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Quantizations of Poisson Lie groups as noncommutative manifolds
∗
S. Neshveyev
Department of Mathematics, University of Oslo,
P.O. Box 1053 Blindern, NO-0316 Oslo, Norway
E-mail: sergeyn@math.uio.no
L. Tuset
Faculty of Engineering, Oslo University College,
P.O. Box 4 St. Olavs plass, NO-0130 Oslo, Norway
E-mail: Lars.Tuset@iu.hio.no
On any q-deformation of a simply connected simple compact Poisson Lie group we construct an equivariant spectral triple which is an isospectral deformation of that defined
by the Dirac operator on the original group. Our quantum Dirac operator is defined using
a Drinfeld twist which relates the q-deformed compact quantum group to the original
group, and thus a priori depends on the choice of the twist, but it turns out that the
spectral triple is nevertheless unique up to unitary equivalence.
Introduction
By work of Drinfeld and Belavin all simply connected simple compact Poisson Lie
groups have been classified. Drinfeld further showed that all of these do admit
quantization by q-deformation. The quantization of the standard Poisson bracket
on G is the compact quantum group Gq one usually encounters. The quantizations
corresponding to the other brackets are obtained by twisting the coproduct of Gq
or G with a 2-cocycle q iu , where u is a self-adjoint element of h ∧ h.
In [1] we constructed a quantum Dirac operator Dq on Gq that defines a biequivariant spectral triple which is an isospectral deformation of that defined by the Dirac
operator D on G. To do this we relied on the existence of a special element F in the
group von Neumann algebra W ∗ (G×G), and an isomorphism ϕ : W ∗ (Gq ) → W ∗ (G)
satisfying certain properties. The existence of the pair (ϕ, F) follows from work by
Kazhdan and Lusztig [2, 3] and is an analytic version of a result by Drinfeld [4, 5].
From the outset Dq and the associated spectral triple depend on the choice of (ϕ, F),
but by a uniqueness result for Drinfeld twists, which we established in [6], we showed
that the spectral triple is unique up to unitary equivalence.
∗ The
note is different in two lines from the published version, correcting an annoying mistake in
the definition of the r-matrix; January 6, 2012
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In this note we briefly describe the above results and show that one can construct
Dirac operators providing equivariant spectral triples for all the twisted versions as
well.
1. Quantum groups
Let G be a compact connected simply connected simple Lie group, g its complexified
Lie algebra. Let h ⊂ g be the Cartan subalgebra defined by a maximal torus T in G.
Fix a system {α1 , . . . , αr } of simple roots. Let (aij )1≤i,j≤r be the Cartan matrix
and d1 , . . . , dr be the coprime positive integers such that (di aij )i,j is symmetric. Let
hi ∈ h be such that αj (hi ) = aij . Denote by hR the R-linear span of hi , i = 1, . . . , r.
Define a bilinear form on h∗ by (αi , αj ) = di aij . The dual form on h extends to a
symmetric invariant form on g. Denote by t ∈ g ⊗ g the corresponding g-invariant
element. Consider the decomposition g = n+ ⊕ h ⊕ n− and write t = t+− + t0 + t−+
with t+− ∈ n+ ⊗ n− , t−+ ∈ n− ⊗ n+ and t0 ∈ h ⊗ h. Define r0 ∈ g ⊗ g by
r0 = i(t+− − t−+ ).
It is known that any self-adjoint skew-symmetric solution of the modified classical Yang-Baxter equation can be written, up to inner automorphisms of G, as
ar0 + u for some a ∈ R and u ∈ ∧2 hR . This means that any Poisson Lie group
structure on G is given by the bracket {·, ·} : C[G] × C[G] → C[G] defined by
ˆ
{f, g} = (f ⊗ g)([ar0 + u, ∆(·)]),
ˆ
up to inner automorphisms of G and a complex rescaling of the bracket, where ∆
is the comultiplication on U g. They all admit quantization. For a = 1 and u = 0
the quantization is the standard q-deformation, and that for a 6= 1 (and u = 0) just
means a reparametrization, i.e. change of q, of the standard one. We shall focus on
the q-deformations for 0 < q < 1 associated to a = 1 and u ∈ ∧2 hR ; the case a = 0
is analogous but easier, one essentially has to replace Uq g everywhere by U g.
The quantized universal enveloping algebra Uq g is generated by elements Ei , Fi ,
Ki , 1 ≤ i ≤ r, satisfying the relations
−aij
a
Ki Kj = Kj Ki , Ki Ej Ki−1 = qi ij Ej , Ki Fj Ki−1 = qi
Ei Fj − Fj Ei = δij
1−aij
X
(−1)k
k=0
1−aij
X
k=0
Ki − Ki−1
,
qi − qi−1
1 − aij
1−a −k
Eik Ej Ei ij = 0,
k
q
i
1 − aij
1−a −k
(−1)
Fik Fj Fi ij = 0,
k
q
k
i
Fj ,
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[m]qi !
q n − qi−n
m
=
where
, [m]qi ! = [m]qi [m − 1]qi . . . [1]qi , [n]qi = i
k q
[k]qi ![m − k]qi !
qi − qi−1
i
di
and qi = q .
ˆ q defined by
This is a Hopf ∗-algebra with coproduct ∆
ˆ q (Ki ) = Ki ⊗ Ki , ∆
ˆ q (Ei ) = Ei ⊗ 1 + Ki ⊗ Ei , ∆
ˆ q (Fi ) = Fi ⊗ K −1 + 1 ⊗ Fi
∆
i
and involution
Ki∗ = Ki , Ei∗ = Fi Ki , Fi∗ = Ki−1 Ei .
Let P be the lattice of integral weights. If V is a finite dimensional Uq g-module and
λ(h )
λ ∈ P , denote by V (λ) the space of vectors v ∈ V of weight λ, that is, Ki v = qi i v.
The module V is called admissible if V = ⊕λ∈P V (λ).
The quantized algebra of regular functions C[Gq ] ⊂ (Uq g)∗ is the Hopf ∗-algebra
of matrix coefficients of finite dimensional admissible Uq g-modules.
Denote by C(g, q) the category of finite dimensional admissible Uq g-modules. It is
a semisimple tensor category, with simple objects Vλ indexed by dominant integral
weights λ ∈ P+ . Therefore, if F : C(g, q) → Vec is the forgetful functor, then
U(Gq ) := Nat(F, F ) ∼
=
Y
End(Vλ ).
λ∈P+
The algebra U(Gq ) is a completion of Uq g. It can also be identified with the algebra
of closed densely defined operators affiliated with the von Neumann algebra W ∗ (Gq )
of Gq . The category C(g, q) is braided, with braiding σ = ΣR, where Σ is the flip
and R ∈ U(Gq × Gq ) := Nat(F ⊗2 , F ⊗2 ) is the universal R-matrix.
The algebra U(T ) embeds into U(Gq ) by identifying hi with a unique self-adjoint
element Hi ∈ U(Gq ) such that Ki = qiHi . If u ∈ ∧2 hR we can therefore consider the
element q iu as an element H ∈ U(Gq × Gq ). Since u∗ = u and u is skew-symmetric,
ˆ ⊗ ι)u = 1 ⊗ u + (ι ⊗ ∆)u,
ˆ
we have H∗ = H−1 = H21 . We also have u ⊗ 1 + (∆
so
that by commutativity of U(T ) we get
ˆ q ⊗ ι)(H) = (1 ⊗ H)(ι ⊗ ∆
ˆ q )(H).
(H ⊗ 1)(∆
ˆ q ) with H−1 = H21 . Thus we
Therefore H is a unitary 2-cocycle for (U(Gq ), ∆
ˆ u,q : U(Gq ) → U(Gq × Gq ) by ∆
ˆ u,q = H∆
ˆ q (·)H−1 . It
can define a new coproduct ∆
defines a new tensor structure on finite dimensional admissible Uq g-modules. Denote
by C[Guq ] the corresponding Hopf ∗-algebra of matrix coefficients. In other words,
C[Guq ] coincides with C[Gq ] as a coalgebra, but has a new ∗-algebra structure defined
by the twist H, see [7]. The compact quantum group Guq is the q-deformation of
the Poisson Lie group G with the Poisson structure associated to u (and a = 1).
The category of finite dimensional representations of Guq is braided with braiding
defined by the R-matrix Ru = H21 RH−1 .
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2. Drinfeld twist
Let ~ ∈ iR be such that q = eπi~ . Assume V1 , V2 , V3 are finite dimensional g-modules
and put V = V1 ⊗ V2 ⊗ V3 . Consider End(V )-valued solutions of the equation
t12
t23
0
w =~
+
w
x
x−1
on (0, 1). There exist unique solutions G0 and G1 such that the functions
G0 (x)x−~t12 and G1 (1 − x)x−~t23 extend to holomorphic functions in the unit disc
with value 1 at x = 0. Hence there exists ΦKZ ∈ GL(V ) such that
G0 (x) = G1 (x)ΦKZ for all x ∈ (0, 1).
The operators ΦKZ for different V1 , V2 , V3 define an element of U(G × G × G). It is
ˆ
a unitary 3-cocycle for (U(G), ∆).
As irreducible representations of G and Gq are both parametrized by dominant
integral weights, we have a canonical identification of the centers of U(G) and U(Gq ).
Since the dimensions of irreducible modules with the same highest weight do not
depend on q, this identification extends to a ∗-isomorphism ϕ : U(Gq ) → U(G).
Furthermore, the dimensions of the weight spaces do not depend on q either, which
implies that we can arrange that ϕ(Ki ) = qihi . In particular, (ϕ ⊗ ϕ)(H) = q iu .
Theorem 2.1. There exists a unitary F ∈ U(G × G) such that
−1
ˆ u,q = F ∆ϕ(·)F
ˆ
(i) (ϕ ⊗ ϕ)∆
;
(ii) (ε̂ ⊗ ι)(F) = (ι ⊗ ε̂)(F) = 1, where ε̂ is the trivial representation of G;
(iii)(ϕ ⊗ ϕ)(Ru ) = F21 q t F −1 ;
−1
ˆ
ˆ ⊗ ι)(F).
(iv) ΦKZ = (ι ⊗ ∆)(F
)(1 ⊗ F −1 )(F ⊗ 1)(∆
Proof. For u = 0 the existence of F follows by work of Kazhdan and Lusztig [2, 3],
see [8]. Denote this element by F0 . For general u the required element is F := q iu F0 .
Indeed, the only nontrivial property is (iv). Since the elements Ki are group-like,
ˆ = (ϕ ⊗ ϕ)∆
ˆ q ϕ−1 on U(T ). It follows that F0 commutes with elements of
we have ∆
ˆ
the form ∆(w),
w ∈ U(T ). Then (iv) for F follows from the corresponding property
ˆ
of F0 and the fact that q iu is a 2-cocycle for (U(T ), ∆).
We call F a unitary Drinfeld twist for Guq . We have the following uniqueness
result.
Theorem 2.2. Suppose E and F are two unitary Drinfeld twists for Guq for the
same ∗-isomorphism ϕ. Then there exists a unitary central element c in U(G) such
ˆ −1 .
that E = (c ⊗ c)F ∆(c)
Proof. The elements q −iu E and q −iu F are unitary Drinfeld twists for Gq . Therefore
the result follows from [6, Theorem 5.2]. Briefly, the reason is that
G := (ϕ−1 ⊗ ϕ−1 )(q −iu EF −1 q iu )
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ˆ q ), that is, RG = G21 R
is a symmetric invariant unitary 2-cocycle for (U(Gq ), ∆
ˆ
and G commutes with elements of the form ∆q (w). Hence G is the coboundary of a
central element in U(Gq ) by [6, Theorem 2.1].
3. Quantum Dirac operator
Denote by Cl(g) the complex Clifford algebra of g and by γ : g → Cl(g) the canonical
embedding, so Cl(g) is generated by γ(x), x ∈ g, and γ(x)2 = (x, x)1. The adjoint
action of G on g extends to an action of G on Cl(g) which lifts to a homomorphism
G → Spin(g). On the Lie algebra level it is given by
1
f
g 3 x 7→ ad(x)
:= −
4
X
γ(xi )γ([x, xi ]),
i
where {xi }i is a basis in g and {xi }i is the dual basis. Fix a spin module, an
irreducible ∗-representation s : Cl(g) → End(S). Identifying the smooth sections of
the spin bundle S = G × S with C ∞ (G) ⊗ S, the Dirac operator D : C ∞ (G) ⊗ S →
C ∞ (G) ⊗ S is given by
X
1
f i )) ,
D=
∂(xi ) ⊗ sγ(xi ) − ⊗ s(γ(xi )ad(x
2
i
where ∂ is the representation of U g by left-invariant differential operators. This can
be written as D = (∂ ⊗ s)(D), where
D = −(ι ⊗ γ)(t) −
X1
i
2
f i)
⊗ γ(xi )ad(x
is an element of the non-commutative Weil algebra U g ⊗ Cl(g).
Fix a ∗-isomorphism ϕ : U(Guq ) = U(Gq ) → U(G) as in the previous section and
choose a unitary Drinfeld twist F ∈ U(G × G) for Guq . Define
−1
f
f
Dqu = (ϕ−1 ⊗ ι)((ι ⊗ ad)(F)D(ι
⊗ ad)(F
)) ∈ U(Guq ) ⊗ Cl(g).
The quantum Dirac operator Dqu is the unbounded operator on L2 (Guq ) ⊗ S defined
by
Dqu = (∂qu ⊗ s)(Dqu ),
where ∂qu is the right regular representations of U(Guq ) on L2 (Guq ).
Theorem 3.1. The triple (C[Guq ], L2 (Guq ) ⊗ S, Dqu ) is a Guq -biequivariant spectral
triple of the same parity as the dimension of G. It does not depend on the choice
of ϕ and F up to unitary equivalence.
The proof is identical to that of [1, Theorem 3.7] and [6, Theorem 6.1].
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References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
S. Neshveyev and L. Tuset, arXiv:math/0703161v2 [math.OA] (2007).
D. Kazhdan and G. Lusztig, J. Amer. Math. Soc. 7, 335 (1994).
D. Kazhdan and G. Lusztig, J. Amer. Math. Soc. 7, 383 (1994).
V. G. Drinfeld, Leningrad Math. J. 1, 1419 (1990).
V. G. Drinfeld, Leningrad Math. J. 2, 829 (1991).
S. Neshveyev and L. Tuset, arXiv:0902.2365v1 [math.QA] (2009).
S. Levendorskii and Y. Soibelman, Commun. Math. Phys. 139, 141 (1991).
S. Neshveyev and L. Tuset, arXiv:0711.4302v1 [math.QA] (2007).