A note on free quantum groups

We study the free complexification operation for compact quantum groups, $G\to G^c$. We prove that, with suitable definitions, this induces a one-to-one correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying $G=G^c$.


Introduction
In this paper we present some advances on the notion of free quantum group, introduced in [3].We first discuss in detail a result mentioned there, namely that the free complexification operation G → G c studied in [2] produces free unitary quantum groups out of free orthogonal ones.Then we work out the injectivity and surjectivity properties of G → G c , and this leads to the correspondence announced in the abstract.This correspondence should be regarded as being a first general ingredient for the classification of free quantum groups.
We include in our study a number of general facts regarding the operation G → G c , by improving some previous work in [2].The point is that now we can use general diagrammatic techniques from [4], new examples, and the notion of free quantum group [3], none of them available at the time of writing [2].
The paper is organized as follows: 1 contains some basic facts about the operation G → G c , and in 2-5 we discuss the applications to free quantum groups.

Free complexification
A fundamental result of Voiculescu [6] states if (s 1 , . . ., s n ) is a semicircular system, and z is a Haar unitary free from it, then (zs 1 , . . ., zs n ) is a circular system.This makes appear the notion of free multiplication by a Haar unitary, a → za, that we call here free complexification.This operation has been intensively studied since then.See Nica and Speicher [5].
This operation appears as well in the context of Wang's free quantum groups [7], [8].The main result in [1] is that the universal free biunitary matrix is the free complexification of the free orthogonal matrix.In other words, the passage O + n → U + n is nothing but a free complexification: Moreover, some generalizations of this fact are obtained, in an abstract setting, in [2].
In this section we discuss the basic properties of A → Ã, the functional analytic version of G → G c .We use an adaptation of Woronowicz's axioms in [9].Definition 1.1.A finitely generated Hopf algebra is a pair (A, u), where A is a C * -algebra and u ∈ M n (A) is a unitary whose entries generate A, such that define morphisms of C * -algebras (called comultiplication, counit and antipode).
In other words, given (A, u), the morphisms ∆, ε, S can exist or not.If they exist, they are uniquely determined, and we say that we have a Hopf algebra.
The basic examples are as follows: (1) The algebra of functions A = C(G), with the matrix u = (u ij ) given by g = (u ij (g)), where G ⊂ U n is a compact group.
Let T be the unit circle, and let z : T → C be the identity function, z(x) = x.Observe that (C(T), z) is a finitely generated Hopf algebra, corresponding to the compact group T ⊂ U 1 , or, via the Fourier transform, to the group Z =< 1 >.Definition 1.2.Associated to (A, u) is the pair ( Ã, ũ), where Ã ⊂ C(T) * A is the C * -algebra generated by the entries of the matrix ũ = zu.
It follows from the general results of Wang in [7] that ( Ã, ũ) is indeed a finitely generated Hopf algebra.Moreover, ũ is the free complexification of u in the free probabilistic sense, i.e. with respect to the Haar functional.See [2].
A morphism between two finitely generated Hopf algebras f : where A s ⊂ A and B s ⊂ B are the dense * -subalgebras generated by the elements u ij , respectively v ij .Observe that in order for a such a morphism to exist, u, v must have the same size, and that if such a morphism exists, it is unique.See [2].
Proof.All the assertions are clear from definitions, see [2] for details.
Proof.By using the Fourier transform isomorphism Then, a careful examination of generators gives the isomorphism Γ Z * Λ. See [2] for details.
At the dual level, we have the following question: what is the compact quantum group G c defined by C(G c ) = C(G)?There is no simple answer to this question, unless in the abelian case, where we have the following result.
, where L is the image of G in the projective unitary group P U n .
Proof.The embedding G ⊂ U n , viewed as a representation, must come from a generating system G =< g 1 , . . ., g n >.It routine to check that the subgroup Λ ⊂ G constructed in Theorem 1.4 is the dual of L, and this gives the result.

Free quantum groups
Consider the groups S n ⊂ O n ⊂ U n , with the elements of S n viewed as permutation matrices.Consider also the following subgroups of U n : (1) S n = Z 2 × S n , the permutation matrices multiplied by ±1.
(3) P n = T × S n , the permutation matrices multiplied by scalars in T.
(4) K n = T S n , the permutation matrices with coefficients in T.
Observe that H n is the hyperoctahedral group.It is convenient to collect the above definitions into a single one, in the following way.Definition 2.1.We use the diagram of compact groups where S * denotes at the same time S and S .
In what follows we describe the free analogues of these 7 groups.For this purpose, we recall that a square matrix u ∈ M n (A) is called: (1) Orthogonal, if u = ū and u t = u −1 .
(2) Cubic, if it is orthogonal, and ab = 0 on rows and columns.
(3) Magic', if it is cubic, and the sum on rows and columns is the same.
(5) Biunitary, if both u and u t are unitaries.
(7) Magik, if it is cubik, and the sum on rows and columns is the same.
Here the equalities of type ab = 0 refer to distinct entries on the same row, or on the same column.The notions (1, 2, 4, 5) are from [7,3,8,7] Proof.The case G = OHSU is discussed in [7,3,8,7], and the case G = S KP follows from it, by identifying the corresponding subgroups.
We proceed with liberation: definitions will become theorems and vice versa.
Definition 2.3.A g (n) with g = ohs * ukp is the universal C * -algebra generated by the entries of a n × n orthogonal, cubic, magic*, biunitary, cubik, magik matrix.

Theorem 2.4. We have the diagram of Hopf algebras
where s * denotes at the same time s and s .
Proof.The morphisms in Definition 1.1 can be constructed by using the universal property of each of the algebras involved.For the algebras A ohsu this is known from [7,3,8,7], and for the algebras A s kp the proof is similar. ( Proof.The case g = ohs is discussed in [3], and the case g = u is discussed in [4].In the case g = s kp we can use the following formulae: The commutation conditions condition, and the extra relations ∪ ∩ ∈ End(u) and ∪ ∩ ∈ End(ū) correspond to the magik condition.Together with the fact that orthogonal plus magik means magic', this gives all the g = s kp assertions.
We can color the diagrams in several ways: either by putting the sequence xyyxxyyx . . . on both rows of points, or by putting α, β on both rows, then by replacing a → xy, b → yx.We say that the diagram is colored if all the strings match, and half-colored, if there is an even number of unmatches.Theorem 3.3.For g = ohs * ukp we have CA g (n) = span(D g ), where: (1) D s (k, l) is the set of all diagrams between 2k points and 2l points.
(2) D s (k, l) = D s (k, l) for k − l even, and D s (k, l) = ∅ for k − l odd.
Proof.This is clear from the above lemma, by composing diagrams.The case g = ohsu is discussed in [3,4], and the case g = s kp is similar.Theorem 3.4.We have the following isomorphisms: Proof.It follows from definitions that we have arrows from left to right.Now since by Theorem 3.3 the spaces End(u ⊗ ū ⊗ u ⊗ . ..) are the same at right and at left, Theorem 5.1 in [2] applies, and gives the arrows from right to left.
Observe that the assertion (1), known since [1], is nothing but the isomorphism U + n = O +c n mentioned in the beginning of the first section.
It follows from Theorem 3.3 that the algebras A ohs * ukp are free.
In the orthogonal case u = ū we say that A is free orthogonal, and in the general case, we also say that A is free unitary.Theorem 4.2.If A is free orthogonal then Ã is free unitary.
Proof.It is shown in [2] that the tensor category of Ã is generated by the tensor category of A, embedded via alternating words, and this gives the result.We know from Theorem 4.2 that the operation A → Ã is well-defined, between the algebras in the statement.Moreover, since by Tannakian duality an orthogonal algebra of infinite level is determined by the spaces Hom(u k , u l ) with k − l even, we get that A → Ã is injective, because these spaces are: It remains to prove surjectivity.So, let A be free unitary satisfying A = Ã.We have CA = span(D) for certain sets of diagrams D(α, β) ⊂ D s (α, β), so we can define a collection of sets D 2 (k, l) ⊂ D s (k, l) in the following way: (1) For k − l even we let D 2 (k, l) = D(γ k , γ l ).

Definition 4 . 1 .
A finitely generated Hopf algebra (A, u) is called free if:
, and (3, 6, 7) are new.The terminology is of course temporary: we have only 7 examples of free quantum groups, so we don't know exactly what the names name.
n ) with G = OHS * U KP is the universal commutative C * -algebra generated by the entries of a n × n orthogonal, cubic, magic*, biunitary, cubik, magik matrix.