The RFD and KAC quotients of the Hopf*-algebras of universal orthogonal quantum groups

We determine the Kac quotient and the RFD (residually finite dimensional) quotient for the Hopf *-algebras associated to universal orthogonal quantum groups.


Introduction
Compact quantum groups of Woronowicz [Wor87] are often studied via their associated Hopf * -algebras, the so-called CQG algebras [DK94].The CQG algebra carries all the grouptheoretic information about the associated quantum group, such as its representation theory, the lattice of quantum subgroups (described via the lattice of the CQG quotients of the original algebra), or Kac property, but also for example encodes approximation properties of the natural operator algebraic completions.
When studying a particular property describing a 'simpler' class of objects, it is natural to ask whether a general object admits a largest subobject with the given property.And thus So ltan, motivated by the considerations concerning quantum group compactifications, showed in [So05] (see also [Tom07]) that every compact quantum group admits a unique maximal subgroup of Kac type; in other words, every CQG algebra admits a maximal Kac type quotient.He also computed such Kac quotients in some explicit examples, including the universal unitary quantum groups U + Q of Wang and Van Daele.The same paper also saw the first seeds of the study of residually finite dimensional CQG algebras, fully developed ten years later by Chirvasitu [Chi15].The latter article shows that every CQG algebra admits the RFD quotient, which roughly speaking is the largest quotient which has 'sufficiently many' finite dimensional representations, discusses various stability results for the RFD property and most importantly proves that the CQG algebras of free unitary and orthogonal quantum groups, U + n and O + n are RFD for all n = 3.The case of n = 3 was established later in [Chi20].One should note that already combining [So05], [Chi15] and [Chi20] leads to the description of the RFD quotient of the CQG algebras of all U + Q .We also refer to these papers and their introduction for further motivation behind studying these concepts.
In this short note we compute the Kac and RFD quotients for the Hopf * -algebras associated to universal orthogonal quantum groups O + F of Wang and Van Daele, exploiting earlier results of Chirvasitu, the classification of O + F up to isomorphism essentially due to Banica and Wang (formulated explicitly in [Rij07]), and the direct computations using the defining commutation relations.The main results are Theorems 3.3 and 3.4.

Preliminaries
We begin by recalling the basic objects and notions studied in this paper.
2.1.Universal compact quantum groups.We will study compact quantum groups in the sense of [Wor87] via the associated CQG (compact quantum group) algebras.These are involutive Hopf algebras which are spanned by the coefficients of their finite-dimensional unitary corepresentations, see, e.g., [KS97,Section 11.3]; each of them admits a unique biinvariant state, called the Haar state.Note that Hopf * -quotients of CQG algebras are again CQG algebras, and the category of CQG algebras admits a natural free product construction (see for example [Wan95]).
The universal compact quantum groups U + Q and O + F were introduced by Van Daele and Wang [WD96].Let N ∈ N, let F ∈ M N (C) be invertible, and put Q = F * F .The universal unitary CQG algebra Pol(U + Q ), also denoted A u (Q), is generated by the N 2 coefficients of its fundamental corepresentation U = (u jk ) 1≤j,k≤N , subject to the conditions that U and This means that for all 1 ≤ j, k ≤ N we have Thus the CQG algebra Pol(U + F * F ) depends only on the positive invertible matrix Q, which, up to isomorphism, we can assume to be diagonal, Q = (δ jk q j ) 1≤j,k≤N , with 0 If F satisfies furthermore F F ∈ RI N , then we define the universal orthogonal CQG algebra Pol(O + F ), also denoted by B u (F ) or A o (F ), as the quotient of Pol(U + Q ) by the additional relation Up to isomorphism of CQG algebras, it is sufficient to consider the following two families, see [Wan02] and [Rij07, Remark 1.5.2].
Case I: F F = I N , and F can be written as Case II: F F = −I N , N is even, and F can be written as , where in case I, 1 is an eigenvalue only if 2k < N .
2.2.Kac quotient and RFD quotient.If A = Pol(G) is the CQG algebra of some compact quantum group, then the Kac ideal of A is defined as the intersection of the (left) null spaces of all tracial states on A: J KAC = {a ∈ A; τ (a * a) = 0 for all tracial states τ on A}, and the Kac quotient is A KAC = A/J KAC .One can show that A KAC is again a CQG algebra, which corresponds to the largest quantum subgroup of G which is of Kac type; the last statement means that the associated Haar state is a trace.
So ltan [So05, Appendix A] [So06, Section 5] worked with the Kac quotient for C * -algebras associated with compact quantum groups, but here we prefer to use a version for CQG algebras, which is also the setting in [Chi15].See Subsection 2.4 below for a brief discussion of the relation between CQG-algebraic and C * -algebraic Kac or RFD quotients.
Motivated by a question about Bohr compactifications of discrete quantum groups, Chirvasitu introduced in [Chi15] the RFD property (where RFD stands for 'residually finite dimensional') for CQG algebras and showed that Pol(U + N ) = A u (I N ) and Pol(O + N ) = B u (I N ) = A o (I N ) have this property, implying that the discrete quantum groups U + N and O + N are maximal almost periodic in the sense of [So05,So06].See also the related more recent paper [BBCW17].
The RFD quotient is defined as the biggest quotient of a CQG algebra that has the RFD property.We recall the relevant definitions from [Chi15].
Definition 2.1.[Chi15, Definition 2.6] A *-algebra A has property RFD, if for any a ∈ A, a = 0, there exists a finite-dimensional representation (i.e. a unital * -homomorphism) π : The RFD quotient A RFD of a *-algebra A is the quotient of A by the intersection of the kernels of all representations π : A → M n (C), with n ∈ N.

In other words, A
One can show that the RFD quotient of a CQG algebra is again a CQG algebra.Note that RFD is a weaker property than inner linearity (defined in [BB10], see also [BFS12]); in general the relationship between various possible notions of residual finiteness for quantum groups remains not fully clarified -see for example the comments in [BBCW17].
Chirvasitu proved the following three results.The proofs in [Chi15] do not include N = 3; this case is dealt with in [Chi20].
The quotient CQG-algebras A RFD and A KAC yield quantum subgroups 2.3.RFD quotient of universal unitary quantum groups.The Kac quotients and the RFD quotients for the universal unitary quantum groups are already known, although the latter result has not been explicitly stated in the literature.
Then the Kac quotient and the RFD quotient of the CQG algebra Pol(U + Q ) are equal to the free product ⋆ r ν=1 Pol(U + Mν ).Remark 2.7.So ltan showed that this is the Kac quotient, cf [So05, Theorem 4.9] and [So06, Section 7].Chirvasitu's results, i.e., Proposition 2.3 and Theorem 2.4, show that this free product is RFD, and therefore it is also the RFD quotient.Again we can define A KAC and A RFD as respective quotients of A by J KAC and J RFD , and again the RFD quotient is a quotient of the Kac quotient.
Since we can restrict tracial states or finite-dimensional representations of A to A, we have In general this inclusion can be proper.If A = C u (G) is the universal C * -algebra of G, then we have equality, since every state and representation on A extends to C u (G).
Example 2.8.[CS19, Proposition 2.4] showed that a compact quantum group is coamenable if and only its reduced C * -algebra admits a finite-dimensional representation.Therefore, using the results of Banica from [Ban96], [Ban97] and [Ban99] we have Banica [Ban97, Theorem 3] showed also that the reduced C * algebra of U + Q admits a unique trace if Q ∈ RI N , and no trace if this is not the case.Thus we get C r (U

RFD quotient of the universal orthogonal quantum groups
Let us now describe the RFD quotients of the free orthogonal quantum groups O + F introduced in the beginning of the last section.
3.1.Two special cases.Let us start with some special cases which will be useful in the next section when we treat the general situation.
Proposition 3.1.Let M ≥ 1 and let J M be the standard symplectic matrix Then the CQG algebra Pol(O + J M ) has property RFD.
Proof.For M = 1, we have O + J 1 = SU (2) and the result is true (as the algebra in question is commutative, see Remark 2.5).
For the general case we can use the same proof as in where z denotes the generator of Z viewed as an element of CZ.
Step 2: The center of Pol(O + J M ) is given by the morphism of CQG algebra γ : Pol(O + J M ) → CZ 2 with γ(u jk ) = δ jk t (where t denotes the generator of Z 2 ).The cocenter (i.e. the Hopf kernel of γ, see [Chi14, Definition 2.10]) is exactly B ′ .Indeed, γ is central, i.e., it satisfies where Σ denotes the flip, and any other central map can be factored through γ.Furthermore, we have Let us consider next the case where F * F has only two eigenvalues: q 2 < 1 < q −2 .
Proposition 3.2.Let M ∈ N, q ∈ (0, 1), ǫ ∈ {−1, 1} and set Then the RFD quotient and the Kac quotient of the CQG algebra Pol(O + F ) are both equal to the CQG algebra Pol(U + M ).Proof.This proof is similar to those of Theorems 3.3 and 3.4 in the next subsection, therefore we will give a rather detailed argument here, and later sketch only the main steps.We decompose the fundamental corepresentation U as So we can write U as and therefore The unitarity condition for U now reads The equalities of upper left corners of (3.1) mean that for all j, k = 1, . . ., M M ℓ=1 Setting j = k and taking the sum, we get (1 − q 4 )u * j+M,ℓ u j+M,ℓ .
If we divide by the *-ideal generated by the coefficients of C, then we see from Equation (3.1) that the remaining generators u jk with 1 ≤ j, k ≤ M , which form the matrix A -or rather their images in the quotient *-algebra -have to satisfy exactly the defining relations of Pol(U + M ), i.e., The result now follows, since Chirvasitu proved that Pol(U + M ) is RFD, cf.Theorem 2.4.

3.2.
Case I: F F = I N .We now look at the case F F = I N , where we can assume that F has the form given in Equation (2.1).But we will permute the rows and columns of F to organize F in blocks corresponding to the eigenvalues of F * F .
Theorem 3.3.Let F be of the form Then the RFD quotient and the Kac quotient of the CQG algebra Pol(O + F ) are both equal to the free product Proof.The proof is similar to that of Proposition 3.2.Writing U as a block matrix and using the relation between the blocks that follow from (H), we can express U as , where furthermore the coefficients of Z are hermitian, i.e., Z = Z.
If we look at the diagonal blocks of the unitarity condition U * U = I N = U U * , we get for every µ = 1, . . ., r r ρ=1 is a matrix with coefficients in some *-algebra A and τ is a tracial state on A, then we have So if τ is a tracial state on Pol(O + F ) and we apply τ • Tr to Equation (3.5), then we get If we now take the sum over µ of the difference between the left-hand-side and the righthand-side in Equations (3.3) and (3.4), and apply τ • Tr, then we get r ρ,µ=1 Adding these two relations and taking Equation (3.6) into account, we get τ Tr(C * ρµ C ρµ ) = 0 for all ρ, µ ∈ {1, . . ., r}; by positivity this means that all the generators that appear in the C-blocks are contained in the Kac ideal J KAC .
By (3.3), we then also have τ Tr(X * µ )X µ = τ Tr(R * µ R µ ) , so, plugging this into (3.6), and we get τ Tr(X * µ X µ ) = 0 = τ Tr(R * µ R µ ) for µ = 1, . . ., r, since all terms in the above sum are non-negative.Once again using the fact that τ is positive we deduce that all the generators that appear in the X-and R-blocks are contained in the Kac ideal J KAC .
Denote by the matrix obtained from U by deleting the generators in the even rows and columns in the block decomposition in Equation (3.2), as well as the last row and column.
If we divide the *-algebra Pol(O + F ) by the *-ideal generated by all C ρµ , X µ and R µ , then unitarity relation U * U = I N = U U * reduces to This means that the quotient Pol(O F )/ C ρµ , X µ , R µ : ρ, µ = 1, . . ., r is equal to the free product of a copy of Pol(U + D ), generated by the coefficients of the A ρµ , ρ, µ = 1, . . ., r, and a copy of Pol(O + N −2K ), generated by the coefficients of Z. Now we can conclude with Theorem 2.6.Theorem 3.4.Let N be an even positive integer and let F ∈ M N be of the form The RFD quotient and the Kac quotient of the CQG algebra Pol(O + F ) are both equal to the free product Proof.Like in the proofs of Proposition 3.2 and Theorem 3.3, we write U as a block matrix.Since U = F U F , we can write .
Denote by J the *-ideal generated by the u jk that have been regrouped in the blocks C µν with (µ, ν) = (r, r), and in the blocks A µν with µ = ν.
2.4.CQG-algebraic quotients vs. C * -algebraic quotients.Let G = (A, ∆) be a compact quantum group with C * -algebra A and CQG algebra A. The C * -algebraic Kac ideal and RFD ideal are J KAC = {a ∈ A; τ (a * a) = 0 for all tracial states τ on A}, with J KAC = A if A has no tracial states, and J RFD = {a ∈ A; π(a) = 0 for all fin.-dim.repr.π of A}, with J RFD = A if A has no finite-dimensional representations.

3. 3 .
Case II: F F = −I N .Let us now consider the case F F = −I N and F a matrix of the form given in Equation (2.2).