Isomorphisms and Fusion Rules of Orthogonal Free Quantum Groups and their Complexifications

We show that all orthogonal free quantum groups are isomorphic to variants of the free orthogonal Wang algebra, the hyperoctahedral quantum group or the quantum permutation group. We also obtain a description of their free complexification. In particular we complete the calculation of fusion rules of all orthogonal free quantum groups and their free complexifications.

A subspace of Hom(( n ) ⊗k , ( n ) ⊗l ) is by definition spanned by partitions if it is linearly generated by a family (T P ) where P runs through some subset of Part(k, l). In [7] the free unitary Wang algebra A u (n) ∶= C * (u ij , 1 ≤ i, j ≤ n (u ij ) ij , (u * ij ) ij are unitary) and the free orthogonal Wang algebra were introduced. Moreover in [8] the quantum permutation group is unitary and u ij are partial isometries summing up to one in every row and every column ⎞ ⎟ ⎠ was defined. Note that "are partial isometries" can be replaced by "are projections". The three last named algebras are compact matrix quantum groups in the sense of Woronowicz [9]. The following class of quantum groups will be of interest in this paper. • The intertwiner spaces Hom(U i 1 ⊠ ⋯ ⊠ U i k , U j 1 ⊠ ⋯ ⊠ U j l ), i α , j β ∈ {1, } are spanned by partitions, where U = (u * ij ) is the conjugate corepresentation of U and ⊠ denotes the tensor product of corepresentations.
If the first condition is strengthened by requiring that the morphism (A o (n), U o ) → (A s (n), U s ) factors through (A, U ), then A it is called orthogonal free.
In [5] the following classification was achieved. Theorem 1.2 There are exactly six orthogonal free quantum groups. Namely (i) The free orthogonal Wang algebra.
(ii) The quantum permutation group.
(iv) The bistochastic quantum group is unitary and u ij sum up to one in every row and every column ⎞ ⎟ ⎠ .
(v) The symmetrized bistochastic quantum group is unitary and u ij sum up to the same element in every row and every column (vi) The symmetrized quantum permutation group is unitary and u ij are partial isometries summing up to the same element in every row and every column The fusion rules of (1) were calculated in [1], those of (2) in [3] and those of (3) in [6]. We show that the remaining examples are slight modifications of A o (n) and A s (n). In particular we can derive their fusion rules and find that A b' (n) and A s' (n) are counterexamples to a conjecture by Banica and Vergnioux given in [6].
In [4] the free complexification of orthogonal free quantum groups was considered. If (A, U ) is a orthogonal free quantum group, then its free complexification (Ã,Ũ ) is by definition the sub-C*algebra of the free product A * C(S 1 ) generated by the entries ofŨ ∶= U ⋅ id S 1 = (u ij ⋅ id S 1 ). Here id S 1 denotes the canonical generator of C(S 1 ). As Banica shows in [4] the intertwiners between tensor products of the fundamental corepresentation and its conjugate can be described by the intertwiners of the orthogonal free quantum group it comes from. With additional requirements we can calculate the fusion rules of the free complexification from the fusion rules of the original orthogonal free quantum group. These additional requirements are fulfilled by A o (n) and A h (n), which gives the fusion rules of A k (n) = A h (n). Those of A u (n) = A o (n) are known from [2]. From [4] we know that A b (n) = A b' (n) and A s (n) = A s' (n). We denote A b (n) =∶ A c (n) and A s (n) =∶ A p (n). They can be decomposed and described in terms of A o (n) and A s (n) again. Acknowledgment: I want to thank both Thomas Timmermann for suggesting to work on fusion rules of free quantum groups and Stefaan Vaes for helpful discussions about this article, especially on the last section. Moreover, I want to thank the referee for helpful comments.

Preliminaries
We will mainly work with compact matrix quantum groups as defined by Worono-wicz in [9]. If A is a *-algebra and U ∈ M n (A) we denote by U the matrix whose entries are conjugated, i.e.
• the matrix U is invertible.
A morphism of compact matrix quantum groups (A, U ) φ → (B, V ) is a *-homo-morphism A → B such that φ(u ij ) = v ij where U and V must have the same size. There is at most one morphism from one quantum group to another. If there is a morphism (A, U ) → (B, V ) then we say that (B, V ) is a quantum subgroup of (A, U ). Every compact matrix quantum group is also a compact quantum group, i.e. a C*-algebra A with a *-homomorphism ∆ ∶ A → A ⊗ min A such that Every morphism of compact matrix quantum groups is also a morphism of compact quantum groups. We will also refer to a quantum group (A, U ) or (A, ∆) as A.
In particular a one dimensional corepresentation matrix is just a unitary group-like element of A.

Free fusion rings
In this section we will introduce free fusion rings and prove that they are free unital rings.
We will use the following notation for words in free monoids. Let M = mon(S) be a free monoid over a set S. If w ∈ M is a word of length k, then we write w i for the i-th letter of w, 1 ≤ i ≤ k. Hence w = w 1 w 2 w 3 . . . w k−1 w k . Definition 3.1 A free fusion monoid is a free monoid M = mon(S) over a set S with a fusion ⋅ ∶ S × S → S ∪ {∅} and a conjugation ∶ S → S. They must satisfy the following conditions.
(i) The fusion ⋅ is associative, where we make the convention that s ⋅ s ′ is the empty set if one of s, s ′ is the empty set.
(ii) The conjugation is involutive, i.e. s = s for all s ∈ S.
(iii) Fusion and conjugation are compatible in the following sense. For all s 1 , s 2 , s 3 ∈ S we have A set S equipped with fusion and conjugation is called a fusion set. The fusion and conjugation of S induce a fusion and a conjugation on M via If M = mon(S) is a free fusion monoid, we can turn M into an associative ring by Here w, w ′ are words in M , a w and a w ′ are the corresponding elements in M , xy, yz and xz denote the concatenation of words and the second term in the sum is by convention always ignored if the fusion x ⋅ z is empty. Actually condition (3) of the previous definition is a necessary condition for making M associative, as it can be seen by considering (a s 1 ⋅ a s 2 ) ⋅ a s 3 = a s 1 ⋅ (a s 2 ⋅ a s 3 ) for s 1 , s 2 , s 3 ∈ S. A *-ring isomorphic to M for some fusion monoid M is called a free fusion ring. From the point of view of rings, free fusion rings are very easy. Actually they are free. The proof of the following lemma was already given in [6] in some special cases.
Lemma 3.2 A free fusion ring over a fusion set S is the free unital ring over a s , s ∈ S.
Proof Let M be the fusion ring over a fusion set S. It suffices to show that M is a free -module with the basis a s 1 ⋯a s k with k ∈ AE and s 1 , . . . , s k ∈ S. So it suffices to express the elements of the -basis a w , w ∈ M as -linear combinations of the elements a s 1 ⋯a s k with k ∈ AE and s 1 , . . . , s k ∈ S and to show that {a s 1 ⋯a s k k ∈ AE, s 1 , . . . , s k ∈ S} is -linearly independent.
There are coefficients C w This shows that all a w , w ∈ M are linear combinations of a s 1 ⋯a s k with k ∈ AE and s 1 , . . . , s k ∈ S. ◻ Remark 3.3 Free fusion rings can be used to describe fusion rules very shortly and there is hope to use free fusion rings as a starting point for proofs of several properties of quantum groups. See section 10 of [6] for a comment on these possibilities. However in order to justify the concept of free fusion rings intrinsically it would be good to answer the following question affirmatively. Is every fusion ring of a compact quantum group that is free as a unital ring a free fusion ring?

Some isomorphisms of combinatorial quantum groups
In this section we will consider combinatorial quantum groups A * (n) for * ∈ {b, b ′ , s ′ , c, p}. They are free products or direct sums of known quantum groups. For * ∈ {b ′ , s ′ , c, p} it turns out that their fusion rings are not free.
Theorem 4.1 We have the following isomorphisms of compact quantum groups (not necessarily preserving the fundamental corepresentation).
(iv) A p (n) is isomorphic to the free product A s (n) * C(S 1 ).
(v) A c (n) is isomorphic to the free product A b (n) * C(S 1 ).

Remark 4.2
Note that in the case n ≤ 3 we have the isomorphisms A s (n) ≅ C(S n ) and A o (1) ≅ C({−1, 1}). So the given descriptions can be further simplified. The key observation for the rest of 4.1 is the following lemma.
Proof Consider * = b ′ , s ′ first. The element z = ∑ i u ij is easily seen to be a unitary group-like element, so it corresponds to a one dimensional unitary corepresentation of A * (n). Consider the group S n ⊕ 2 ⊂ U n as permutation matrices with entries +1 and −1. Let U Sn⊕ 2 be the canonical fundamental corepresentation of C(S n ⊕ 2 ). Then the image of z under the map For * = p, c consider z ∶= id S 1 as coming from the copy of C(S 1 ). This copy is contained in A * (n), since the trivial corepresentation is contained in the fundamental corepresentation of A b (n) and A s (n). Using the relations of A * (n) we can check the rest of the claim by simple calculations. ◻

Remark 4.4
The last lemma shows, that the fusion rules of neither of the quantum groups A * (n) for ∈ {b ′ , s ′ , c, p} can be described by a free fusion ring. Actually in a free fusion ring any element a ≠ 1 satisfies a ⋅ a * ≠ 1. This gives two counterexamples to the conjecture that for n ≥ 4 the fusion rules of all free orthogonal quantum groups can be described by a free fusion ring, which was stated in [6].

Remark 4.5
The fundamental corepresentation of any matrix quantum group that has (A s (n), U s ) as a sub quantum group cannot be the sum of more than two irreducible corepresentations. In particular the last lemma already gives a decomposition U ≃ U z ⊞ V with U z non-trivial and one dimensional and V irreducible, where U is the fundamental corepresentation of A * (n).
Proof (Proof of Theorem 4.1) The isomorphism of (2) is given by This map exists since z is central in A s' (n) as an easy calculation shows. The inverse map is given by In order to prove (3) we use again an orthogonal matrix T ∈ M n ( ) such that T (1, 0, ..., 0) t = (1 √ n, ..., 1 √ n) t . Then a matrix U ∈ M n (A) for some C * -algebra A satisfies the relations of U b' if and only if T t U T is a block matrix with a self-adjoint unitary in the upper left corner and an orthogonal (n − 1) × (n − 1) matrix in the lower right corner. This proves The isomorphism of (4) is given by The isomorphism of (5) is given by All the isomorphisms respect the comultiplication, since z is group-like. Hence, they are isomorphisms of quantum groups. ◻ 5 Fusion rules for free products and the quantum group A k (n) In this section we describe the fusion rules of the free complexification A k (n) ≅ A h (n). Instead of referring to A k (n) explicitly, we will work in a more general setting and deduce its fusion rules as a corollary. Roughly the main statement of this section is given by the following theorem. See theorem 5.5 for a precise statement.
Theorem 5.1 Let (A, U ) be an orthogonal compact matrix quantum group, i.e. U = U , such that its fusion rules are free. Assume further that 1 ∉ U ⊠2k+1 for any k ∈ AE. Then the fusion rules of (Ã,Ũ ) are free and can be described in terms of the fusion rules of (A, U ).
The following theorem is due to Wang [8]. The following observation will be useful when studying the fusion rules of a free complexification.

Remark 5.3
Let A * B be a free product of compact quantum groups with irreducible corepresentations W γ 1 ⊠ ⋯ ⊠ W γn and W δ 1 ⊠ ⋯ ⊠ W δm as in the last theorem. Then (i) If W γn and W δ 1 are not corepresentations of the same factor of the free product, then W γ 1 ⊠ ⋯ ⊠ W γn ⊠ W δ 1 ⊠ ⋯ ⊠ W δm is an irreducible corepresentation of A * B.
(ii) If W γn and W δ 1 are corepresentations of the same factor and W γn ⊠W δ 1 = ∑ k i=1 W ǫ i +δ W γn ,W δ 1 ⋅ 1 is the decomposition into irreducible corepresentations, then and the first k summands of this decomposition are irreducible.
For the rest of this section fix an orthogonal compact matrix quantum group (A, U ) such that its fusion rules are described by a free fusion ring over the fusion set S. Assume further that 1 ∉ U ⊠2k+1 for any k ∈ AE.
Note that the fusion ring ofÃ is the fusion subring of Rep(A * C(S 1 )) that is generated by U ⊠ z, where z denotes the identity on the circle. We will construct the free complexificationS of S and prove that the fusion rules of (Ã,Ũ ) are described byS. We begin by constructingS. Let Rep irr even (respectively Rep irr odd ) be the set of classes of irreducible corepresentations of A that appear as subrepresentations of an even (respectively odd) tensor power of U . We have Rep irr even ∩ Rep irr odd = ∅ due to Frobenius duality and the requirement 1 ∉ U 2k+1 for all k ∈ AE. Let S even ⊂ S (resp. S odd ⊂ S) be the set of elements corresponding to corepresentations from Rep irr even (resp. Rep irr odd ). The setS is then by definition the disjoint union S even ⊔ S even ⊔ S odd ⊔ S odd . Denote the first copy of S even (resp. S odd ) by S   odd . Note that S even = S even and S odd = S odd , i.e. the conjugation onS is well defined. A fusion onS can be defined according to the following table.
The row gives the element which is fused from the right with an element coming from the set indicated by the column. The fusion is empty if this is indicated by the table and is otherwise the usual fusion of two elements of S lying in the part ofS indicated by the table. Note that this definition makes sense, since S even ⋅ S even , S odd ⋅ S odd ⊂ S even ∪ {∅} and S even ⋅ S odd , S odd ⋅ S even ⊂ S odd ∪ {∅}. It is easy to see thatS with this structure is a fusion set. Now we can state a precise version of 5.1.
Theorem 5.5 Let (A, U ) be an orthogonal compact matrix quantum group such that its fusion rules are described by a free fusion ring over the fusion set S. Assume further that 1 ∉ U ⊠2k+1 for any k ∈ AE. Then the fusion rules of (Ã,Ũ ) are given by the free complexificationS of S.
We construct a complete set of corepresentations ofÃ. In order to do so we associate an irreducible corepresentations of (Ã,Ũ ) to any element ofR ∶= Rep irr even ⊔ Rep irr even ⊔ Rep irr odd ⊔ Rep irr odd . We denote the i-th copy of Rep irr even (Rep irr odd ) by Rep irr,(i) even (Rep irr,(i) odd ). Let V be a irreducible corepresentation in Rep irr even . Then V and z * ⋅ V ⋅ z are corepresentations ofÃ. Actually, if V is an irreducible subrepresentation of U ⊠2k then V is an irreducible subrepresentation of (Ũ ⊠Ũ ) ⊠k and z * ⋅ V ⋅ z is an irreducible subrepresentation of (Ũ ⊠Ũ ) ⊠k . We consider V as an element of Rep irr, (1) even and z * ⋅ V ⋅ z as an element of Rep irr, (2) even . Similarly we see that if V ∈ Rep irr odd then we can associate with it corepresentations V ⋅ z ∈ Rep irr,(1) odd and z * ⋅ V ∈ Rep irr, (2) odd . Note that elements s fromS give corepresentationsŨ s by this identification. Consider a word w = w 1 . . . w k with letters inR. We say that w is reduced if in the sequenceŨ w 1 , . . . ,Ũ wn a z is never followed by z * and U x is always followed by z or z * . In formal terms: Any such reduced word w = w 1 . . . w k gives rise to an irreducible corepresentation ofÃ byŨ w ∶= U w 1 ⊠ . . . ⊠Ũ w k and different reduced words give rise to inequivalent corepresentations by 5.2. Since any iterated tensor product ofŨ andŨ decomposes as a sum of irreducible corepresentations of the typeŨ w , where w is a reduced word with letters inR, any irreducible corepresentation ofÃ is equivalent to someŨ w .
Definition 5.6 Consider now a word w = w 1 . . . w k with letters inS. It is called connected if every z is followed by a z * . Formally: odd ) The following definition says how we can associate irreducible corepresentations ofÃ to words with letters inS.
Definition 5.7 If w is an arbitrary word with letters inS then it has a unique decomposition w = x 1 . . . x l into maximal connected words. This gives rise to a unique reduced word w ′ with letters inR. We setŨ w ∶=Ũ w ′ Next we have to do some preparations in order to prove theorem 5.5. Remark 5.9 Note that if x is a connected word with letters in S then according to remark 5.4 it can be written as z i 0 ⋅x ⋅ z i 1 , i 0 , i 1 ∈ {0, 1, −1} and we haveŨ x = z i 0 ⊠ Ux ⊠ z i 1 .
Definition 5.10 Let x, y be connected words with letters inS. We say that (x, y) fits together if xy is a connected word.
Lemma 5.11 Let x = x 1 . . . x m and y = y 1 . . . y n be connected words with letters inS such that (x m , y 1 ) fits together. WriteŨ Proof Since (x, y) fits together, we have z i 1 ⊠ z j 0 = 1. So by remark 5.3 the first equation follows. We have to prove that for all x = ac, y = cb In order to prove (1), note that ab is connected, since a, b are connected and (a, b) fits together. So (1) follows from the way irreducible corepresentations are associated to connected words remarked in 5.9. For (2) note that, since (a, b) fits together,ǎ ⋅b = ∅ if and only if a ⋅ b = ∅. If a ⋅ b ≠ ∅ then it is connected and (2) follows by remark 5.9 again. ◻ Now we can give the proof of Theorem 5.5 Proof (Proof of Theorem 5.5) Let x = x 1 . . . x k and y = y 1 . . . y l be words with letters inS. We have to show thatŨ x ⊠Ũ y = x=ac,y=cbŨ ab ⊞Ũ a⋅b Let x = u 1 . . . u m and y = v 1 . . . v n be the decomposition in maximal connected words. We identify them with letters inR. Theñ We are going to consider the two cases (x k , y 1 ) do or do not fit together. Assume that (x k , y 1 ) do not fit together. This means z i m+1 ⋅ z j 0 ≠ 1. ThenŨ x ⊠Ũ y is irreducible by Theorem 5.2. Moreover, xy = u 1 . . . u m v 1 . . . v n is a decomposition in maximal connected words. SoŨ x ⊠Ũ y =Ũ xy . On the other hand (x k , y 1 ) not fitting together implies x k ≠ y 1 and x k ⋅y 1 = ∅. So ∑ x=ac,y=cbŨab ⊞Ũ a⋅b =Ũ xy . This completes the proof for the first case. Assume now that (x k , y 1 ) fits together. This means z i m+1 ⋅ z j 0 = 1. By Lemma 5.11 By applying the induction hypothesis to the term we obtainŨ x ⊠Ũ y = x=ac,y=cbŨ ab ⊞Ũ a⋅b . ◻ We are now going to deduce the fusion rules of A k (n). The following result is proven in [6] and describes the fusion rules of A h (n).
Theorem 5.12 Let S h ∶= {u, p} with fusion u ⋅ u = p ⋅ p = p, u ⋅ p = p ⋅ u = u and trivial conjugation. The fusion rules of (A h (n), U h ) are given by the free fusion ring over S h in such a way that U u ≃ U h and U p ⊞ 1 ≃ (u 2 ij ).
Using this theorem we obtain the following corollary in the case A = A k (n).

Corollary 5.13
The irreducible corepresentations of A k (n) are described by the fusion set S k ∶= {u, v, p, q} with fusion given by and conjugation u = v, p = p, q = q. The elements of S k correspond to the following corepresentations.
• The class of the fundamental corepresentation U is U u .
• The class of U is U v .
• The class of the corepresentation (u * ij ⋅ u ij ) is U p ⊞ 1 • The class of the corepresentation (u ij ⋅ u * ij ) is U q ⊞ 1 Proof We only have to prove the part about the concrete description of U u , U v , U p and U q . The fact that U u is the class of the fundamental corepresentation is obvious from the construction. U v ≃ U follows directly. It is easy to check that (u * ij ⋅ u ij ) and (u ij ⋅ u * ij ) are corepresentation of A k (n). We have the decomposition U ⊠ U ≃ U uv ⊞ U p ⊞ 1. Moreover the construction in this section shows that U uv is n 2 − n dimensional and U p is n − 1 dimensional. Since (u ij ⋅ u * ij ) is non trivial, it suffices to give at least two linearly independent intertwiners from the n dimensional corepresentation (u ij ⋅ u * ij ) to U ⊠ U . Two such intertwiners are n → ( n ) ⊗2 ∶ e i ↦ e i ⊗ e i and n → ( n ) ⊗2 ∶ e i ↦ ∑ j e j ⊗ e j . The proof for (u ij ⋅ u * ij ) works similarly. ◻