 •· · · ·^{2 n−}•¹ •ⁿ or •¹ •· · · ·^{2 n−}•¹ •ⁿ

•_? •_?

Here either α=α2 orα =αn−1. In any case dα = 1, bαλ=bλα =−1 and ϕ(λ, λ) = 2, λ=λ_α,(λ, λ) = 2(n−1)

n+ 1 , x= 4 n+ 1.

Soλ =λ_α2 or λ =λ_αn−1 together with x= _n+1⁴ extend the matrix of A_n to D_n+1.

A₁ →G₂:

• ≡≡1 V•^? or •¹ W≡≡•^?

In any caseα=α₁ and d_α = 1. The left diagram meansb_αλ=−1, b_λα=−3 and

ϕ(λ, λ) = 2

3, λ=−λ_α1,(λ, λ) = 1

2, x= 1 6.

The right diagram means b_αλ=−3, b_λα=−1 and ϕ(λ, λ) = 6, λ= 3λ_α1,(λ, λ) = 9

2, x= 3 2.

Soλ =λ_α1, x= ¹₆ orλ = 3λ_α1, x= ³₂ extend the matrix ofA₁ to G₂. A₅ →E₆:

•1 •² •³ •⁴ •⁵

•?

Hereα =α₃ and d_α = 1, b_αλ =b_λα=−1. This implies ϕ(λ, λ) = 2, λ =λ_α3,(λ, λ) = 3

2, x= 1 2.

A6 →E7:

•1 •² •³ •⁴ •⁵ •⁶ or •¹ •² •³ •⁴ •⁵ •⁶

•_? •_?

α=α₃ orα =α₄. Furthermored_α = 1, b_αλ=b_λα=−1. This implies ϕ(λ, λ) = 2, λ=λ_α,(λ, λ) = 12

7 , x= 2 7.

A₇ →E₈:

•1 •² •³ •⁴ •⁵ •⁶ •⁷ or •¹ •² •³ •⁴ •⁵ •⁶ •⁷

• •

? ?

α=α₃ orα =α₅. Furthermored_α = 1, b_αλ=b_λα=−1. This implies ϕ(λ, λ) = 2, λ=λ_α,(λ, λ) = 15

8 , x= 1 8.

4.4. Results on the Gelfand-Kirillov dimension 105

B_n→B_n+1, n ≥2:

•? •¹ •· · · ·^{2 n−1}• ==⇒ⁿ• We haveα =α₁, d_α= 2, b_αλ =b_λα=−1 and this means

ϕ(λ, λ) = 4, λ=λα,(λ, λ) = 2, x= 2.

B₂ →C₃:

•1 ==⇒•² •^?

In this case α=α₂ and d_α = 1, b_αλ=b_λα =−1. This means ϕ(λ, λ) = 2, λ=λ_α,(λ, λ) = 1, x= 1.

B₃ →F₄:

•1 •² ==⇒•³ •^? Hereα=α3 and dα = 1, bαλ=bλα =−1. This means

ϕ(λ, λ) = 2, λ=λ_α,(λ, λ) = 1, x= 1.

All the other cases follow the same idea and are omitted.

It remains to show that the data from the table lead to Nichols algebras of finite Gelfand-Kirillov dimension. It is clear that the braiding is of strong exponential type in every case. Moreover in each line the extended Cartan matrix (b_ij) is of finite type and thus the Nichols algebra has finite Gelfand-Kirillov dimension by Theorem 4.4.4.

Im Dokument Braided Hopf algebras of triangular type (Seite 101-110)