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Acta Polytechnica Vol. 51 No. 1/2011

Quantum Moment Map and Invariant Integration Theory on
Quantum Spaces

O. Osuna Castro, E. Wagner

Abstract

It is shown that, on the one hand, quantum moment maps give rise to examples for the operator-theoretic approach
to invariant integration theory developed by K.-D. Kürsten and the second author, and that, on the other hand, the
operator-theoretic approach to invariant integration theory is more general since it also applies to examples without a
well-defined quantum moment map.

Keywords: quantum spaces, invariant integration theory, quantum moment map.

1 Introduction
A noncommutative analogue of an (infinitesimal)
group action on a topological space is described by
the action of a Hopf algebra on a noncommutative
function algebra. In this setting, a generalization of
the classical Haar measure is given by an invariant
integral, that is, a positive linear functional with cer-
tain invariance properties. Usually the noncommu-
tative function algebra is generated by a finite set of
generators which are considered as coordinate func-
tions on the quantum space. As in the classical case,
one does not expect that polynomials in the coor-
dinate functions on locally compact quantum spaces
are integrable. This leads to theproblemthatonehas
to associatealgebrasof integrable (anddifferentiable)
functions to the noncommutative polynomial algebra
in an appropriate way. In the algebraic approach
(see e.g. [2, 7]), one associates function algebras by
imposing commutation relations with the generators
and defines the invariant integral by Jackson-type
integrals. A more rigorous method was developed
by Kürsten and the second author in [3], based on
Hilbert space representations and (unbounded) oper-
atoralgebras. Theadvantageof thismethodbecomes
apparent in the examples of [5], where the algebraic
approach would fail.
The first step of the operator-theoretic approach

is to express the action of a Hopf *-algebra U on a
*-algebra A by algebraic relations of Hilbert space
operators. It should be noted that the operators de-
scribing the action do not have to satisfy the com-
mutation relations of U. On the other hand, any
joint representation of U and A on the same Hilbert
space allows one to equip A with a U-action, given
by the formulas of the adjoint action, provided that
A is invariant under these algebraic expressions [6].
This will be automatically the case if there is a

*-homomorphism from U into A. Then one only has
to consider *-representations of A on a Hilbert space
and the methods from [3] will apply without restric-
tions. In [2], Korogodsky called a *-homomorphism
from U into A intertwining the (adjoint) action a
“quantum moment map”.
The aim of this paper is to show that, on the one

hand, quantum moment maps give rise to examples
for the operator-theoretic approach to invariant inte-
gration theory. On the other hand, we demonstrate
that the operator-theoretic approach to invariant in-
tegration theory is more general, since it also applies
to caseswhere the operators describing the action do
not satisfy the commutation relations of U and hence
do not define a quantum moment map.

2 Operator-theoretic
approach to invariant
integration theory

For details on quantum groups and related notions,
we refer the reader to [1]. Let U be a Hopf *-algebra
withHopf structureΔ, ε and S, whereΔ : U → U⊗U
and ε : U → C are *-homomorphisms and S : U → U
is an anti-homomorphism satisfying certain condi-
tions. Wewill use Sweedler-Heinemannnotation and
write Δ(f)= f(1) ⊗ f(2).
A *-algebra X is called a left U-module *-algebra

if there is a left U-action � on X such that
f � (xy) = (f(1) � x)(f(2) � y),

(f � x)∗ = S(f)∗ � x∗, (1)

x, y ∈ X , f ∈ U.
For unital algebras, one also requires f � 1 = ε(f)1.
By an invariant integral we mean a positive linear
functional h on X satisfying

h(f � x)= ε(f)h(x), x ∈ X , f ∈ U. (2)

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Acta Polytechnica Vol. 51 No. 1/2011

Givenadense linear subspace D ofaHilbert space
H, consider the *-algebra

L+(d) := { x ∈ End(D); D ⊂ D(x∗), x∗D ⊂ D }

with involution x �→ x∗ ⇁D. An (unbounded) *-re-
presentation of X is a *-homomorphism π : X →
L+(d). If for each f ∈ U there exists a finite number
of operators Li, Ri ∈ L+(d) such that

π(f � x)=
∑

i
Liπ(x)Ri, x ∈ X , (3)

then we say that we have an operator expansion of
the action. Obviously, it suffices to know the opera-
tors Li, Ri for a set of generators of U.
Let A denote the *-subalgebra of L+(d) gener-

ated by π(X) and the operators Li, Ri for a set of
generators of U. Set

S(A) := { t ∈ L+(d); t̄H ⊂ D, t̄∗H ⊂ D,
atb ∈ L1(H) ∀a, b ∈ A }, (4)

where the bar denotes the closure of closeable oper-
ators on D, and L1(H) is the Schatten class of trace
class operators on H.
The *-algebra S(A) will be considered as an al-

gebra of differentiable functions which vanish suffi-
ciently rapidly at “infinity”. If the operators from
the operator expansion satisfy convenient commuta-
tion relations (but not necessarily the defining rela-
tions of U), then the U-action can be expanded to
S(A). In favorable cases, one can define an invari-
ant integral by a weighted trace on S(A), where the
weight is easily guessed from the operator expansion
of the action by analogy to the well-known quantum
trace (see [3, 5]).
AHopfalgebraU actsalwayson itself by the (left)

adjoint action:

adL(f)(x) := f(1) x S(f(2)), f, x ∈ U, (5)

In [2], L. I. Korogodsky defined a quantum moment
map as a *-homomorphism ρ : U → X such that
ρ(adL(f)(x)) = f � ρ(x) for all f, x ∈ U. Then any
*-representation π : X → L+(D) leads to a *-re-
presentation π ◦ ρ : U → L+(D), and it follows easily
from the Hopf algebra structure of U that

adL(f)(X) := π(ρ(f(1)))X π(ρ(S(f(2)))),

f ∈ U, X ∈ L+(D), (6)

defines a left U-action on L+(D) turning it into a U-
module *-algebra. Moreover, the algebra S(A) is in-
variant under this action. Suppose furthermore that
U denotes the QuantizedUniversal EnvelopingAlge-
bra of a semisimple Lie algebra. Then there exists a
distinguished element Γ in U such that Γf = S2(f)Γ
for all f ∈ U. By the definition of S(A), the traces

tr(π(ρ(Γ))f) are well-defined and we can state the
following theorem:
Theorem 1 Let ρ : U → X be a quantum moment
map and π : X → L+(D) a *-representation such
that ±π(ρ(Γ)) is a non-negative selfadjoint operator.
Then

h(f) := ±tr(π(ρ(Γ))f), f ∈ S(A), (7)

defines an invariant integral on the U-module
*-algebra S(A).

Proof. The invariance of h follows from the same for-
mulasas in theproofof the invarianceof thequantum
trace in [1, Section 7.1.6] by applying the trace prop-
erty tr(af) = tr(f a) which continues to hold for all
f ∈ S(A) and a ∈ A, see [3]. �

3 Example: A quantum
hyperboloid

Let q ∈ (0,1)and s ∈ [−1,1). Following [2],wedefine
the two-sheet quantum hyperboloid X := Oq(Xs,1)
(after a slight reparametrization) as the *-algebra
generated by y, y∗ and x = x∗ with commutation
relations

yx = q2xy, xy∗ = q2y∗x,

y∗y = (q−2x − s)(q−2x −1),
yy∗ = (x − s)(x −1).

The Hopf *-algebra U := Uq(su1,1) is generated by
E, F , K and its inverse K−1 with relations

KE = q2EK, F K = q2KF,

EF − F E = (q − q−1)−1(K − K−1),

with Hopf structure

Δ(E) = E ⊗1+ K ⊗ E,
Δ(F) = F ⊗ K−1 +1⊗ F,
Δ(K) = K ⊗ K,
ε(E) = ε(F)= 0,

ε(K) = 1, S(E)= −K−1E,
S(F) = −F K, S(K)= K−1,

and with involution K∗ = K, E∗ = −KF .
The quantum hyperboloid X becomes a U-module
*-algebrawith the action defined by

K � y = q2y, E � y =0,

F � y = q1/2((1+ q−2)x − (1+ s)),
K � x = x, E � x = q1/2y, F � x = q5/2y∗,

K � y∗ = q−2y∗,

E � y∗ = q−3/2((1+ q−2)x − (1+ s)), F � y∗ =0.

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Acta Polytechnica Vol. 51 No. 1/2011

Let I be an at most countable index set, H0 a
Hilbert space, and H = ⊕

i∈I
H0. We denote by ηi

the vector of H which has the element η ∈ H0 as
its i-th component and zero otherwise. It is under-
stood that ηi = 0 whenever i /∈ I. Let U be a
unitary operator on H0, and let A and B be self-
adjoint operators on the Hilbert space H0 such that
spec(A) ⊂ [q2,1], spec(B) ⊂ [q2, s], q2 is not an
eigenvalue of A and B, and s is not an eigenvalue
of B. Set λn :=

√
(q2n − s)(q2n −1) and λn(t) :=√

(q2nt − s)(q2nt −1). Then a list of non-equivalent
*-representations of X is given by the following for-
mulas (suppressing the letter π of the representation).

s ∈ [−1,1) : xηn = q−2nηn, yηn = λ−(n+1)ηn+1
on H = ⊕

n∈N0
H0.

s ∈ [0,1) : xηn = −q2(n+1)Aηn, yηn = λn(−A)ηn−1
on H = ⊕

n∈ZH0.
s ∈ (0,1) : xηn = q2(n+1)sηn, yηn = λn(s)ηn−1

on H = ⊕
n∈N0

H0;
x =0, y = sU on H0.

s ∈ (q2,1) : xηn = q−2nsηn, yηn = λ−(n+1)(s)ηn+1
on H = ⊕

n∈N0
H0;

xηn = q
2(n+1)

ηn, yηn = λnηn−1

on H = ⊕
n∈N0

H0;
xηn = q

2(n+1)
Bηn, yηn = λn(B)ηn−1

on H = ⊕
n∈ZH0.

s = q2 : x = q2, y =0 on H0.
s =0 : x = y =0 on H0.
s ∈ [−1,0) : xηn = q−2nsηn, yηn = λ−(n+1)(s)ηn+1

on H = ⊕
n∈N0

H0.

The domain D of the representation can be cho-
sen, for instance, to be the linear span of the ηn’s.
If one imposes some well-behavedness conditions, for
instance that x̄ is self-adjoint and that yf(x̄) ⊂ f(x̄)y
for all boundedmeasurable functions (with respect to
the spectral measure of x̄), then this list is complete
in the sense that eachwell-behaved representation is
a direct sum of representations from the above list.
A single representation is irreducible if and only if
H0 = C. In this case A, B and U become com-
plex numbers such that A ∈ (q2,1], B ∈ (q2, s) and
|U| =1. For the proof of these claims, see [4].
Given a *-representation such that x is invertible

in L+(D), set

e := q−1/2(q − q−1)−1x−1y,
f := −q1/2(q − q−1)−1y∗, k := qx−1.

Direct computations show that

K � z = kzk−1, E � z = ez − kzk−1e,
F � z = f zk − zf k (8)

for z = x, y, y∗. Using the relations

ke = q2ek, f k = q2kf,

ef − f e = (q − q−1)−1(sk − k−1), (9)

one easily proves that (8) defines a U-action on
L+(D) turning it into a left U-module *-algebra.
With A being the *-subalgebra of L+(D) generated
by y, y∗, x and x−1, the *-algebra S(A) defined in
(4) becomes a left U-module *-subalgebra of L+(D).
Since the traces of elements from S(A) are well-
defined, we can state the following proposition:
Proposition 2 If ±x is a non-negative selfadjoint
operator, then

h(f) := ±tr(k−1f)

defines an invariant integral on S(A).

Proof. The invariance follows from the trace prop-
erty tr(af) = tr(f a) for all f ∈ S(A) and a ∈ A. As
an example, we show the invariance with respect to
E,

h(E � z) = ±tr(k−1ez − zk−1e)=
±tr(k−1ez − k−1ez)= 0= ε(E)h(z).

The positivity of h is clear by the positivity of
±k−1 = ±q−1x. �
Note that Equation (8) is invariant under the

rescaling k �→ tk and f �→ t−1f. If t ∈ R \ {0},
the rescaling does not affect the involution, i.e., we
have k∗ = k and e∗ = −kf. From (9), it follows that

ρ(K)= s1/2k, ρ(E)= e, ρ(F)= s−1/2f

defines a moment map ρ : U → A if and only if
s ∈ (0,1). In this situation, Proposition 2 is an im-
mediate consequence of Theorem1 togetherwith the
formula of the quantum trace.
However, we emphasize that Proposition 2 holds

for all s ∈ [−1,0), even if the operators k, e and
f do not satisfy the defining relations of Uq(su1,1).
This shows that the operator-theoretic approach to
invariant integration theory is more general than the
method based on a quantum moment map. We also
would like to point out that our approach works for
all representations from the above list where x �= 0,
even for those where x has a continuous spectrum,
whereas in the algebraic approach, one usually con-
siders functions in x which are supported on a dis-
crete set [2, 7].

References

[1] Klimyk,A.U., Schmüdgen,K.: QuantumGroups
and Their Representations. Berlin : Springer-
Verlag, 1997.

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Acta Polytechnica Vol. 51 No. 1/2011

[2] Korogodsky, L. I.: Representation of quan-
tum algebras arising from non-compact quantum
groups: quantum orbit method and super-tensor
products, Ph.D. Thesis, Massachusetts Institute
of Technology, Dept. of Math., 1996. Compli-
mentary series representations and quantumorbit
method. ArXiv:q-alg/9708026v1.

[3] Kürsten, K.-D., Wagner, E.: An operator-
theoretic approach to invariant integrals onquan-
tum homogeneous SUn,1-spaces. Publ. Res. Inst.
Math. Sci. 43 (1), 2007, p. 1–37.

[4] LucioPeña,P.C.,OsunaCastro,O.,Wagner,E.:
Invariant integration theory on the quantum hy-
perboloid, in preparation.

[5] Osuna Castro, O., Wagner, E.: An operator-
theoretic approach to invariant integrals on quan-
tum homogeneous sl(n + 1, R)-spaces. ArXiv:
math.QA/0904.0669v1.

[6] Schmüdgen, K., Wagner, E.: Hilbert space rep-
resentations of cross product algebras. J. Funct.
Anal. 200 (2), 2003, p. 451–493.

[7] Shklyarov, D. L., Sinel’shchikov, S. D., Vaks-
man, L. L.: Integral representations of func-
tions in the quantum disk. I. (Russian) Mat. Fiz.
Anal. Geom. 4 (3), 1997, p. 286–308. Quantum
Matrix Balls: Differential and Integral Calculi.
ArXiv:math.QA/9905035.

Osvaldo Osuna Castro
E-mail: osvaldo@ifm.umich.mx
Department of Physics and Mathematics
University of Michoacan
Morelia, Mexico

Elmar Wagner
E-mail: elmar@ifm.umich.mx
Department of Physics and Mathematics
University of Michoacan
Morelia, Mexico

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