Invariants of the half-liberated orthogonal group

Abstract

The half-liberated orthogonal group $O_n^*$ appears as intermediate quantum
group between the orthogonal group $O_n$, and its free version $O_n^+$. We
discuss here its basic algebraic properties, and we classify its irreducible
representations. The classification of representations is done by using a
certain twisting-type relation between $O_n^*$ and $U_n$, a non abelian
discrete group playing the role of weight lattice for $O_n^*$, and a number of
methods inspired from the theory of Lie algebras. We use these results for
showing that the discrete quantum group dual to $O_n^*$ has polynomial growth.

Full text

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arXiv:0902.2719v2 [math.QA] 15 Oct 2009
INVARIANTS OF THE HALF-LIBERATED ORTHOGONAL GROUP
TEODOR BANICA AND ROLAND VERGNIOUX
Abstract. The half-liberated orthogonal group O∗
tum group between the orthogonal group On, and its free version O+
here its basic algebraic properties, and we classify its irreducible representations. The
classification of representations is done by using a certain twisting-type relation be-
tween O∗
nand Un, a non abelian discrete group playing the role of weight lattice
for O∗
n, and a number of methods inspired from the theory of Lie algebras. We use
these results for showing that the discrete quantum group dual to O∗
growth.
nappears as intermediate quan-
n. We discuss
nhas polynomial
Introduction
The quantum groups introduced by Drinfeld in [13] have played a prominent role
in various areas of mathematics and physics.
Woronowicz’s axiomatization in [23], [24] of the compact quantum groups has been
very influential as well and opened the way for the search of new examples. In particu-
lar it allowed the discovery by Wang of the free quantum groups [20], which have been
subject of several systematic investigations. Since then other families of examples have
been discovered, building up a fast evolving area:
(1) The first two quantum groups are O+
a number of general developments, including the study of connections with
subfactors, noncommutative geometry and free probability [1], [5], [14] and a
number of advances in relation with operator algebras [16], [17], [18].
(2) The third quantum group is S+
amazing world of quantum permutation groups, heavily investigated in the last
few years. These quantum groups allowed in particular a clarification of the
relation with subfactors [3], noncommutative geometry [9] and free probability
[15].
(3) The fourth quantum group is H+
group gave rise as well to a number of new investigations, which are currently
under development. Let us mention here the opening world of quantum reflec-
tion groups [8], and the new formalism of easy quantum groups [6].
In addition to Drinfeld’s discovery,
n, U+
n, introduced in [20]. These led to
n, introduced in [21]. This led to the quite
n, recently introduced in [4]. This quantum
2000 Mathematics Subject Classification. 20G42 (16W30, 46L65).
Key words and phrases. Quantum group, Maximal torus, Root system.
1

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2 TEODOR BANICA AND ROLAND VERGNIOUX
In this paper we study the fifth “new” quantum group, namely the half-liberated
orthogonal group O∗
predecessors O+
namely, that of the “root systems” for the compact quantum groups.
The quantum group O∗
functions uij: On→ C. These commute with each other, form an orthogonal matrix,
and generate the algebra C(On). Wang’s algebra C(O+
the “commutativity” assumption. As for obtaining the “half-liberated” algebra C(O∗
the commutativity condition ab = ba with a, b ∈ {uij} should be replaced by the weaker
condition abc = cba, for a, b, c ∈ {uij}.
Summarizing, the quantum group O∗
Wang’s original relations in [20]. Observe that we have On⊂ O∗
One can prove that, under a suitable “easiness” assumption, O∗
group between Onand O+
name “half-liberated”, and provides a first motivation for the study of O∗
was introduced in as one of the 15 easy intermediate quantum groups Sn⊂ G ⊂ O+
n, constructed in [6]. We believe that, as it was the case with its
n, U+
n, S+
n, H+
n, this quantum group will open up as well a new area:
nis constructed as follows. Consider the basic coordinate
n) is obtained by simply removing
n),
nappears via a kind of tricky “‘weakening” of
n⊂ O+
nis the only quantum
n.
n. This abstract result, to be proved in this paper, justifies the
n. In fact O∗
n
n.
In this paper we perform a systematic study of O∗
of representations. After discussing the first features of the definition, we describe in
Sections 2 – 4 some Hom-spaces of this category in terms of Brauer diagrams and derive
two consequences: a connection with the group Unvia the projective version PO∗
the uniqueness result mentionned above.
Then we undertake the classification of irreducible representations of O∗
technical novelty is the use of diagonal groups and root systems in a quantum frame-
work. Diagonal groups are meant to be replacements for maximal torii in good situa-
tions and are introduced in Section 5. In the case of O∗
noncommutative weight lattice which is used in Sections 6 and 7 together with a subtle
relation to the classical group Unto classify representations of O∗
Finally we derive in Sections 8, 9 some applications of the classification of irreducible
representations to fusion rules, Cayley graph and growth. Let us mention here a quite
surprising feature of our results: although O∗
two orthogonal groups On, O+
noncommutative and its exponent of polynomial growth is the same as for SUn. This
shows also that O∗
compact quantum group so far, in particular it is the first original example of a compact
quantum group with exponential growth as considered in [7].
n, and in particular of its category
n, and
n. The main
nthe diagonal group provides a
n.
nis an intermediate subgroup between
nwith commutative fusion rules, its fusion rules are
nis not monoidally equivalent, in the sense of [10], to any known

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INVARIANTS OF THE HALF-LIBERATED ORTHOGONAL GROUP3
1. Half-liberation
Given a compact group G, the algebra of complex continuous functions C(G) is a
Hopf algebra, with comultiplication, counit and antipode given by:
∆(ϕ) = ((g,h) → ϕ(gh))
ε(ϕ) = ϕ(1)
S(ϕ) = (g → ϕ(g−1))
Consider in particular the orthogonal group On. This is a real algebraic group, and
we denote by xij: On→ R its basic coordinates, xij(g) = gij.
The matrix x = (xij) is by definition orthogonal, in the sense that all its entries
are self-adjoint, and we have xxt= xtx = 1. Moreover, it follows from the Stone-
Weierstrass theorem that the elements xijgenerate C(On) as a C∗-algebra.
These observations lead to the following presentation result.
Theorem 1.1. C(On) is the universal unital commutative C∗-algebra generated by the
entries of an n × n orthogonal matrix x. The maps given by
?
ε(xij) = δij
S(xij) = xji
∆(xij) =xik⊗ xkj
are the comultiplication, counit and antipode of C(On).
Proof. The first assertion is a direct application of the classical theorems of Stone-
Weierstrass and Gelfand. The second assertion follows by transposing the usual rules
for the matrix multiplication, unit and inversion.
?
The following key definition is due to Wang [20].
Definition 1.2. Ao(n) is the universal unital C∗-algebra generated by the entries of
an n × n orthogonal matrix u. The maps given by
?
ε(uij) = δij
S(uij) = uji
∆(uij) =uik⊗ ukj
are the comultiplication, counit and antipode of Ao(n).
It is routine to check that Ao(n) satisfies the general axioms of Woronowicz in [23].
This tells us that we have the heuristic formula Ao(n) = C(O+
compact quantum group, called free version of On. See [20].
It is known that we have Ao(2) = C(SU−1
n ≥ 2 arbitrary shares many properties with the algebra C(SU2). See [1], [5].
The following definition is from the recent paper [6].
n), where O+
nis a certain
2). More generally, the algebra Ao(n) with

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4 TEODOR BANICA AND ROLAND VERGNIOUX
Definition 1.3. The half-liberated orthogonal quantum algebra is
A∗
o(n) = Ao(n)/?abc = cba??a,b,c ∈ {uij}?
with comultiplication, counit and antipode coming from those of Ao(n).
It is routine to check that the comultiplication, counit and antipode of Ao(n) factorize
indeed, and that A∗
In order to get some insight into the structure of A∗
commutative version”. We have the following analogue of Definition 1.3.
o(n) satisfies the general axioms of Woronowicz in [23].
o(n), we first examine its “co-
Definition 1.4. We consider the discrete group
Ln= Z∗n
2/?abc = cba??a,b,c ∈ {gi}?
where g1,...,gnwith g2
i= 1 are the standard generators of Z∗n
2.
Observe that we have surjective group morphisms Z∗n
Proposition 1.6 below, these morphisms are not isomorphisms in general.
We recall that for any discrete group Γ, the group algebra C∗(Γ) is a Hopf algebra,
with comultiplication, counit and antipode given by:
2
→ Ln→ Zn
2. As shown in
∆(g) = g ⊗ g
ε(g) = 1
S(g) = g−1
The interest in the above group Lncomes from the following result.
Proposition 1.5. We have quotient maps as follows:
Ao(n)
??
??A∗
o(n)
??
??C(On)
??
C∗(Z∗n
2)
??C∗(Ln)
??C∗(Zn
2)
Proof. The vertical maps can be defined indeed by uij→ δijgi, by using the universal
property of the algebras on top. Observe that these maps are indeed Hopf algebra
morphisms, because the formulae of ∆, ε, S in Definition 1.2 reduce to the above
cocommutative formulae, after performing the identification uij= 0 for i ?= j.
?
The group Lnwill play an important role in the present paper in relation with the
representation theory of O∗
isomorphism Ln≃ Zn−1⋊ Z2. Let us start with a simpler statement that we use for
Theorem 1.7. Note that for n = 2 this already gives L2= D∞= Z ⋊ Z2.
n. In particular we will obtain in section 6 an abstract
Proposition 1.6. The groups Lnare as follows:
(1) At n = 2 we have Z∗2
(2) At n ≥ 3, we have Z∗n
2= L2?= Z2
2?= Ln?= Zn
2.
2.

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INVARIANTS OF THE HALF-LIBERATED ORTHOGONAL GROUP5
Proof. We denote by g, h the standard generators of Z∗2
(1) We know that L2appears as quotient of Z∗2
b, c ∈ {g,h}. In the case a = b or b = c this is a trivial relation (of type k = k), and in
the case a ?= b, b ?= c we must have a = c, so once again our relation is trivial (of type
klk = klk). Thus we have L2= Z∗2
(2) The first assertion is clear, because the equality g1g2g3= g3g2g1doesn’t hold in
Z∗n
2. Observe now that we have a quotient map L3→ Z∗2
g3 → h. This shows that L3 is not abelian. We deduce that Lnwith n ≥ 3 is not
abelian either, so in particular it is not isomorphic to Zn
2.
2by the relations abc = cba, with a,
2, which gives the result.
2, given by g1→ g, g2→ g,
2.
?
Theorem 1.7. The algebras A∗
(1) At n = 2 we have Ao(2) = A∗
(2) At n ≥ 3, we have Ao(n) ?= A∗
o(n) are as follows:
o(2) ?= C(O2).
o(n) ?= C(On).
Proof. The three non-equalities in the statement follow from the three non-equalities
in Proposition 1.6. In order to prove the remaining statement A∗
the fundamental corepresentation of Ao(2):
?x y
o(2) = Ao(2), consider
u =
zt
?
The elements x, y, z, t are by definition self-adjoint, and satisfy the relations making
u unitary. These unitary relations can be written as follows:
x2
y2
= t2
= z2
= 1x2+ y2
xy + zt = 0
xz + yt = 0
With these relations in hand, the verification of the relations of type abc = cba with
a, b, c ∈ {x,y,z,t} is routine, and we get A∗
o(2) = Ao(2).
?
2. Brauer diagrams
In this section and in the next one we discuss some basic properties of A∗
“interpolating” between some well-known results regarding Ao(n) and C(On).
For k, l with k+l even we consider the pairings between an upper sequence of k points,
and a lower sequence of l points. We make the convention that the k+l points of a pair-
ing p are counted counterclockwise, starting from bottom left. As an example we draw
hereafter the diagram corresponding to the pairing {{1,9},{2,4},{3,7},{5,8},{6,10}}
for k = l = 5. These pairings, also called Brauer diagrams, are taken as usual up to
planar isotopy. See e.g. [22].
o(n), by