Drinfeld double of quantum groups, tilting modules and $\mathbb{Z}$-modular data associated to complex reflection groups

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Z -modular data associated to complex reflection groups

Abel Lacabanne

To cite this version:

Abel Lacabanne. Drinfeld double of quantum groups, tilting modules andZ-modular data associated to complex reflection groups. 2018. �hal-01827849v3�

AND

_Z

-MODULAR DATA ASSOCIATED TO COMPLEX REFLECTION GROUPS

by

A

BEL

L

ACABANNE

In this article, we construct a categorification of some Fourier matrices associated to complex reflections groups by Malle [Mal95]. To any spetsial imprimitive complex re- flection group

W

, he attached a set of unipotent characters, which is in bijection with the set of unipotent characters of the corresponding finite reductive group when

W

is a Weyl group. He also defined a partition of these characters into families, the analogue of Lusztig’s nonabelian Fourier transform and eigenvalues of the Frobenius. In [Cun07], Cuntz showed that for each family, this defines a so-called

Z

-modular datum. Therefore, one can associate a

_Z

-algebra, free of finite rank over

_Z

. If the stucture constants of such an algebra

R

are positive, it is a classical problem to find a tensor category, with some ex- tra structure, whose Grothendieck ring is precisely

R

. However, these structure constants belong in general to

Z

and the categorification is more involved.

In the case of cyclic groups, Bonnafé and Rouquier [BR17] constructed a tensor trian- gulated category which categorifies the modular datum attached to the non-trivial family of the cyclic spets. In the present article, we construct fusion categories with symmetric center equivalent to

Rep(G,z)

for

G

a cyclic group and

z ∈G

satisfying

z² =1

. If the order of

G

is even, we can therefore construct a new category, which is enriched over superspaces, so that its Grothendieck group has negative structure constants in general.

As shown in [Lac18], this defines a

Z

-modular datum.

Our construction is as follows: we consider the representation theory of a quantum double associated to the complex simple Lie algebra

^sln+1

at a root of unity of even or- der. Similarly to the usual construction for

Uq(g)

at a root of unity

ξ

, we consider the semisimplification of the full subcategory of tilting modules. A partial modularization of an integral subcategory gives then the category with symmetric center

Rep(G,z)

, which is denoted by

Z(T_ξ)oS

.

We investigate this construction in full generality considering any simple complex Lie algebra

g

and any root of unity. For certain values of

ξ

, we are able to compute explicitely the symmetric center of the category

Z(T_ξ)oS

, see Theorem 3.14. When

^g

is a simple Lie algbera of type

A

and

ξ

is a root of unity of even order, the category

Z(T_ξ)oS

gives rises to a

_Z

-modular datum which we relate to some of the Fourier matrices constructed by Malle. The main result of this paper is given by the following theorem:

Theorem A. —

Let

n≥1

and

d≥n

be two integers, and

^g=sl_n+1

.

If

n

is even, the symmetric center of

Z(T_ξ)oS

is equal to the category

Rep(Z/(n+1)Z)

. If

n

is odd, the symmetric center of

Z(T_ξ)oS

is equal to the category

Rep Z/(n+1)Z,ⁿ⁺₂¹

.

In both cases, there exists a family

F

of unipotent characters for the complex reflection group

G€

d, 1,ⁿ⁽ⁿ₂⁺¹⁾Š

such that the Malle’s

Z

-modular data coincide with the

Z

-modular data arising from the category

_Z(Tξ)oS

.

Moreover, some modular data constructed by Broué, Malle and Michel for spetsial ex- ceptional complex reflection groups can be categorified by considering similar categories in type

B

.

This article is organized as follows: in the first Section, we study the algebra

Dq(g)

when

q

is a generic parameter. Then, in the second Section, we study some finite dimen- sional representations of

Dq(g)

and extra structure arising from the Hopf algebra

Dq(g)

: tensor product, braiding, pivotality. The third Section contains the main constructions of this paper: we specialize the algebra

_Dq(g)

at a root of unity

_ξ

and consider tilting modules for this algebra, which allows us to construct the category

Z(T_ξ)oS

. In the fourth Section, we recall the Malle’s

_Z

-modular datum and compare it with the modular datum arising from the category

Z(T_ξ)oS

. Finally, in the last Section, we study some exceptional examples arising from complex reflection groups.

Acknowledgements. — I warmly thank my advisor C. Bonnafé for many fruitful dis- cussions and his constant support and R. Rouquier for suggesting me to consider super- categories in this work. I also thank G. Malle for many valuable comments on a first version of this paper. The paper is partially based upon work supported by the NSF un- der grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester.

1. Quantum double of Borel algebras

In this section, we recall the construction of the quantum enveloping algebra

Uq(g)

as- sociated to a simple complex Lie algebra

g

. Our main object of study will be the quantum double of a Borel algebra

^b

of

^g

, which will be denoted by

Dq(g)

.

1.1. Notations. — Let

g

be a simple complex Lie algebra of rank

n

. We fix a Cartan subalgebra

^h

of

^g

. Let

Φ⊆h^∗

be the set of roots of

^g

relative to

^h

,

Π={α1, . . . ,αn}

be a base of

Φ

,

Φ⁺

be the positive roots relative to

Π

. Let

b⁺

(resp.

b⁻

) be the Borel subalgebra of

g

relative to

Π

(resp.

−Π

). We denote by

h

the Coxeter number of

^g

and by

h^∨

the dual Coxeter number of

g

.

Fix a symmetric bilinear form

_〈·,·〉

on

h^∗

normalized such that

〈α,α〉=2

for short roots.

For

α∈Φ

, we define

α^∨∈h^∗

by

α^∨= 2α

〈α,α〉.

Let

Q

be the root lattice,

P

the weight lattice,

P⁺

the cone of dominant weights,

Q^∨

the coroot lattice and

P^∨

the coweight lattice

P ={λ∈h^∗| 〈λ,α^∨〉 ∈Z, ∀α∈Π}

P⁺={λ∈h^∗| 〈λ,α^∨〉 ∈N, ∀α∈Π}

P^∨={λ∈h^∗| 〈λ,α〉 ∈Z, ∀α∈Π}

Let

($i)1≤i≤n

be the dual basis of

(α^∨_i)1≤i≤n

with respect to the form

〈·,·〉

and

ρ

be the half

sum of positive roots. We call the

$i

’s the fundamental weights. The element

ρ

is in

P

and is equal to the sum of fundamental weights [Bou68, VI.1.10 Proposition 29].

We denote by

Q⁺

the monoid spanned by

Π

. We denote by

≤

the usual partial order on

P

:

λ≤µ

if and only if

µ−λ∈Q⁺

.

For any

α∈Φ

, denote by

s_α

the reflection in the hyperplane orthogonal to

α s_α(v) =v− 〈v,α^∨〉α=v− 〈v,α〉α^∨, ∀v∈h^∗.

For

1≤i≤n

denote by

s_i

the reflection

s_αi

. All these reflections generate the Weyl group

W

of

g

and for any

w ∈W

,

w(Φ) =Φ

. The form

_〈·,·〉

is then invariant with respect to

W

. We denote by

l(w)

the length of

w∈W

relative to the generating set

(s_i)1≤i≤n

.

For

q

an indeterminate, define the following elements of

_Z[q,q⁻¹] [n]q =qⁿ−q⁻ⁿ

q−q⁻¹ ,n∈Z, [n]q!=

n

Y

k=1

[k]q,n∈N

and

•n k

˜

q

=

k

Y

i=1

[n+1−i]q

[i]q

,n∈Z,k∈N.

We denote by

[n]_ξ

(resp.

[n]_ξ!

, resp.

•n k

˜

ξ

) the evaluation of

[n]q

(resp.

[n]q!

, resp.

•n k

˜

q

) at an invertible element

_ξ

of a ring. We will work over the field

_Q(q)

and the ring

_{A =} Z[q,q⁻¹]

. We define

q_i=q^{〈αi2,αi〉}

.

1.2. Positive and negative parts of the usual quantum enveloping algebra. — We start by defining the usual positive and negative part of the quantum group

Uq(g)

.

Definition 1.1. —

Let

Uq(b⁺)

be the associative unital

Q(q)

-algebra with generators

K_i^±1

,

E_i

,

1≤i≤n

, and defining relations

K_iK_j=K_jK_i, K_iK_i⁻¹=1=K_i⁻¹K_i, ∀1≤i,j≤n, K_iE_j =q^{〈αi,αj〉}E_jK_i, ∀1≤i,j≤n,

1−〈αi,α^∨j〉

X

r=0

(−1)^r

1− 〈αi,α^∨_j〉 r

qi

E^{1−〈αi,α}

∨ j〉−r

i E_jE_i^r=0, ∀1≤i6=j≤n.

There exist several Hopf algebra structures on

Uq(b⁺)

, and we choose the following comultiplication

∆

, counit

"

and antipode

S

:

∆(K_i) =K_i⊗K_i "(K_i) =1 S(K_i) =K_i⁻¹,

∆(E_i) =1⊗E_i+E_i⊗K_i "(E_i) =0 S(E_i) =−E_iK_i⁻¹.

Definition 1.2. —

Let

_Uq(b⁻)

be the associative unital

_Q(q)

-algebra with generators

L_i^±1

,

F_i

,

1≤i≤n

, and defining relations

L_iL_j =L_jL_i, L_iL⁻¹_i =1=L_i⁻¹L_i, ∀1≤i,j≤n, L_iF_j =q^{−〈αi,αj〉}F_jL_i, ∀1≤i,j≤n,

1−〈αi,α^∨j〉

X

r=0

(−1)^r

1− 〈αi,α^∨_j〉 r

q_i

F^{1−〈αi,α}

∨j〉−r

i F_jF_i^r=0, ∀1≤i6=j≤n.

There exist severals Hopf algebra structures on

Uq(b⁻)

, and we choose the following comultiplication

∆

, counit

"

and antipode

S

:

∆(L_i) =L_i⊗L_i "(L_i) =1 S(L_i) =L_i⁻¹,

∆(F_i) =L⁻¹_i ⊗F_i+F_i⊗1 "(F_i) =0 S(F_i) =−L_iF_i.

For

λ=Pn

i=1λiαi∈Q

, we denote by

K_λ

(resp.

L_λ

) the element

^Qn_i=1K_i^λi

(resp.

^Qn_i=1L_i^λi

).

1.3. A Hopf pairing between

Uq(b⁺)

and

Uq(b⁻)

. — The following is due to Drinfeld [Dri87, Section 13] and has been studied by Tanisaki [Tan92].

Proposition 1.3. —

There exists a unique bilinear form

(·,·):Uq(b⁺)×Uq(b⁻)→Q(q)

satisfying for all

x,x⁰∈ Uq(b⁺)

,

y,y⁰∈ Uq(b⁻)

and

1≤i≤n

1.

(x,y y⁰) = (∆(x),y⊗y⁰)

, 2.

₍x x⁰,y) = (x⁰⊗x,∆(y))

, 3.

(x, 1) ="(x)

,

(1,y) ="(y)

,

4.

(K_i,L_j) =q^{〈αi,αj〉}

, 5.

₍K_i,F_j) =0= (E_i,L_j)

, 6.

(E_i,F_j) =_q^δi,j

i−q_i⁻¹

.

This bilinear form endows the tensor product

Uq(b⁻)⊗ Uq(b⁺)

with a Hopf algebra structure, which we denote by

Dq(g)

, see [KS97, Section 8.2]. Using Sweedler notation for the coproduct, the product of

y₁⊗x₁

and

y₂⊗x₂

in

_Uq(b⁻)⊗ Uq(b⁺)

is given by

(y₁⊗x₁)(y₂⊗x₂) = X

(x1)(y2)

(x₁⁰,y₂⁰)(x₁⁰⁰⁰,S(y₂⁰⁰⁰))y₁y₂⁰⁰⊗x₁⁰⁰x₂.

Consequently, both

Uq(b⁻)⊗1

and

1⊗Uq(b⁺)

are Hopf subalgebras of

Dq(g)

. In the follow- ing, we identify

Uq(b⁻)

and

Uq(b⁺)

with their images in

Dq(g)

and will write

y x

instead of

y ⊗x

.

Proposition 1.4. —

The algebra

Dq(g)

is the associative unital

Q(s)

-algebra with generators

K_i^±1

,

L^±1_i

,

E_i

and

F_i

,

1≤i≤n

and defining relations

K_iK_j=K_jK_i, K_iK_i⁻¹=1=K_i⁻¹K_i, L_iL_j=L_jL_i, L_iL⁻¹_i =1=L_i⁻¹L_i,

K_iL_j =L_jK_i,

K_iE_j =q^{〈αi,αj〉}E_jK_i, K_iF_j =q^{−〈αi,αj〉}F_jK_i, L_iE_j =q^{〈αi,αj〉}E_jL_i, L_iF_j =q^{−〈αi,αj〉}F_jL_i,

[E_i,F_j] =δi,j

K_i−L_i⁻¹ q_i−q_i⁻¹,

1−〈αi,α^∨_j〉

X

r=0

(−1)^r

1− 〈αi,α^∨_j〉 r

q_i

E^{1−〈αi,α}

∨j〉−r

i E_jE_i^r=0, 1≤i6=j≤n,

1−〈αi,α^∨j〉

X

r=0

(−1)^r

1− 〈αi,α^∨_j〉 r

qi

F^{1−〈αi,α}

∨j〉−r

i F_jF_i^r=0, 1≤i6=j≤n.

The elements

z_i=K_iL⁻¹_i

are central and

Uq(g)

is the quotient of

Dq(g)

by the Hopf ideal generated by

(z_i−1)1≤i≤n

.

Denote by

Dq(g)^<0

(resp.

Dq(g)⁰

, resp.

Dq(g)^>0

) the subalgebra of

Dq(g)

generated by

(E_i)1≤i≤n

(resp.

(K_i,L_i)1≤i≤n

, resp.

(F_i)1≤i≤n

). Multiplication yields an isomorphism of vector spaces

Dq(g)^<0⊗Dq(g)⁰⊗Dq(g)^>0' Dq(g)

. It is worth mentioning that the coproduct and antipode of

Dq(g)

do not restrict to

Dq(g)^<0

or

Dq(g)^>0

.

The square of the antipode is given by the conjugation by any element of the form

L_λK_2ρ−λ

,

λ∈Q

as it is easily checked on the generators.

1.4. Graduation. — There exists a

Q

-graduation on

Dq(g)

given by

deg(K_i) =0=deg(L_i)

,

deg(E_i) =αi

and

deg(F_i) =−αi

. For

λ∈Q

, we denote by

Dq(g)_λ

the homogeneous elements of degree

λ

. We have

Dq(g)_λ=

v∈ Dq(g)

K_iv=q^〈λ,αi〉v K_i, ∀1≤i≤n .

The coproduct, the counit and the antipode respect the grading. Moreover, the same lemma as [Jan96, lemma 4.12] shows that for

µ∈Q

,

µ≥0

∆€

Dq(g)^>_µ⁰Š

⊆ M

0≤ν≤µ

Dq(g)^>_ν⁰⊗K_νDq(g)^>_µ−ν⁰

and

∆€

Dq(g)^<0_−µŠ

⊆ M

0≤ν≤µ

L⁻¹_ν Dq(g)^<0_−(µ−ν)⊗ Dq(g)^<0_−ν.

Therefore, for

x∈ Dq(g)^>0_µ

, there are elements

r_i(x)

and

r_i⁰(x)

in

Dq(g)^>0_µ−αi

such that

(1)

∆(x)∈1⊗x+

n

X

i=1

E_i⊗K_ir_i(x) + X

0<ν≤µ ν6∈Π

Dq(g)^>_ν⁰⊗K_νDq(g)^>_µ−ν⁰ ,

and

(2)

∆(x)∈x⊗K_µ+ Xn

i=1

r_i⁰(x)⊗K_µ−αiE_i+ X

0<ν≤µ µ−ν6∈Π

Dq(g)^>0_ν ⊗K_νDq(g)^>0_µ−ν.

Similarly, we can define

ρi(y)

and

ρ⁰_i(y)

for

y ∈ Dq(g)^<_−µ⁰

by the following

∆(y)∈L⁻¹_µ ⊗y+

n

X

i=1

L⁻¹_µ−α

iF_i⊗ρi(y) + X

0<ν≤µ µ−ν6∈Π

L_ν⁻¹Dq(g)^<_−(µ−ν)⁰ ⊗ Dq(g)^<_−ν⁰,

and

∆(y)∈y⊗1+ Xn i=1

L_α⁻¹

iρ_i⁰(y)⊗F_i+ X

0<ν≤µ ν6∈Π

L⁻¹_ν Dq(g)^<0_−(µ−ν)⊗ Dq(g)^<0_−ν.

The values of

r_i

,

r_i⁰

,

ρi

and

ρ⁰_i

can be computed by induction. Compare with [Jan96, 6.14,6.15].

Lemma 1.5. —

Let

µ,µ⁰∈Q

,

µ≥0

and

µ⁰≥0

. 1. For all

x∈ Dq(g)^>_µ⁰

and

x⁰∈ Dq(g)^>_µ0⁰

,

r_i(x x⁰) =q^{−〈αi,µ〉}x r_i(x⁰) +r_i(x)x⁰, r_i⁰(x x⁰) =x r_i⁰(x⁰) +q^{−〈αi,µ0〉}r_i⁰(x)x⁰.

2. For all

y ∈ Dq(g)^<0_−µ

and

y⁰∈ Dq(g)^<0_−µ0

,

ρi(y y⁰) =yρi(y⁰) +q^{−〈αi,µ0〉}ρi(y)y⁰, ρ_i⁰(y y⁰) =q^{−〈αi,µ〉}yρ⁰_i(y⁰) +ρ⁰_i(y)y⁰.

These elements can also be used to compute some commutators.

Lemma 1.6. —

Let

µ∈Q

,

µ≥0

. Let

x∈ Dq(g)^>_µ⁰

and

y ∈ Dq(g)^<_−µ⁰

. Then

x F_i−F_ix=K_ir_i(x)−r_i⁰(x)L_i⁻¹

q_i−q_i⁻¹

and

E_iy −y E_i=ρi(y)K_i−L_i⁻¹ρ_i⁰(y) q_i−q_i⁻¹ .

Proof. — We only show the first formula, the second one is proven similarly. If

x =1

or

x=E_i

, the formula is satisfied. Now, supposing that it is true for

x

and

x⁰

, we show it for

x x⁰

:

x x⁰F_i−F_ix x⁰=x(x⁰F_i−F_ix⁰) + (x F_i−F_ix)x⁰

= (q_i−q_i⁻¹)⁻¹(x(K_ir_i(x⁰)−r_i⁰(x⁰)L⁻¹_i ) + (K_ir_i(x)−r_i⁰(x)L_i⁻¹)x⁰)

= (q_i−q_i⁻¹)⁻¹(K_i(q^{−〈αi,µ〉}x r_i(x⁰) +r_i(x)x⁰)−(x r_i⁰(x⁰) +q^{−〈αi,µ0〉}r_i⁰(x)x⁰))

= (q_i−q_i⁻¹)⁻¹(K_ir_i(x x⁰)−r_i⁰(x x⁰)L⁻¹_i ),

where we used Lemma 1.5 in the last equality.

1.5. Some properties of the pairing. — We gather here some well known properties of the pairing.

Proposition 1.7. —

Let

x∈ Dq(g)^>0

,

y ∈ Dq(g)^<0

and

λ,µ∈Q

. Then 1.

(K_λx,L_µy) = (K_λ,L_µ)(x,y)

.

2. Suppose

λ,µ≥0

. If

x

is of weight

λ

and

y

of weight

−µ

then

(x,y) =0

if

λ6=µ

.

Proof. — For the first equality, it suffices to show it for

x

and

y

homogeneous. It then follows easily from (1), (2) and the fact that

(K_λ,y⁰) =0= (x⁰,L_µ)

for any

x⁰∈ Dq(g)^>0

and

y⁰∈ Dq(g)^<0

.

For the second assertion, we proceed by induction on

ht(λ) = Pn

i=1λi

, where

λ = Pn

i=1λiαi

. We can suppose that

x

is a product of the generators

E_i

. The case

ht(λ) =1

, i.e.

λ∈Π

is easily proved by induction on

ht(µ)

. Then, writing

x=E_ix⁰

and

∆(y) = X

0≤ν≤µ

L⁻¹_ν y_−(µ−ν)⁰ ⊗y_−ν⁰⁰, y_−ν⁰ ,y_−ν⁰⁰ ∈ Dq(g)^<_−ν⁰,

one has

(E_ix⁰,y) = X

0≤ν≤µ

(x⁰,L⁻¹_ν y_−(µ−ν)⁰ )(E_i,y_−ν⁰⁰) = (x⁰,y_−(µ−α⁰

i))(E_i,y_−α⁰⁰

i).

As

λ6=µ

, the two terms can not be simultaneously non-zero.

Using these two facts and an easy induction, one can show that, for

1≤i≤n

, and

r∈N

,

(E_i^r,F_i^r) =q⁻

r(r−1) 2

i [r]q_i! (q_i−q_i⁻¹)^r .

We now turn to the compatibility between the pairing and the

r_i

,

r_i⁰

,

ρi

and

ρ_i⁰

defined

in section 1.4.

Lemma 1.8. —

Let

x∈ Dq(g)^>_µ⁰

and

y ∈ Dq(g)^<_−µ⁰

. One has

(x,F_iy) = (E_i,F_i)(r_i(x),y), (x,y F_i) = (E_i,F_i)(r_i⁰(x),y), (E_ix,y) = (E_i,F_i)(x,ρ⁰_i(y)), (x E_i,y) = (E_i,F_i)(x,ρi(y)).

Proof. — It follows easily from (1), (2) and Proposition 1.7.

We end this section with [Tan92, Proposition 2.1.4].

Proposition 1.9. —

The restriction of the pairing to

Dq(g)^>0× Dq(g)^<0

is non-degenerate.

1.6. Basis for

Dq(g)

. — We introduce the divided power

E_i^(r)

and

F_i^(r)

for

1≤i ≤n

and

r ≤0

:

E_i^(r)= E_i^r [r]qi

and F_i^(r)= F_i^r [r]qi

.

The following is the analogue of [Lus90, Theorem 3.1] for

Dq(g)

.

Proposition 1.10. —

For all

1≤i≤n

, there exists a unique

Q(q)

-algebra isomorphism

T_i

such that:

T_i(K_λ) =K_si(λ) T_i(L_λ) =L_si(λ), ∀λ∈Q,

and, for

1≤j≤n

, setting

r =−〈αj,α^∨_i〉

,

T_i(E_j) =







−F_iL_i

if

i=j,

r

X

k=0

(−1)^kq_i^−kE_i^(r−k)E_jE_i^(k)

otherwise

,

T_i(Fj) =







−K_i⁻¹E_i

if

i=j, Xr

k=0

(−1)^kq_i^kF_i^(k)F_jF_i^(r−k)

otherwise

,

We also have the analogue of [Lus90, Theorem 3.2] for

_Dq(g)

.

Proposition 1.11. —

The

₍T_i)1≤i≤n

satisfy the braid relations and therefore define a morphism from the braid group of

W

to the algebra automorphisms of

Dq(g)

.

Choose now a reduced decomposition

w₀=s_i1s_i2· · ·s_ir

of the longest element

w₀

of

W

. The positive roots are then

Φ⁺=

s_i1s_i2· · ·s_ik(αik+1)

0≤k≤r −1 .

For any

α=s_i1s_i2· · ·s_ik(αik+1)∈Φ⁺

define

E_α=T_i1T_i2· · ·T_ik(Ei_k+1) and F_α=T_i1T_i2· · ·T_ik(Fi_k+1).

Note that

E_α

(resp

F_α

) is homogeneous of degree

α

(resp.

−α

). The choice of a reduced decomposition of

w₀

gives an order on

_Φ⁺

, namely

s_i1s_i2· · ·s_ik(αik+1)´s_i1s_i2· · ·s_ik+1(αik+2)

for every

k

. All products will be ordered with respect to this order.

Proposition 1.12

([Lus90, Theorem 6.7]). — The elements

Y

α∈Φ⁺

E_α^(nα), n_α∈N

form a

Q(q)

-basis of

Dq(g)^>0

.

The elements

Y

α∈Φ⁺

F_α^(nα), n_α∈N

form a

Q(q)

-basis of

Dq(g)^<0

.

We compute now the dual basis of

_Dq(g)^>0

with respect to

(·,·)

. The following can be found in [Jan96, 8.29].

Proposition 1.13. —

For any

α∈Φ⁺

, let

a_α,b_α∈N

. We have

‚ Y

α∈Φ⁺

E_α^aα,Y

α∈Φ⁺

F_α^bα

Œ

= Y

α∈Φ⁺

δa_α,b_α

q⁻

aα(aα−1)

α 2 [a_α]q_α! (q_α−q_α⁻¹)^aα .

Therefore, the dual basis of

^€Q_α∈Φ+E_α^(nα)Š

(n_α)∈N^Φ+

is

‚ Y

α∈Φ⁺

q

nα(nα−1)

α 2 [n_α]q_α!(q_α−q_α⁻¹)^nαF_α^(nα)

Œ

(nα)∈N^Φ+

.

1.7. Quasi-

R

-matrix. — The algebra

Dq(g)

is not quasi-triangular. Nevertheless, we con- struct a quasi-

R

-matrix which will endow the usual category of modules with a braiding (see section 2.3). We adapt the exposition of Jantzen [Jan96, Chapter 7] to the case of

Dq(g)

.

For any

µ ∈Q

,

µ≥0

, fix a basis

(u_i^µ)i∈I_µ

of

Dq(g)^>0_µ

and let

(v_i^µ)i∈I_µ

the dual basis in

Dq(g)^<0_−µ

with respect to the pairing

(·,·)

. Set

Θ_µ=X

i∈I_µ

u_i^µ⊗v_i^µ∈ Dq(g)⊗ Dq(g).

Note that this does not depend on the choice of the basis of

_Dq(g)_µ

. A homogeneous basis of

Dq(g)^>0

and its dual with respect to

(·,·)

have already been computed above. Following [Ros90, A.1], we define an algebra automorphism

Ψ

of

Dq(g)⊗ Dq(g)

by

Ψ(K_i⊗1) =K_i⊗1, Ψ(1⊗K_i) =1⊗K_i, Ψ(L_i⊗1) =L_i⊗1, Ψ(1⊗L_i) =1⊗L_i, Ψ(E_i⊗1) =E_i⊗L⁻¹_i , Ψ(1⊗E_i) =K_i⁻¹⊗E_i,

Ψ(F_i⊗1) =F_i⊗L_i, Ψ(1⊗F_i) =K_i⊗F_i. Proposition 1.14. —

Let

µ∈Q

,

µ≥0

. We have

Θ_µ(K_i⊗K_i) =Ψ(K_i⊗K_i)Θ_µ, Θ_µ(L_i⊗L_i) =Ψ(L_i⊗L_i)Θ_µ,

Θ_µ(1⊗E_i) +Θ_µ−αi(E_i⊗K_i) =Ψ(E_i⊗1)Θ_µ−αi+Ψ(K_i⊗E_i)Θ_µ, Θ_µ(F_i⊗1) +Θ_µ−αi(L⁻¹_i ⊗F_i) =Ψ(1⊗F_i)Θ_µ−αi+Ψ(F_i⊗L_i⁻¹)Θ_µ.

Proof. — We follow closely the proof of [Jan96, Lemma 7.1]. The first two equations are trivial. We will use the following fact: for any

µ∈Q

,

µ≥0

, and any

x ∈ Dq(g)^>_µ⁰

and

y ∈ Dq(g)^<_−µ⁰

we have

x=X

i

(x,v_i^µ)u_i^µ and y =X

i

(u^µ_i,y)v_i^µ.

We set

c_i= (q_i−q_i⁻¹)⁻¹

, and we have

(1⊗E_i)Θ_µ−Θ_µ(1⊗E_i) =X

j

u^µ_j ⊗€

E_iv^µ_j −v_j^µE_iŠ

=c_iX

j

u^µ_j ⊗€ ρi

€v_j^µŠ

K_i−L⁻¹_i ρ_i⁰€ v_j^µŠŠ

by Lemma 1.6

=c_iX

j,k

u^µ_j ⊗”€

u_k^µ−αi,ρi

€v_j^µŠŠ

v_k^µ−αiK_i−€

u_k^µ−αi,ρ⁰_i€ v_j^µŠŠ

L⁻¹_i v_k^µ−αi—

=c_iX

j,k

u^µ_j ⊗”€

u_k^µ−αiE_i,v_j^µŠ

v_k^µ−αiK_i−€

E_iu^µ−α_k ⁱ,v_j^µŠ

L_i⁻¹v_k^µ−αi—

by Lemma 1.8

=X

k

u_k^µ−αiE_i⊗v_k^µ−αiK_i−E_iu_k^µ−αi⊗L⁻¹_i v_k^µ−αi

=Θ_µ−αi(Ei⊗K_i)− E_i⊗L⁻¹_i Θ_µ−αi,

as expected because

Ψ(E_i⊗1) =E_i⊗L⁻¹_i

and

Ψ(K_i⊗E_i) =1⊗E_i

. A similar calculation shows the fourth formula.

Define, as in [Lus10, Chapter 4], the completion

Dq(g)⊗Dˆ q(g)

of

Dq(g)⊗ Dq(g)

with re- spect to the descending sequence of spaces

(Dq(g)⊗Dq(g))N=Dq(g)⊗ X

ht(µ)≥N

Dq(g)^>0Dq(g)⁰Dq(g)^<_−µ⁰+ X

ht(µ)≥N

Dq(g)^<0Dq(g)⁰Dq(g)^>_µ⁰⊗Dq(g).

The morphism

_Ψ

extends by continuity to

_Dq(g)⊗Dˆ q(g)

and we consider the following element of

_Dq(g)⊗Dˆ q(g)

Θ=X

µ∈Qµ≥0

Θ_µ.

Then we can rewrite Proposition 1.14 as

Θ∆(u) = Ψ◦∆^op (u)Θ

for any

u∈ Dq(g)

.

We also have the analogue of [Jan96, Lemma 7.4].

Lemma 1.15. —

For

µ∈Q

,

µ≥0

, we have

(∆⊗id)(Θ_µ) = X

0≤ν≤µ

(Θ_ν)13(1⊗K_ν⊗1)(Θ_µ−ν)23,

and

(id⊗∆)(Θ_µ) = X

0≤ν≤µ

(Θ_ν)13(1⊗L⁻¹_ν ⊗1)(Θ_µ−ν)12.

Proof. — First, note that for any

x∈ Dq(g)^>_µ⁰

∆(x) = X

0≤ν≤µ i,j

€x,v_i^νv_j^µ−νŠ

u_i^ν⊗K_νu^µ−ν_j ,

and for any

y ∈ Dq(g)^<0_−µ

∆(y) = X

0≤ν≤µ i,j

€u^ν_iu^µ−ν_j ,yŠ

L⁻¹_ν v_j^µ−ν⊗v_i^ν.