Table Of ContentQUANTUM COADJOINT ACTION AND THE 6j-SYMBOLS OF U sl
q 2
1
1 STE´PHANEBASEILHAC
0
2 Dedicated to my advisor, Claude Hayat-Legrand, on the occasion of her 65th birthday.
n
a Abstract. Wereviewtherepresentation theoryofthequantumgroup Uεsl2Cat aroot of
J
unity ε of odd order, focusing on geometric aspects related to the 3-dimensional quantum
8 hyperbolic field theories (QHFT). Our analysis relies on the quantum coadjoint action of
1
De Concini-Kac-Procesi, and the theory of Heisenberg doubles of Poisson-Lie groups and
] Hopf algebras. We identify the 6j-symbols of generic representations of Uεsl2C, the main
T ingredients of QHFT, with a bundle morphism defined over a finite cover of the algebraic
G quotient PSL2C//PSL2C, of degree two times the order of ε. It is characterized by a
non Abelian 3-cocycloid identity deforming the fundamental five term relation satisfied by
.
h theclassical dilogarithm functions,thatrelates thevolumeofhyperbolic3-polyhedraunder
t retriangulation, and more generally, the simplicial formulas of Chern-Simons invariants of
a
m 3-manifolds with flat sl2C-connections.
[
1
v 1. Introduction
0
4 After more than twenty years of outstanding efforts, the geometry of quantum groups, es-
4
pecially the relationships between their representation theories at roots of unity and the
3
. underlying Lie groups, remains a prominent matter of quantum topology. It stands, for in-
1
stance, in the background of the geometric realization problem of the combinatorially defined
0
1 statespacesoftheReshetikhin-TuraevTQFT([BHMV],[T]),thedeformationquantizationof
1 character varieties of 3-manifolds via skein modules [BFK, PS], or the asymptotic expansion
:
v of quantum invariants ([Prob, Ch. 7], [MN]).
i Inrecentyears, newandcompletely unexpectedinteractions betweenthenon restrictedquan-
X
tum group U = U sl C and 3-dimensional hyperbolic geometry have been revealed by the
r q q 2
a volume conjecture [Ka], and the subsequent development of the quantum hyperbolic field
theories (QHFT) [BB1, BB2, BB3] and quantum Teichmu¨ller theory [BL]. The global pic-
ture is that many of the fondamental invariants of hyperbolic (ie. PSL C) geometry, like the
2
volume of hyperbolic manifolds, and more generally (some of the) Chern-Simons invariants
of 3-manifolds with flat sl C-connections, should be determined by the semi-classical limits
2
of these quantum algebraic objects.
Like any topological quantum field theory, the QHFT are symmetric tensor functors defined
on a category of 3-dimensional bordisms. For QHFT, these are equipped with additional
structures given by refinements of holonomy representations in PSL C, and links up to
2
isotopy. The QHFT are built from ”local” basic datas, universally encoded by the moduli
space of isometry classes of “flattened” (a kind a framing) hyperbolic ideal 3-simplices, and
the associativity constraint, or 6j-symbols, of the cyclic representations of a Borel subalgebra
Date: January 31, 2010 and,in revised form, XXX..
2010 Mathematics Subject Classification. Primary 17B37, 57R56; Secondary 14L24.
Keywords andphrases. quantumgroups,Poisson-Liegroups,coadjoint action,geometricinvarianttheory,
dilogarithm functions, TQFT, invariants of 3-manifolds.
This work was supported by the grant ANR-08-JCJC-0114-01 of the French Agence Nationale de la
Recherche.
2 STE´PHANEBASEILHAC
U bofU ataprimitiverootofunityεofoddorder. Workingwithsuchamodulispaceallows
ε ε
one to define the QHFT also for 3-dimensional bordisms equipped with holonomies having
singularities along the links, like cusped hyperbolic manifolds (the link being “at infinity”),
and to get surgery formulas. For cylindrical bordisms the QHFT are related by this way to
the local version of quantum Teichmu¨ller theory ([BB3],[Bai, BBL]).
Similarly, the Chern-Simons gauge theory for flat sl C-connections, which was originally
2
derived from the integral complex-valued invariant “volume” 3-form of SL C via secondary
2
characteristic class theory,definesafunctorbymeansofthesamemodulispaceastheQHFT,
usingNeumann’ssimplicial formulas [N]in placeof quantumstate sums,whereadilogarithm
function formally corresponds to the cyclic 6j-symbols of U b ([BB2], see also [Ma]). In fact,
ε
both maps are 3-cocycloids in a natural way, the latter on the category of cyclic U b-modules
ε
with (partially defined) tensor product, and the former on the group PSL C with discrete
2
topology, via Neumann’s isomorphism of H (BPSL Cδ;Z) with a certain extension of the
3 2
Bloch group.
In this paper we consider this interplay of Abelian vs. non Abelian cohomological structures,
which certainly concentrates a key part of the quantization procedure relating the Chern-
Simons theory for PSL C to the QHFT. We describe in detail the non Abelian part of the
2
story, that is, how the simple U -modules “fiber” over PSL C, and the 6j-symbols of regular
ε 2
U -modules (rather than the cyclic U b-modules). By the way we indicate common features
ε ε
and discrepancies with the 6j-symbols of the color modules of the restricted quantum group
U¯ .
ε
We point out also that the cyclic 6j-symbols of U b, or (basic) matrix dilogarithms, coincide
ε
with the regular 6j-symbols of U . More precisely, we define a bundle Ξ(2) of regular U -
ε ε
modules over a covering of degree n2 (n being the order of ε) of a smooth subset of a
Poisson-Lie group H2 dual to PSL C2, endowed with an action of an infinite dimensional
2
Lie group derived from the quantum coadjoint action of De Concini-Kac-Procesi, originally
defined for U . We have (see Theorem 6.13, 6.14 and 6.16 for precise statements):
ε
Theorem 1.1. The regular 6j-symbols of U and the matrix dilogarithms coincide and define
ε
a bundle morphism : Ξ(2) Ξ(2) equivariant under the quantum coadjoint action.
R −→
Since the quantum coadjoint action lifts the adjoint action of PSL C via an unramifield
2
2-fold covering H PSL C0 of the big cell of PSL C, it will follow that descends to
2 2
→ R
a morphism of a vector bundle of rank n2 over a 2n-fold covering of the algebraic quotient
PSL C//PSL C.
2 2
Theremarkabledependenceofthematrixdilogarithms oncross-ratios of4-tuplesofpointson
P1, which was previously known by direct computation and allowed the QHFT to be defined
on the moduli space of flattened hyperbolic ideal tetrahedra, is a consequence of Theorem
1.1. This alternative description is presented in Section 6.4.
In order to achieve our goals we have to review quite a lot of material, starting from the basic
properties of U (Section 2), and developing in detail its representation theory and quantum
ε
coadjoint action (Section 4 and 3, respectively). The 6j-symbols of the color modules of U¯
ε
and of the regular modules of U are defined in Section 5. Theorem 1.1 is proved in Section
ε
6. There we make a crucial use of fundamental results of Semenov-Tian-Shansky [STS],
Weinstein-Xu [WX],andLu[Lu1]onPoisson-Liegroupsandtheirdoublesandquantizations.
There should be no obstruction to extend Theorem 1.1 to the quantum groups U g of arbi-
ε
trary complex simple Lie algebras g. The corresponding quantum coadjoint action theory is
described in [DCK], [DCKP], [DCP], and [DCPRR]. Recent works of Geer and Patureau-
Mirand [GP] show that the QHFT setup extends to ”relative homotopy quantum field theo-
ries”, including TQFT associated to the categories of finite dimensional weight U g-modules.
ε
QUANTUM COADJOINT ACTION AND THE 6j-SYMBOLS OF Uqsl2 3
A challenging problem is to relate them to the three-dimensional Chern-Simons theory for
flat g-connections.
Finally, let us note that a similar approach can be used to describe “holonomy” R-matrices
for the regular U g-modules, in the spirit of [KaRe].
ε
We will often meet notions from the theory of Poisson-Lie groups. The reader will find the
needed material in standard textbooks, like [CP], [ES] and [KS].
Thispaper is based on my lectures at theworkshop “Interactions Between HyperbolicGeom-
etry, QuantumTopology andNumberTheory”heldat ColumbiaUniversity (June3rd -19th,
2009). I am grateful to the organizers for their hospitality and wonderful working conditions
during my stay.
2. The quantum group U sl
q 2
2.1. Definition & PBW basis. We fix our ground ring to be C, and denote by q a complex
number such that q = 1,0,1. When q is constrained to be a root of unity we denote it by
6 −
ε, and we assume that ε has odd order n 3.
≥
Definition 2.1. The quantum group U = U sl is the algebra generated over C by elements
q q 2
E, F, K and K−1, with defining relations KK−1 = K−1K = 1 and
K K−1
(2.1) KEK−1 = q2E , KFK−1 = q−2F , [E,F] = − .
q q−1
−
The algebra U is a Hopf algebra with coproduct ∆: U U U , antipode S: U U
q q q q q q
and counit η: U C defined on generators by → ⊗ →
q
→
∆(K)= K K
⊗
∆(E) = E 1+K E, ∆(F) = 1 F +F K−1
(2.2) ⊗ ⊗ ⊗ ⊗
S(K)= K−1, S(E) = K−1E, S(F) = FK
− −
η(K) = 1, η(E) = η(F) =0.
To make sense of this definition, let us just recall here that being a Hopf algebra means that
∆ and η are morphisms of algebras satisfying the coassociativity and counitality constraints
(2.3) (∆ id) ∆ = (id ∆) ∆ , (η id) ∆ = (id η) ∆ = id.
⊗ ◦ ⊗ ◦ ⊗ ◦ ⊗ ◦
Here thealgebra structureof U U is by componentwise multiplication, andU is identified
q q q
with U C in the canonical w⊗ay. The antipode S is the inverse of the identity for the
q
⊗
convolution product, that is, it satisfies
(2.4) µ (id S) ∆ = µ (S id) ∆ = η1,
◦ ⊗ ◦ ◦ ⊗ ◦
where µ: U U U is the product. In particular, this implies S(1) = 1, η S = η,
q q q
⊗ → ◦
S(xy) = S(y)S(x) for all x, y U , and
q
∈
(2.5) (S S)∆ = τ ∆ S,
⊗ ◦ ◦
where τ(u v) = v u is the flip map ([Ks, Ch. III]).
⊗ ⊗
Remark 2.2. A simply-connected version of U is often considered in the litterature (see
q
eg. [DCP, 9]). It is obtained by adding to the generators a square root of K acting by
§
conjugation on E and F by multiplication by q and q−1, respectively.
Exercise 2.3. Define an algebra U′ with generators E, F, K and K−1 satisfying the same
q
relations as in U except the last one in (2.1), and with a further generator L such that
q
[E,F] = L , (q q−1)L = K K−1
(2.6) − −
[L,E] = q(EK +K−1E) , [L,F] = q−1(FK +K−1F).
−
4 STE´PHANEBASEILHAC
Show that we have isomorphisms U = U′, U′ = U[K]/(K2 1), U = U′/(K 1), where
q ∼ q 1 ∼ − ∼ 1 −
U = Usl is the universal enveloping algebra of sl .
2 2
Thisexerciseshowsthat,assuggestedbythenotation,U isagenuinedeformationdepending
q
on the complex parameter q of the universal enveloping algebra of sl . Like the latter, we
2
can think of U as a ring of polynomials (in non commuting variables). In particular, we have
q
the following fundamental result (see [J, Th. 1.5-1.8] or [Ks, Th. VI.1.4] for a proof):
Theorem 2.4. 1) (PBW basis) The monomials FtKsEr, where t, r N and s Z, make a
∈ ∈
linear basis of U .
q
2)ThealgebraU hasnozerodivisorsandisgivenagradingbystipulatingthateachmonomial
q
FtKsEr is homogeneous of degree r t.
−
Notethattherelations(2.1)arehomogeneousofdegree1, 1and0,respectively. Productsof
−
monomials can bewritten in thebasis FtKsEr r,t∈N,s∈Z by usingthe two firstcommutation
{ }
relations in (2.1), together with
min(r,s)
(2.7) ErFs = Fs−ih Er−i, r,s N,
i
∈
i=0
X
where the h are Laurent polynomials in C[K,K−1] given by
i
i
r s
h = [i]! [K;i+j (r+s)].
i i i −
(cid:20) (cid:21)(cid:20) (cid:21) j=1
Y
Here we assume that the product is 1 for i = 0, and we put
Kql K−1q−l
(2.8) [K;l] = − .
q q−1
−
We use also the standard notations for q-integers, q-factorials, and q-binomial coefficients
ql q−l l [l]!
(2.9) [l] = − , [l]!= [l][l 1]...[1] , = , l Z,
q q−1 − m [m]![l m]! ∈
− (cid:20) (cid:21) −
with [0]! = 1 by convention. Note that [l] = ql−1+...+q1−l, so [l] Z[q,q−1]. Also,
∈
l l l
(2.10) = = 1 and [l] divides if 1 < m < l and l is odd.
0 l m
(cid:20) (cid:21) (cid:20)(cid:21) (cid:20) (cid:21)
The elements (2.8) appear in all computations involving commutators. In the sequel we will
use (2.7) when r = 1 and s is arbitrary, or r is arbitrary and s = 1:
(2.11) EFs = FsE +[s]Fs−1[K;1 s] , FEr = ErF [r]Er−1[K;r 1].
− − −
Another important identity is the q-binomial formula, which holds for any u, v such that
vu = q2uv [Ks, Ch. VI (1.9)]:
r
r
(2.12) (u+v)r = qj(r−j)ujvr−j.
j
j=0(cid:20) (cid:21)
X
l
In particular, it implies that Z[q,q−1].
m ∈
(cid:20) (cid:21)
Exercise 2.5. (a) Prove (2.11) by induction, using the relations (2.1).
(b) Show that the second relation in (2.11) follows from the first one by applying the Cartan
automorphism of the algebra U , defined by
q
(2.13) ω(E) = F , ω(F) = E , w(K) = K−1.
QUANTUM COADJOINT ACTION AND THE 6j-SYMBOLS OF Uqsl2 5
2.2. Center. Theorem 2.4 gives a lot of informations on the center Z of U . Indeed, denote
q q
by U the degree d piece of U . For any u U we have
d q d
∈
(2.14) KuK−1 = q2du.
Hence, when q is not a root of unity, U is the eigenspace of ad with eigenvalue q2d. In
d K
particular, Z U .
q 0
⊂
When q = ε is a root of unity of odd order n 3, Z is much bigger. Let us show that
ε
≥
it is generated by En, Fn and Z U . Since the relations (2.1) are homogeneous, the
ε 0
∩
homogeneous parts of a central element are central. Hence it is enough to prove our claim
for a central element u U . By (2.14), if u U is central then n divides d. Now, if d = ln
d d
for some l Z 0 wi∈th l > 0 (resp. l < 0∈), U is spanned by the monomials FtKsEt+ln
d
∈ \{ }
(resp. Ft+lnKsEt). Hence u = u′Eln (resp. u= Flnu′) for some u′ U . The relations
0
∈
KlE = q2lEKl , KlF = q−2lFKl
(2.15) , l N
KEl = q2lElK , KFl = q−2lFlK ∈
imply that K±n Z . Together with [n] = 0 and (2.11) for s = n and r = n, they give also
ε
⊂
En, Fn Z . Since U has no zero divisors, we get that any u U Z can be decomposed
ε ε d ε
⊂ ∈ ∩
as a product of central elements Eln and Fln with a central element in U .
0
Definition 2.6. The subalgebra Z of the center Z of U is generated by En, Fn and K±n.
0 ε ε
The algebra Z plays a key role in the structure of U sl . In particular, the next theorem
0 ε 2
shows that any element of Z is a root of a monic polynomial with coefficients in Z .
ε 0
Theorem 2.7. 1) The algebra U is a free Z -module of rank n3, with basis the monomials
ε 0
FtKsEr, 0 t,s,r n 1.
≤ ≤ −
2) The center Z of U is a finitely generated algebra, and is integrally closed over Z .
ε ε 0
Proof. The first claim follows directly from Theorem 2.4 (note in particular that En, Fn
and K±n are independent). It implies that U is a noetherian Z -module, and so the Z -
ε 0 0
submodule Z is finitely generated. Hence Z is integrally closed over Z [AMcD, Th. 5.3].
ε ε 0
Then, by Hilbert’s basis theorem Z is finitely generated as an algebra. (cid:3)
ε
In another direction, we will see in the next section that Z is the basic link between U and
0 ε
the Lie group PSL C. This depends on the fact that it is a (commutative) Hopf algebra.
2
To prove this, let us introduce for future reference a prefered set x,y,z of generators of Z ,
0
{ }
given by
(2.16) x = (ε ε−1)nEnK−n , y = (ε ε−1)nFn , z = Kn.
− − −
We also put
(2.17) e= (ε ε−1)nEn , f = (ε ε−1)nFnKn.
− − −
Note that
x = T(y) , f = T(e), x = S(e) , y = S(f)
whereS istheantipodeof U ,andT thebraid group automorphism ofthealgebraU , defined
ε ε
by
(2.18) T(E) = FK , T(F)= K−1E , T(K) = K−1 , T(K−1) = K.
− −
Lemma 2.8. The subalgebra Z of U is a Hopf subalgebra.
0 ε
6 STE´PHANEBASEILHAC
Proof. We have to show that ∆(Z ) Z Z and S(Z ) Z . Because these maps are
0 0 0 0 0
⊂ ⊗ ⊂
morphisms of algebras, it is enough to prove this on the generators x, y and z of Z . We
0
claim that
∆(z) = z z
⊗
∆(x) =x z−1+1 x, ∆(y) = 1 y+y z−1
(2.19) ⊗ ⊗ ⊗ ⊗
S(z) = z−1, S(x) = zx, S(y)= yz
− −
η(z) = 1, η(x) = η(y) = 0.
The proof of the formulas for S is immediate. As for ∆, they are consequences of (2.10), the
q-binomial identity (2.12), and the fact that [n]= 0 when q = ε. (cid:3)
So far, we have only considered in (2.14) the invariance of the center of U under conjugation
q
by K. In order to describe the effect of the conjugation action by E and F, let us introduce
the automorphisms γ of C[K,K−1] given by
l
(2.20) γ (K)= qlK, l Z.
l
∈
By (2.1), for all h C[K,K−1] we have
∈
hE = Eγ (h) and hF = Fγ (h).
2 −2
It is easy to check that u = Fih Ei U satisfies uE = Eu and uF = Fu in U if and
i≥0 i ∈ 0 q
only if for all i,
P
(2.21) h γ (h )= [i+1][K; i]h .
i −2 i i+1
− −
SinceC[K,K−1]is anintegral domain, h is thus determinedinductively byh if [j+1] = 0
i+1 0
6
for all 0 j i. This condition is satisfied for all i when q is not a root of unity, and for
≤ ≤
i < n 1 when q = ε. Hence the projection map
−
π : U C[K,K−1]
(2.22) ∞ F0rh Er −→ h
r=0 r 7−→ 0
is injective over Zq ⊂ U0 when q Pis not a root of unity, and injective over Zε∩U0′ when q = ε,
where
n−1
(2.23) U′ = Fih Ei h ,...,h C[K,K−1] U .
0 i | 0 n−1 ∈ ⊂ 0
( )
i=0
X
In fact, Ker(π) = FU E is a two-sided ideal in U , so π is a homomorphism of algebras,
0 0
the Harish-Chandra homomorphism. It is easy to check that Z is generated by Fn, En and
ε
Z U′. On another hand, (2.21) is solved by
ε∩ 0
qK +q−1K−1
h = , h = 1 , h = 0 if r 2,
0 (q q−1)2 1 r ≥
−
which defines the Casimir element
qK +q−1K−1
(2.24) Ω = FE + Z .
(q q−1)2 ∈ q
−
Note that (see (2.20))
K +K−1
γ π(Ω) = .
−1◦ (q q−1)2
−
We will see in Section 4 that Z is actually generated by Ω, and Z U′ by Ω and K±n (see
q ε∩ 0
Theorem 4.9; for an alternative approach based on induction on i in (2.21), see [J, Prop.
2.20]). More precisely:
QUANTUM COADJOINT ACTION AND THE 6j-SYMBOLS OF Uqsl2 7
Theorem 2.9. (1) If q is not a root of unity, Z is a polynomial algebra over C generated
q
by Ω, and γ π :Z C[K +K−1] is an isomorphism.
−1 q
(2) If q = ε is ◦a root of→unity of odd order n 3, then Z is the algebra over C generated by
ε
≥
En, Fn, K±n and Ω with relation
n−1 Kn+K−n 2
(2.25) (Ω c ) =EnFn+ −
− j (ε ε−1)2n
j=0 −
Y
where
εj+1+ε−j−1
c = .
j (ε ε−1)2
−
Statement (2) is part of [DCK, Th. 4.2]. It shows that Z is a finite extension of Z built on
ε 0
the image C[Ω] of the injective homomorphism
(2.26) (γ π)−1: C[K +K−1] Z .
−1 ε
◦ −→
The domain C[K +K−1] is the fixed point set of C[K,K−1] under the involution s(K±1) =
K∓1, which can be identified with a generator of the Weyl group of PSL C.
2
3. The quantum coadjoint action for U
ε
Consider the set Spec(Z ) of algebra homomorphisms from the center Z of U to C. An
ε ε ε
element of Spec(Z ) is called a central character of U . The set Spec(Z ) has a very rich
ε ε ε
Poisson geometry related to the adjoint action of PSL C, that we are going to describe. It
2
will be used in Section 4.
Since Z is finitely generated, by Hilbert’s nullstellensatz Spec(Z ) is an affine algebraic set.
ε ε
The inclusion Z Z induces a regular (restriction) map
0 ε
⊂
(3.1) τ: Spec(Z ) Spec(Z ).
ε 0
−→
We have seen in Proposition 2.7 that Z is integrally closed over Z . Hence τ is finite: any
ε 0
y Im(τ) has a finite number of preimages. From this property it follows that τ is surjective
∈
[Sh, 5.3].
§
3.1. A Poisson-Lie group structure on Spec(Z ). We need some explicit formulas. De-
0
note by T, U and PB the subgroups of PSL C = SL C/( 1) of diagonal, upper/lower
± ± 2 2
±
unipotent and upper/lower triangular matrices up to sign, with their natural structures of
affine algebraic groups induced by the adjoint action on sl C. The action
2
a b aw+b
: w , w P1
c d 7→ cw+d ∈
(cid:18) (cid:19)
identifies PSL C with the group Aut(P1) of automorphisms of the Riemann sphere, and T
2
with the subgroup of maps w zw, z C∗. Consider the group
7→ ∈
H = (tu ,t−1u ) t T,u U PB PB .
+ − ± ± + −
{ | ∈ ∈ }⊂ ×
By evaluating on the generators x, y and z of Z (see (2.16)) we get a map
0
ψ′ : Spec(Z ) C2 C∗
(3.2) 0 −→ ×
g (x ,y ,z ).
g g g
7−→
For all g Spec(Z ), consider the automorphisms of P1 given in ψ′-coordinates by
0
∈
ψ (g): w z w x z , ψ (g): w w/( z y w+z ), w P1.
1 g g g 2 g g g
7→ − 7→ − ∈
8 STE´PHANEBASEILHAC
Under the above isomorphism of Aut(P1) with PSL C, they correspond respectively to
2
√zg xg√zg 1/√zg 0
− PB , PB
± 0 1/√zg ∈ + ± yg√zg √zg ∈ −
(cid:18) (cid:19) (cid:18) − (cid:19)
for any choice of square root √zg. Define
ψ : Spec(Z ) H
0
−→
g (ψ (g),ψ (g)).
1 2
7−→
Clearly, ψisanisomorphismofalgebraicvarieties. ItgivesH aPoisson-Liebracketasfollows.
By using the PBW basis FtKsEr r,t∈N,s∈Z we can identify Uε as a linear subspace of Uq,
considered as a family of{algebras o}ver C with varying parameter q. Hence, for any given
element u U we can specify a lift u˜ U , such that
ε q
∈ ∈
u= u˜ mod(qn q−n)U .
q
−
We can write this as u= lim u˜. Then, for all a Z and u U , put
q→ε 0 ε
∈ ∈
[a˜,u˜]
D (u) = lim .
a q→εn(qn q−n)
−
Note that
1 a˜ 1 a˜
(3.3) D (u) = lim ,u˜ = lim ,u˜
a (ε ε−1) q→ε n[n] (ε ε−1)n q→ε [n]!
− (cid:20) (cid:21) − (cid:20) (cid:21)
where we use the formula [n 1]! = n(ε ε−1)1−n.
− −
Proposition 3.1. (i) The map ψ: Spec(Z ) H is an isomorphism of algebraic groups.
0
→
(ii) The maps D : U U are well-defined derivations of U preserving Z and Z . At the
a ε ε ε 0 ε
→
generators e= xz and z in (2.17) we have:
−
(ε ε−1)n−1 (ε ε−1)n
(3.4) D (F) = − [K;1]En−1 , D (K±1)= − K±1En
e e
n ∓ n
1 1
(3.5) D (E) = KnE , D (F) = FKn.
z z
n −n
where [K;l] is given by (2.8) with q = ε.
Proof. (i) This follows from classical duality arguments in Hopf algebra theory (see [Mo,
§
9]). Because Z is a commutative Hopf algebra (Lemma 2.8), the algebraic set Spec(Z ) has
0 0
a canonical group structure identifying Z with the algebra of regular functions on Spec(Z ).
0 0
It is defined dually by stipulating that for any u Z and g, h Spec(Z ),
0 0
∈ ∈
(3.6) u(gh) = µ ∆(u)(f,g) , u(g−1)= S(u)(g) , u(11) = η(u).
◦
Here, 11 Spec(Z ) is the identity and µ the product of Z . The associativity of the product
0 0
∈
of Spec(Z ) follows from the coassociativity of ∆ (see (2.3)). The inverse is well-defined by
0
u(11) = η(u) and the computation
u(gg−1)= µ (id S) ∆(u)(g,g) =η(u)1(g) = η(u),
◦ ⊗ ◦
where 1 Z is identified with the constant function on Spec(Z ) with value 1, and we use
0 0
∈
(2.4) in the second equality. By (2.19) and (3.6) we get in the ψ′-coordinates (x ,y ,z ) of
g g g
the point g of Spec(Z ):
0
x = x +x z−1 , y = y +y z−1 , z = z z
gh h g h gh h g h gh g h
(3.7) x = z x , y = y z , z = z−1
g−1 − g g g−1 − g g g−1 g
x = 0 , y = 0 , z = 1.
11 11 11
QUANTUM COADJOINT ACTION AND THE 6j-SYMBOLS OF Uqsl2 9
Now we have
ψ (g)ψ (h)(w) =z (z w z x ) z x
1 1 g h h h g g
− −
=z z w z z (x +x z−1),
g h − g h h g h
which is equal to ψ (gh) by (3.7). A similar computation gives ψ (g)ψ (h)(w) = ψ (gh).
1 2 2 2
Hence ψ is a group homomorphism.
(ii) The maps D are well-defined because a Z implies [a˜,u˜] (qn q−n)U . They are
a ε q
∈ ∈ −
derivations of U because the commutator [ , ] of U satisfies the Leibniz rule. Clearly, they
ε q
map Z to Z , and also Z to Z since for all u Z , D (u) is a sum of monomials in Z .
ε ε 0 0 0 a 0
∈
All this can also be check by direct computation, as follows. By (2.7), the products of basis
elements of U read as
q
min(r,c)
(FtKsEr)(FcKbEa)= q−2((r−i)b+s(c−i))Fc+t−ih Ks+bEr+a−i.
i
i=0
X
Suppose that r, s and t are (possibly 0) multiples of n, say r = r′n, s = s′n and t = t′n.
Then, all the h except h = 1 are divided by [ln] for some l, and so vanish at q = ε. Since
i 0
the coefficient of Fc+tKs+bEr+a in the commutator [FtKsEr,FcKbEa] is
q−2n(r′b+s′c) q−2n(s′a+bt′) = q−n(b(r′+t′)+s′(a+c)[n(s′(a c)+b(t′ r′))],
− − −
and lim [nl]/[n] = l for any integer l, the limit (3.3) is well-defined.
q→ε
The formula for D (F) follows from a straightforward computation using (2.11) and (2.15).
e
Also we have
[(q q−1)nEn,K±1]
D (K±1) =lim −
e q→ε n(qn q−n)
−
(q q−1)n q∓2n 1
=lim − − K±1En
q→ε n qn q−n
−
(ε ε−1)n
= − K±1En.
∓ n
We get D by similar computations. (cid:3)
z
For all u, v Z and f Spec(Z ), put
0 0
∈ ∈
u,v (f) = D (v)(f).
u
{ }
By Proposition 3.1, forall a Z , D : Z Z definesan algebraic vector field on Spec(Z ),
0 a 0 0 0
∈ →
and , is a bivector satisfying the Jacobi identity. Hence:
{ }
Corollary 3.2. The bivector , is a Poisson bracket on Spec(Z ).
0
{ }
Exercise 3.3. (a) Check that
e,x = zx2 , e,y = z z−1 , e,z = xz2
{ } − { } − { }
f,x = z−1 z , f,y = y2z , f,z = yz2
{ } − { } { } −
y,x = 1 xy z−2 , z,x = zx , z,y = yz.
{ } − − { } { } −
(b) Deduce that the Poisson bracket ψ , on H does not depend on the order n of ε.
∗
{ }
(c) Show that for any algebra automorphism φ of U we have
ε
(3.8) φD φ−1(u) = D (u).
a φ(a)
Deduce that the group of algebra automorphisms of U induces a group of Poisson automor-
ε
phisms of (Spec(Z ), , ). By using the braid group automorphism (2.18), compute the
0
{ }
value of the derivation D on E, F and K±1.
f
10 STE´PHANEBASEILHAC
(d) Show that the derivations D obtained from the D by allowing the lift a˜ to be arbitrary
a˜ a
generate a Lie algebra . Show that fits into an exact sequence of Lie algebras 0
L L −→
0 ′ 0, where 0 (resp. ′) is the Lie algebra of innerderivations of U (resp.
ε
L −→ L −→ L −→ L L
of derivations of Z induced by , ) .
0
{ }
Remark 3.4. (Semi-classical geometry, I) By Exercise 3.3 the Poisson bracket ψ , is
∗
{ }
canonically associated to U . Conversally, (H,ψ , ) is dual to the standard Poisson-Lie
q ∗
structure on PSL C, and the rational quantizatio{n o}f the latter is U ([DCKP, 7], [DCP,
2 q
§
11, 14 & 19]).
§
Denote by PSL C0 = U TU the big cell of PSL C; it is the open subset consisting of
2 − + 2
matrices up to sign with non vanishing upper left entry. We have an unramified 2-fold
covering
σ : H PSL C0
2
(3.9) −→
(t−1u ,tu ) u−1t2u .
− + 7−→ − +
Consider the matrices
0 1 0 0 1 0
(3.10) , , .
0 0 1 0 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
They correspondto theChevalley generators of sl Cin its fundamentalrepresentation. Since
2
σ is an unramified covering, the associated (complex) left invariant vector fields on PSL C
2
lift to vector fields e, f and h on H. By using Proposition 3.1 we consider them as vector
fields on Spec(Z ).
0
Recall the generators e, f and z of Z , which define functions on Spec(Z ). Denote by g
0 0
(resp. ˜g) the Lie algebras of vector fields on Spec(Z ) generated by e and f (resp. by D and
0 e
D ).
f
Proposition 3.5. We have
(3.11) D = zf , D = ze , D = zh/2
e f z
−
and
(3.12) [D ,D ] = z2h z2xe+z2yf.
e f
−
Hence the linear spans of g and ˜g coincide at every point of Spec(Z ).
0
Proof. Since all vector fields are algebraic, it is enough to check (3.11) on functions on
Spec(Z ) = H lifting functions on PSL C0, and we can restrict to coordinates of the map σ.
0 ∼ 2
Put
1 0 1 x √z 0
u− = ± y 1 , u+ = ± 0 −1 , t = ± 0 1/√z
(cid:18) − (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
and
z zx
M = σ(tu+,t−1u−) = u−−1t2u+ = ± zy zx−y+z−1 .
(cid:18) − (cid:19)
By Exercise 3.3 (a) we have
z2x 0
D M = .
e ± z2xy+z2 1 z2x
(cid:18) − − (cid:19)
Let a sl C. Denote by a the left invariant vector field on PSL C associated to a, and a
2 ∗ 2
its pul∈l-back to Spec(Z ) via σ. By definition, a(M) = a (M), and for all g PSL(2,C)0,
0 ∗
∈
d d
a (M)(g) = M(geat) = (R∗ M) (g),
∗ dt t=0 dt eat t=0
(cid:0) (cid:1)
