Corepresentation Theory

For compact quantum groups we have a good understanding of corepresentation theory.

This is fundamental to a lot of work in compact quantum groups and there are many pre-sentations of this in the literature, seeMaes & Van Daele(1998) andWoronowicz(1998) for example. However, many conventions used in the case of locally compact quantum groups are different to those used by Woronowicz and van Daele for compact quantum groups, for example in the compact case we tend use the right regular representation where in the locally compact case we tend to use the left regular corepresentation. Addi-tionally it is difficult to find some of the results as stated here in the literature and as such we present a thorough treatment of this subject here.

Throughout this section let pA, ∆q be a C^˚-algebraic quantum group satisfying the conditions of Definition-Theorem 3.2.3 and G the underlying compact quantum group,

that is the reduced C^˚-algebra CpGq is constructed as in Proposition3.2.4.

Definition 3.2.5 A corepresentation ofpA, ∆q on a Hilbert space H is an element U P MpA b B⁰pHqq such that p∆ b idqU “ U¹³U23. If U is unitary then we say this is a unitary corepresentation.

If H is n-dimensional with orthonormal basiste^k | 1 ď k ď nu then for 1 ď i, j ď n we define u_ij P MpAq “ A by u^ij “ pid b ω^ej,eiqpUq and we have

∆pu^i,jq “ pid b id b ω^ej,eiqp∆ b idqpUq “ pid b id b ω^ej,eiqpU¹³U23q.

As ωe_j,e_ipxyq “ pxye^j|eⁱq “řn

k“1pye^j|e^kq pe^k|x^˚eiq “řn

k“1ωe_k,e_ipxqω^ej,e_kpyq for x, y P BpHq then we have

∆pu^i,jq “ ÿn k“1

pid b id b ω^ek,eiqpU¹³qpid b id b ω^ej,ekqpU²³q

“ ÿn k“1

pid b ω^ek,eiqpUq b pid b ω^ej,ekqpUq “ ÿn k“1

ui,k b u^k,j.

We can also similarly show that if this equation holds then so does Definition3.2.5. So equivalently, for a finite dimensional corepresentation, we can consider a matrixpu^ijqⁿi,j“1 P M_npAq such that ∆u^ij “řn

i“1uikb u^kj.

We now move on to consider irreducible corepresentations.

Definition 3.2.6 Let U P MpA b B⁰pHqq be a corepresentation of G. Then a closed subspace K of H is invariant if for e the orthogonal projection from H to K we have p1 b eqUp1 b eq “ Up1 b eq. We say U is irreducible if the only invariant subspaces of H aret0u and H.

Definition 3.2.7 LetU and V be corepresentations ofpA, ∆q on Hilbert spaces H and K respectively. Then we say a linear mapT : H Ñ K intertwines U and V if p1 b T qU “ Vp1 b T q. We say corepresentations U and V are equivalent if there is an invertible

intertwiner fromU to V and unitarily equivalently if there is a unitary intertwiner from U to V .

We now offer some theorems on corepresentations and refer the reader toMaes & Van Daele (1998) for proofs.

Proposition 3.2.8 Any invertible, finite-dimensional corepresentation ofpA, ∆q is equiv-alent to a unitary corepresentation.

The following is one of the most important theorems on corepresentations of compact quantum groups.

Theorem 3.2.9 There exists a maximal family denoted

tU^α P A b BpH^αq | α P Au

of mutually inequivalent, finite-dimensional, irreducible, unitary corepresentations of pA, ∆q such that U^α is contained in the left regular corepresentation for allα P A. Fur-thermore any unitary irreducible corepresentation is equivalent to someU^α.

Let U P MpA b B⁰pHqq be a corepresentation where H is finite dimensional with or-thonormal basisteⁱ | 1 ď i ď nu. Then we have U “ řn

i,j“1uij b e^ij for some u_ij P A such that ∆pu^ijq “ řn

k“1uik b u^kj for all 1 ď i, j ď n. In particular for the maximal family given in the previous theorem we can write this as

u“ pu^αijqⁿi,j^α“1 P M^nαpAq ˇ

ˇ α P A, 1 ď i, j ď n^α( .

We will use this notation throughout the rest of this section for this maximal family.

Notation 3.2.10 We denote by HopfpGq the linear span of u^α_ij ˇ

ˇ α P A, 1 ď i, j ď n^α( inA.

Proposition 3.2.11 We have that HopfpGq is a unital Hopf ˚-algebra that is dense in A such that the Haar state φ is faithful on HopfpGq. Furthermore, for all α P A and 1ď i, j ď n^α, we have the following relations

∆pu^αijq “

nα

ÿ

k“1

u^α_ikb u^αkj, Spu^αijq “ pu^αjiq^˚, ǫpu^αijq “ δ^ij, φpu^αijq “ δ^α,α0

where α0 P A is the unique 1-dimensional corepresentation consisting of the identity 1P A.

Example 3.2.12 We now give examples for constructing corepresentation matrices from given corepresentation matrices. Say pu^ijqⁿi,j“1 P MⁿpAq is a corepresentation matrix, then we have

∆pu^˚ijq “ ∆pu^ijq^˚“ ÿn k“1

u^˚_ikb u^˚kj

and so we have another corepresentation ¯U “ řn

i,j“1u^˚_ij b e^ij. We can show that if U is irreducible then so is ¯U . Clearly the definition of ¯U depends on the choice of or-thonormal basis however given any other oror-thonormal basis we can show that the two corepresentations obtained are equivalent.

Say u P MⁿpAq is a corepresentation. Similarly to Example 2.2.21 we have a C^˚ -algebraic quantum group pA, ∆^opq where ∆^op “ σ ˝ ∆ for σ the flip map on A b^minA.

Then we have

∆^opppu^tq^ijq “ ∆^oppu^jiq “ ÿn k“1

ukib u^jk “ ÿn k“1

pu^tq^ikb pu^tq^kj

and so u^t P MⁿpAq is a corepresentation matrix of pA, ∆^opq. Similarly u^˚ “ p¯uq^t P M_npAq is a corepresentation matrix of pA, ∆^opq.

We can show that for ann-dimensional unitary irreducible corepresentation U that ¯U is equivalent to a unitary corepresentation. Then there exists an invertible matrixT P Mⁿ such thatV “ p1 b T q ¯Up1 b T^´1q is unitary. Then we have ¯U is invertible with inverse

U¯^´1 “ p1 b T^´1qV^˚p1 b T q. Also U^t “ p ¯Uq^˚is invertible with

pU^tq^´1 “ pp ¯Uq^˚q^´1 “ p1 b T^˚qV p1 b pT^´1q^˚q “ p1 b T^˚Tq ¯Up1 b pT^˚Tq^´1q.

It then follows thatpU^tq^´1 is a corepresentation as

p∆ b idqpU^tq^´1 “ p1 b 1 b T^˚Tqp∆ b idqp ¯Uqp1 b 1 b pT^˚Tq^´1q

“ p1 b 1 b T^˚Tqp ¯Uq¹³p1 b 1 b pT^˚Tq^´1q

ˆ p1 b 1 b T^˚Tqp ¯Uq²³p1 b 1 b pT^˚Tq^´1q

“ ppU^tq^´1q¹³ppU^tq^´1q²³

and thusp1 b T^˚Tq ¯U “ pU^tq^´1p1 b T^˚Tq.

We offer a proof of the next few results as they differ slightly from that found in the liter-ature. We first record the following which comes from Lemma 6.3 inMaes & Van Daele (1998).

Lemma 3.2.13 LetU and V be corepresentations ofpA, ∆q on Hilbert spaces H and K respectively, and letxP B⁰pH, Kq and let

y“ pφ b idqpV^˚p1 b xqUq.

Theny P B⁰pH, Kq and V^˚p1 b yqU “ 1 b y.

Proof

Because x P B⁰pH, Kq we have p1 b xqU P B⁰pH, Kq b A and thus V^˚p1 b xqU P B₀pH, Kq b A also. In particular y P B⁰pH, Kq.

Also we have

p∆ b idqpV^˚p1 b xqUq “ V23^˚V₁₃^˚p1 b 1 b xqU¹³U23

and then applying φb id b id we get the result. ✷

Lemma 3.2.14 (Schur’s Lemma) LetU and V be finite-dimensional corepresentations ofpA, ∆q on H and K respectively and let T an intertwiner from U to V . Then

(i) KerT is invariant for U and Image T is invariant for V ;

(ii) If eitherU or V is irreducible then either U and V are inequivalent and there is only the 0 intertwiner fromU to V or U is equivalent to V and there is some invertible T P BpH, Kq such that the set tλT | λ P Cu gives all the intertwiners from U to V . In particular we have that the intertwiners fromU to itself are given bytλ id | λ P Cu.

Proof

Let e denote the orthogonal projection of H onto Ker T and f the orthogonal projection of K onto Image T . Then 0“ p1 b T eq and thus 0 “ V p1 b T eq “ p1 b T qUp1 b eq and so we havep1 b eqUp1 b eq “ Up1 b eq. Similarly we have

p1 b fqV p1 b T q “ p1 b fT qU “ p1 b T qU “ V p1 b T q

and as fpKq “ T H then we have p1 b fqV p1 b fq “ V p1 b fq.

It follows that if U is irreducible then either Ker T “ t0u or Ker T “ U, that is T is either injective (and thus an isomorphism as it is finite dimensional) or 0. Similarly if V is irreducible then Image T “ t0u or Image T “ K, that is T is 0 or surjective (and thus an isomorphism as it is finite dimensional). In either case if U and V are inequivalent then the only intertwiner is 0. On the other hand if T is a non-zero intertwiner from U to V then T is bijective and U and V are equivalent.

Now say S is a non-zero intertwiner from U to V , then for all λP C we have λT ´ S is an intertwiner from U to V and so is either bijective (and thus an isomorphism) or 0.

Choose λ such that detpλT ´ Sq “ 0 and then we have S “ λT for some λ P C as required. ✷

The diagonalisation argument in the following theorem was originally given in Daws (2010).

Theorem 3.2.15 For all α P A there exists a unique invertible positive definite matrix F^α P M^nα with Tr F^α “ Tr pF^αq^´1 such that for all α, β P A, 1 ď k, l ď n^α and 1ď i, j ď n^β we have

φpu^αijpu^βklq^˚q “ δ^αβδik

F_lj^α

Λ^α, φppu^αijq^˚u^β_klq “ δ^αβδjlppF^αq^´1q^ki

Λ^α (3.2)

where Λ^α “ Tr F^α. We can assume that the maximal family of corepresentations is chosen such that for all α P A we have F^α “ diagpλ^α1, . . . , λ^α_nαq with λ^αi ą 0 for all i and Λ^α “ řnα

i“1λ^α_i “ řnα

i“1pλ^αiq^´1 ą 0. Furthermore we have that F^α intertwines U¯^α andppU^αq^tq^´1 andpF^αq^´1 intertwinesU^α andκ²_nαpU^αq. We call these matrices the F -matrices ofpA, ∆q.

Proof

Let α P A. We have from Example 3.2.12that there is some T P M^nα such that V “ p1bT q ¯U^αp1bT^´1q is unitary, ¯U^αis invertible and 1bT^˚T intertwines ¯U^αwithppU^αq^tq^´1 as corepresentations of pA, ∆q. We have T^˚T is positive definite as T is invertible. We define F^α “ λpT^˚Tq^t where we choose λ such that Tr F^α “ Tr pF^αq^´1 ą 0 and we have that pF^αq^t intertwines ¯U^α andppU^αq^tq^´1. As ¯U^α is irreducible it follows from Schur’s Lemma (3.2.14) that F^α is the unique operator such that pF^αq^t intertwines ¯U^α andppU^αq^tq^´1and such that Tr F^α “ Tr pF^αq^´1.

Let α, β P A, i, j P N⁰ such that 1 ď i ď n^α, 1 ď j ď n^β and let x “ e^β,αi,j be the nβ ˆ n^α-matrix in BpH^α, Hβq with 1 in the i, j-th position and 0 elsewhere. Then by

Lemma3.2.13we have an intertwiner given by

and so for j ‰ l we have ν^jl “ 0 and if j “ l then ν^jj is independent of j so we let ν_jj “ ν

Tr F^α and we have the second equation in (3.2).

The first equation is proved similarly by consideringpφ b idq`

U^αp1 b e^αjlqpU^αq^˚˘

Using similar techniques we can show that V^αis unitary and satisfies Equation (3.2) with Fij the diagonal matrix diagpλ^α1, . . . , λ^α_nαq.

Finally we show that F^α intertwines U^α and S_n²_αpU^αq. Above we have a T P M^nα such that we have a unitary matrix V “ p1bT q ¯U^αp1bT^´1q. Then using that SⁿαpW q^t“

pW^˚q^t “ ¯W for any corepresentation W P MⁿαpAq of pA, ∆q (where we’ve used No-tation 1.1.1 for the antipode S) and so S_nαp ¯Wq^t “ Sn²αpW q. Using these equations it follows that

p1 b ¯TqU^αp1 b ¯T^´1q “ ¯V “ S^nαpV q^t“ pp1 b T qS^nαp ¯U^αqp1 b T^´1qq^t

“ p1 b pT^´1q^tqS^nαp ¯U^αq^tp1 b T^tq “ p1 b pT^´1q^tqSn²αpU^αqp1 b T^tq

and so rearranging we get

p1 b T^tT¯qU^α “ Sn²αpU^αqp1 b T^tT¯q

and using that T^tT¯“ pT^˚Tq^twe are done. ✷

Definition 3.2.16 LetpA, ∆q be a compact quantum group with tU^α | α P Au the max-imal family of mutually inequivalent, finite-dimensional, irreducible, unitary corepresen-tations. We definefz : HopfpGq Ñ C for all z P C as the map u^αij ÞÑ ppF^αq^zq^ij, that is thei, j-th entry of the matrixpF^αq^z.

We can calculate the modular automorphism group from Definition-Theorem1.4.16and the scaling group from Definition-Theorem2.2.7for a compact quantum group in terms of the collectionpf^zq^zPC. We have the following properties for the collectionpf^zq^zPC. See Theorem 3.2.19 inTimmermann(2008) for a proof.

Proposition 3.2.17 For allz P C we have that f^z is a character on HopfpGq, that is it is a non-zero homomorphismfz : HopfpGq Ñ C. Furthermore we have the following

(i) For any x P HopfpGq we have that the map w ÞÑ f^wpxq is entire and there exists C ą 0 and d P R such that |fpzq| ď Ce^{dRe z}for allz P C with Re z ą 0;

(ii) f0 “ ε (the counit) and f^z ˚ f^w “ f^z`w where fz ˚ f^w “ pf^z b f^wq ˝ ∆ for all z, w P C;

(iii) fzp1q “ 1, f^zpSpxqq “ f´zpxq and f^itpx^˚q “ f^itpxq for all x P HopfpGq, z P C and tP R.

Proposition 3.2.18 Let σ denote the modular automorphism group and let τ denote the scaling group of the reduced C^˚-algebra CpGq of a compact quantum group G. Then for anyz P C we have HopfpGq Ă Dompσ^zq and HopfpGq Ă Dompτ^zq and furthermore for

and we show that this can be extended to a one-parameter group of˚-automorphisms on the reduced C^˚-algebra CpGq. Using Proposition3.2.11, for αP A and 1 ď i, j ď n^α, we

We may assume F^it is diagonal without loss of generality and so from Equation (3.5) we

and similarly using Proposition3.2.17we have

σ⁰_tppu^αijq^˚q “

From Proposition3.2.17we have that wÞÑ f^wpxq is entire for any x P HopfpGq and thus G is analytic on Spzq and continuous. We have from Equation (3.5) that Gptq “ σ⁰tpu^αijq

and so u^α_ij P Dompσ^zq with

It then follows by linearity that for all x, y P HopfpGq we have φpxσ⁰_´ipyqq “ φpyxq.

Then for all x, yP HopfpGq we have xσ_´i⁰ pyq P HopfpGq and also

φpxσ⁰_´ipyqq “ φpyxq “ φpxσ´ipyqq.

Now using that φ is faithful on HopfpGq it follows that xσ_´i⁰ pyq “ xσ´ipyq for all x, y P HopfpGq and then letting x “ 1 we have σ_´i⁰ pyq “ σ´ipyq for all y P HopfpGq.

So σ⁰_´ipyq “ σ´ipyq for all y P HopfpGq and thus for all y P Dompσ´iq. It then follows from Proposition1.3.11that we have σ “ σ⁰.

Define τ_z⁰ : HopfpGq Ñ HopfpGq be the map u^αij ÞÑřnα

all tě 0. Then it follows that

Corollary 3.2.19 For anyz P C and compact quantum group G we have that HopfpGq is a core forτz. Similarly HopfpGq is a σ-weak core for τ^z on L⁸pGq.

Proof

From Proposition1.3.19we have thattxpnq | x P HopfpGq, n P Nu is a core for Dompτ^zq as HopfpGq is dense in A. Fix n P N and let pu^αijq ˇ

ˇ α P A, 1 ď i, j ď n^α(

be the family from Theorem 3.2.9that is a basis for HopfpGq. Then we need only show that for any αP A and 1 ď i, j ď n^αwe have u^α_ijpnq P HopfpGq.

Corollary 3.2.20 For anyz P C we have that σ^zandτzare automorphisms on HopfpGq.

In document Involutive Algebras and Locally Compact Quantum Groups (Page 111-124)