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Compact quantum group

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In <a href="Mathematics" title="Mathematics">mathematics</a>, compact quantum groups are generalisations of <a href="Compact_group" title="Compact group">compact groups</a>, where the commutative <a href="C*-algebra" title="C*-algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} ^{*}}</annotation>

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</math><img src="./f6dfd7b24c8a39f4acdd9e5c117d1b9d1e6ab5ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathrm {C} ^{*}}" loading="lazy">-algebra</a> of continuous complex-valued functions on a compact group is generalised to an <a href="Abstract_structure" title="Abstract structure">abstract structure</a> on a not-necessarily commutative unital <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} ^{*}}</annotation>

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</math><img src="./f6dfd7b24c8a39f4acdd9e5c117d1b9d1e6ab5ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathrm {C} ^{*}}" loading="lazy">-algebra, which plays the role of the "algebra of continuous complex-valued functions on the compact quantum group".<a href="#cite_note-1">[1]</a>

The basic motivation for this theory comes from the following analogy. The space of complex-valued functions on a compact Hausdorff <a href="Topological_space" title="Topological space">topological space</a> forms a commutative C*-algebra. On the other hand, by the <a href="Gelfand_representation" title="Gelfand representation">Gelfand Theorem</a>, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a>.

<a href="S._L._Woronowicz" title="S. L. Woronowicz">S. L. Woronowicz</a><a href="#cite_note-2">[2]</a> introduced the important concept of compact matrix quantum groups, which he initially called compact pseudogroups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a <a href="Noncommutative_geometry" title="Noncommutative geometry">noncommutative geometry</a>.

<div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2></div>

For a compact <a href="Topological_group" title="Topological group">topological group</a>, G, there exists a C*-algebra homomorphism

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<annotation encoding="application/x-tex">{\displaystyle \Delta :C(G)\to C(G)\otimes C(G)}</annotation>

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</math><img src="./f2584af6ba01e6889d023d3120ae8dc4cfb56e47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.535ex; height:2.843ex;" alt="{\displaystyle \Delta :C(G)\to C(G)\otimes C(G)}" loading="lazy"></dd></dl>

where C(G) ⊗ C(G) is the minimal C*-algebra tensor product — the completion of the algebraic <a href="Tensor_product" title="Tensor product">tensor product</a> of C(G) and C(G) — such that

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<annotation encoding="application/x-tex">{\displaystyle \Delta (f)(x,y)=f(xy)}</annotation>

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</math><img src="./f2b8915121de03dd9083931076730f57000d4c7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.024ex; height:2.843ex;" alt="{\displaystyle \Delta (f)(x,y)=f(xy)}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C(G)}">

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<annotation encoding="application/x-tex">{\displaystyle f\in C(G)}</annotation>

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</math><img src="./05098bf188ef0bbe04b2a465cd4854e5fb607967.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.522ex; height:2.843ex;" alt="{\displaystyle f\in C(G)}" loading="lazy">, and for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in G}">

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<annotation encoding="application/x-tex">{\displaystyle x,y\in G}</annotation>

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</math><img src="./cb3bd958e7750dc36ecaaa12caaacd9b6601af7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.186ex; height:2.509ex;" alt="{\displaystyle x,y\in G}" loading="lazy">, where

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<annotation encoding="application/x-tex">{\displaystyle (f\otimes g)(x,y)=f(x)g(y)}</annotation>

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</math><img src="./9e57f897ffd2c718a689e75f604fc09226ad6baf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.97ex; height:2.843ex;" alt="{\displaystyle (f\otimes g)(x,y)=f(x)g(y)}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in C(G)}">

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<annotation encoding="application/x-tex">{\displaystyle f,g\in C(G)}</annotation>

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</math><img src="./56df82e095201fee4c619710515b6e18c60aacb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.672ex; height:2.843ex;" alt="{\displaystyle f,g\in C(G)}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in G}">

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<annotation encoding="application/x-tex">{\displaystyle x,y\in G}</annotation>

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</math><img src="./cb3bd958e7750dc36ecaaa12caaacd9b6601af7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.186ex; height:2.509ex;" alt="{\displaystyle x,y\in G}" loading="lazy">. There also exists a linear multiplicative mapping

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<annotation encoding="application/x-tex">{\displaystyle \kappa :C(G)\to C(G)}</annotation>

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</math><img src="./a8503f8236c2e23d5846aa8bc17a5fc65c09a6a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.695ex; height:2.843ex;" alt="{\displaystyle \kappa :C(G)\to C(G)}" loading="lazy">,</dd></dl>

such that

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<annotation encoding="application/x-tex">{\displaystyle \kappa (f)(x)=f(x^{-1})}</annotation>

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</math><img src="./d9c3e85652b9fe03c044980fe5fd5f765bb36220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.415ex; height:3.176ex;" alt="{\displaystyle \kappa (f)(x)=f(x^{-1})}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C(G)}">

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<annotation encoding="application/x-tex">{\displaystyle f\in C(G)}</annotation>

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</math><img src="./05098bf188ef0bbe04b2a465cd4854e5fb607967.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.522ex; height:2.843ex;" alt="{\displaystyle f\in C(G)}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in G}">

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<annotation encoding="application/x-tex">{\displaystyle x\in G}</annotation>

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</math><img src="./6d7e0b51bd905f35d11790939139d18014f8b017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.997ex; height:2.176ex;" alt="{\displaystyle x\in G}" loading="lazy">. Strictly speaking, this does not make C(G) into a <a href="Hopf_algebra" title="Hopf algebra">Hopf algebra</a>, unless G is finite.

On the other hand, a finite-dimensional <a href="Group_representation" title="Group representation">representation</a> of G can be used to generate a <a href="*-algebra" title="*-algebra">*-subalgebra</a> of C(G) which is also a Hopf *-algebra. Specifically, if

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<annotation encoding="application/x-tex">{\displaystyle g\mapsto (u_{ij}(g))_{i,j}}</annotation>

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is an n-dimensional representation of G, then

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<annotation encoding="application/x-tex">{\displaystyle u_{ij}\in C(G)}</annotation>

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</math><img src="./3a7617ef494637b456df4e6e6493c966113926d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.05ex; height:3.009ex;" alt="{\displaystyle u_{ij}\in C(G)}" loading="lazy"></dd></dl>

for all i, j, and

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<annotation encoding="application/x-tex">{\displaystyle \Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj}}</annotation>

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</math><img src="./94dd2663f98267cc097ec895baf275ba85f4d453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.315ex; height:5.509ex;" alt="{\displaystyle \Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj}}" loading="lazy"></dd></dl>

for all i, j. It follows that the <a href="*-algebra" title="*-algebra">*-algebra</a> generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle u_{ij}}</annotation>

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</math><img src="./6dda9331f162d678a64d0c1ffabfe67c1ca570b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.807ex; height:2.343ex;" alt="{\displaystyle u_{ij}}" loading="lazy"> for all i, j and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (u_{ij})}">

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<annotation encoding="application/x-tex">{\displaystyle \kappa (u_{ij})}</annotation>

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</math><img src="./78cedf0dcb8c44e1c25b2d79a0b2ad0c01c697ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.955ex; height:3.009ex;" alt="{\displaystyle \kappa (u_{ij})}" loading="lazy"> for all i, j is a Hopf *-algebra: the counit is determined by

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<annotation encoding="application/x-tex">{\displaystyle \epsilon (u_{ij})=\delta _{ij}}</annotation>

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</math><img src="./6c71e2384ada470f22a48f2d120b752469baca3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.168ex; height:3.009ex;" alt="{\displaystyle \epsilon (u_{ij})=\delta _{ij}}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}">

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<annotation encoding="application/x-tex">{\displaystyle i,j}</annotation>

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</math><img src="./f4cbf8bbc622154cda8208d6e339495fe16a1f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>

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</math><img src="./fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"> is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>), the antipode is κ, and the unit is given by

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<annotation encoding="application/x-tex">{\displaystyle 1=\sum _{k}u_{1k}\kappa (u_{k1})=\sum _{k}\kappa (u_{1k})u_{k1}.}</annotation>

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</math><img src="./5f3a6763d9c07fb26e30c4dd5548fd7483c1f27f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.748ex; height:5.509ex;" alt="{\displaystyle 1=\sum _{k}u_{1k}\kappa (u_{k1})=\sum _{k}\kappa (u_{1k})u_{k1}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Compact_matrix_quantum_groups">Compact matrix quantum groups</h2></div>

As a generalization, a compact matrix quantum group is defined as a pair (C, u), where C is a C*-algebra and

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<annotation encoding="application/x-tex">{\displaystyle u=(u_{ij})_{i,j=1,\dots ,n}}</annotation>

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</math><img src="./eb781909dbf15e89054a2ef0d4ba65767ecaeff8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.906ex; height:3.009ex;" alt="{\displaystyle u=(u_{ij})_{i,j=1,\dots ,n}}" loading="lazy"></dd></dl>

is a matrix with entries in C such that

<ul><li>The *-subalgebra, C0, of C, which is generated by the matrix elements of u, is dense in C;</li>

<li>There exists a C*-algebra homomorphism, called the comultiplication, Δ : C → C ⊗ C (here C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \forall i,j:\qquad \Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj};}</annotation>

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</math><img src="./374baf17a4a629a668145923b9f4febb238ec2e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.631ex; height:5.509ex;" alt="{\displaystyle \forall i,j:\qquad \Delta (u_{ij})=\sum _{k}u_{ik}\otimes u_{kj};}" loading="lazy"></dd></dl></dd></dl>

<ul><li>There exists a linear antimultiplicative map, called the coinverse, κ : C0 → C0 such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (\kappa (v*)*)=v}">

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<annotation encoding="application/x-tex">{\displaystyle \kappa (\kappa (v*)*)=v}</annotation>

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</math><img src="./27c93d687c8074770da0a82a047c92b87b772204.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.975ex; height:2.843ex;" alt="{\displaystyle \kappa (\kappa (v*)*)=v}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in C_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle v\in C_{0}}</annotation>

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</math><img src="./7bb97a78031c010a9875647afbfa106aa1bb682d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.684ex; height:2.509ex;" alt="{\displaystyle v\in C_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k}\kappa (u_{ik})u_{kj}=\sum _{k}u_{ik}\kappa (u_{kj})=\delta _{ij}I,}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k}\kappa (u_{ik})u_{kj}=\sum _{k}u_{ik}\kappa (u_{kj})=\delta _{ij}I,}</annotation>

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</math><img src="./a1490e8d3f31f83b9892117ec67bbaa0b8543f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.469ex; height:5.509ex;" alt="{\displaystyle \sum _{k}\kappa (u_{ik})u_{kj}=\sum _{k}u_{ik}\kappa (u_{kj})=\delta _{ij}I,}" loading="lazy"> where I is the <a href="Identity_element" title="Identity element">identity element</a> of C. Since κ is antimultiplicative, κ(vw) = κ(w)κ(v) for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,w\in C_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle v,w\in C_{0}}</annotation>

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</math><img src="./9fbd88427df34aa0b3f1c4703c0105d5e7518688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.382ex; height:2.509ex;" alt="{\displaystyle v,w\in C_{0}}" loading="lazy">.</li></ul>

As a consequence of continuity, the comultiplication on C is coassociative.

In general, C is a bialgebra, and C0 is a Hopf *-algebra.

Informally, C can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group.

<div class="mw-heading mw-heading2"><h2 id="Compact_quantum_groups">Compact quantum groups</h2></div>

For C*-algebras A and B acting on the Hilbert spaces H and K respectively, their minimal tensor product is defined to be the norm completion of the algebraic tensor product A ⊗ B in B(H ⊗ K); the norm completion is also denoted by A ⊗ B.

A compact quantum group<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> is defined as a pair (C, Δ), where C is a unital C*-algebra and

<ul><li>Δ : C → C ⊗ C is a unital *-homomorphism satisfying (Δ ⊗ id) Δ = (id ⊗ Δ) Δ;</li>

<li>the sets {(C ⊗ 1) Δ(C)} and {(1 ⊗ C) Δ(C)} are dense in C ⊗ C.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2></div>

A representation of the compact matrix quantum group is given by a <a href="Coalgebra" title="Coalgebra">corepresentation</a> of the Hopf *-algebra<a href="#cite_note-5">[5]</a> Furthermore, a representation, v, is called unitary if the matrix for v is unitary, or equivalently, if

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<annotation encoding="application/x-tex">{\displaystyle \forall i,j:\qquad \kappa (v_{ij})=v_{ji}^{*}.}</annotation>

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</math><img src="./d41fbd77e01967f30cf4f2121310958ad7e6bae9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.772ex; height:3.343ex;" alt="{\displaystyle \forall i,j:\qquad \kappa (v_{ij})=v_{ji}^{*}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

An example of a compact matrix quantum group is SUμ(2),<a href="#cite_note-6">[6]</a> where the parameter μ is a positive real number.

<div class="mw-heading mw-heading3"><h3 id="First_definition">First definition</h3></div>

SUμ(2) = (C(SUμ(2)), u), where C(SUμ(2)) is the C*-algebra generated by α and γ, subject to

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<annotation encoding="application/x-tex">{\displaystyle \gamma \gamma ^{*}=\gamma ^{*}\gamma ,\ \alpha \gamma =\mu \gamma \alpha ,\ \alpha \gamma ^{*}=\mu \gamma ^{*}\alpha ,\ \alpha \alpha ^{*}+\mu \gamma ^{*}\gamma =\alpha ^{*}\alpha +\mu ^{-1}\gamma ^{*}\gamma =I,}</annotation>

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</math><img src="./0e819f692da2085de6bf40a18e62de6be1cb485c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.363ex; height:3.176ex;" alt="{\displaystyle \gamma \gamma ^{*}=\gamma ^{*}\gamma ,\ \alpha \gamma =\mu \gamma \alpha ,\ \alpha \gamma ^{*}=\mu \gamma ^{*}\alpha ,\ \alpha \alpha ^{*}+\mu \gamma ^{*}\gamma =\alpha ^{*}\alpha +\mu ^{-1}\gamma ^{*}\gamma =I,}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle u=\left({\begin{matrix}\alpha &\gamma \\-\gamma ^{*}&\alpha ^{*}\end{matrix}}\right),}</annotation>

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</math><img src="./5ac385dc9cd8913c4fd06a561106e8b37ae7e547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.641ex; height:6.176ex;" alt="{\displaystyle u=\left({\begin{matrix}\alpha &\gamma \\-\gamma ^{*}&\alpha ^{*}\end{matrix}}\right),}" loading="lazy"></dd></dl>

so that the comultiplication is determined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (\alpha )=\alpha \otimes \alpha -\gamma \otimes \gamma ^{*},\Delta (\gamma )=\alpha \otimes \gamma +\gamma \otimes \alpha ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta (\alpha )=\alpha \otimes \alpha -\gamma \otimes \gamma ^{*},\Delta (\gamma )=\alpha \otimes \gamma +\gamma \otimes \alpha ^{*}}</annotation>

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</math><img src="./65d2f9e83eadc541c6797522d6ccdfd05a8fbd20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.639ex; height:2.843ex;" alt="{\displaystyle \Delta (\alpha )=\alpha \otimes \alpha -\gamma \otimes \gamma ^{*},\Delta (\gamma )=\alpha \otimes \gamma +\gamma \otimes \alpha ^{*}}" loading="lazy">, and the coinverse is determined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (\alpha )=\alpha ^{*},\kappa (\gamma )=-\mu ^{-1}\gamma ,\kappa (\gamma ^{*})=-\mu \gamma ^{*},\kappa (\alpha ^{*})=\alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa (\alpha )=\alpha ^{*},\kappa (\gamma )=-\mu ^{-1}\gamma ,\kappa (\gamma ^{*})=-\mu \gamma ^{*},\kappa (\alpha ^{*})=\alpha }</annotation>

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</math><img src="./2186c7a37094b56a0b5db031fc1144a2e08e4b15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.093ex; height:3.176ex;" alt="{\displaystyle \kappa (\alpha )=\alpha ^{*},\kappa (\gamma )=-\mu ^{-1}\gamma ,\kappa (\gamma ^{*})=-\mu \gamma ^{*},\kappa (\alpha ^{*})=\alpha }" loading="lazy">. Note that u is a representation, but not a <a href="Unitary_representation" title="Unitary representation">unitary representation</a>. u is equivalent to the unitary representation

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<annotation encoding="application/x-tex">{\displaystyle v=\left({\begin{matrix}\alpha &{\sqrt {\mu }}\gamma \\-{\frac {1}{\sqrt {\mu }}}\gamma ^{*}&\alpha ^{*}\end{matrix}}\right).}</annotation>

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</math><img src="./a7c9e70c0e84bebfbe9dd5903afc448a0ef6b2c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.954ex; height:7.843ex;" alt="{\displaystyle v=\left({\begin{matrix}\alpha &{\sqrt {\mu }}\gamma \\-{\frac {1}{\sqrt {\mu }}}\gamma ^{*}&\alpha ^{*}\end{matrix}}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Second_definition">Second definition</h3></div>

SUμ(2) = (C(SUμ(2)), w), where C(SUμ(2)) is the C*-algebra generated by α and β, subject to

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<annotation encoding="application/x-tex">{\displaystyle \beta \beta ^{*}=\beta ^{*}\beta ,\ \alpha \beta =\mu \beta \alpha ,\ \alpha \beta ^{*}=\mu \beta ^{*}\alpha ,\ \alpha \alpha ^{*}+\mu ^{2}\beta ^{*}\beta =\alpha ^{*}\alpha +\beta ^{*}\beta =I,}</annotation>

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</math><img src="./da7120bfbeb60ed01e28308a67dbbc225514d16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.444ex; height:3.176ex;" alt="{\displaystyle \beta \beta ^{*}=\beta ^{*}\beta ,\ \alpha \beta =\mu \beta \alpha ,\ \alpha \beta ^{*}=\mu \beta ^{*}\alpha ,\ \alpha \alpha ^{*}+\mu ^{2}\beta ^{*}\beta =\alpha ^{*}\alpha +\beta ^{*}\beta =I,}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle w=\left({\begin{matrix}\alpha &\mu \beta \\-\beta ^{*}&\alpha ^{*}\end{matrix}}\right),}</annotation>

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</math><img src="./4e1598e7abe86a59a5033b1399f46bb08decf840.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.225ex; height:6.176ex;" alt="{\displaystyle w=\left({\begin{matrix}\alpha &\mu \beta \\-\beta ^{*}&\alpha ^{*}\end{matrix}}\right),}" loading="lazy"></dd></dl>

so that the comultiplication is determined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (\alpha )=\alpha \otimes \alpha -\mu \beta \otimes \beta ^{*},\Delta (\beta )=\alpha \otimes \beta +\beta \otimes \alpha ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta (\alpha )=\alpha \otimes \alpha -\mu \beta \otimes \beta ^{*},\Delta (\beta )=\alpha \otimes \beta +\beta \otimes \alpha ^{*}}</annotation>

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</math><img src="./f2f31cf00e4735c949d76a9b2a359574417b5a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.377ex; height:2.843ex;" alt="{\displaystyle \Delta (\alpha )=\alpha \otimes \alpha -\mu \beta \otimes \beta ^{*},\Delta (\beta )=\alpha \otimes \beta +\beta \otimes \alpha ^{*}}" loading="lazy">, and the coinverse is determined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (\alpha )=\alpha ^{*},\kappa (\beta )=-\mu ^{-1}\beta ,\kappa (\beta ^{*})=-\mu \beta ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \kappa (\alpha )=\alpha ^{*},\kappa (\beta )=-\mu ^{-1}\beta ,\kappa (\beta ^{*})=-\mu \beta ^{*}}</annotation>

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</math><img src="./e75ddd5dd669d6a5878f0ef4d43337c359228aeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.036ex; height:3.176ex;" alt="{\displaystyle \kappa (\alpha )=\alpha ^{*},\kappa (\beta )=-\mu ^{-1}\beta ,\kappa (\beta ^{*})=-\mu \beta ^{*}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (\alpha ^{*})=\alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa (\alpha ^{*})=\alpha }</annotation>

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</math><img src="./2b3c6cb522035a2100c18b1bfff50d1d8595ae5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.276ex; height:2.843ex;" alt="{\displaystyle \kappa (\alpha ^{*})=\alpha }" loading="lazy">. Note that w is a unitary representation. The realizations can be identified by equating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\sqrt {\mu }}\beta }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma ={\sqrt {\mu }}\beta }</annotation>

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</math><img src="./811cb161d625c36a7495e30a40d6526649377672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.03ex; height:3.009ex;" alt="{\displaystyle \gamma ={\sqrt {\mu }}\beta }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Limit_case">Limit case</h3></div>

If μ = 1, then SUμ(2) is equal to the concrete compact group SU(2).

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBanica2023" class="citation book cs1">Banica, Teo (2023). Introduction to Quantum Groups. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-031-23816-1</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Woronowicz, S.L. "Compact Matrix Pseudogrooups", Commun. Math. Phys. 111 (1987), 613-665

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Woronowicz, S.L. "Compact Quantum Groups". Notes from <a rel="nofollow" class="external free" href="http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf">http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf</a>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> van Daele, A. and Maes, Ann. "Notes on compact quantum groups", arXiv:math/9803122

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> a corepresentation of a counital coassiative coalgebra A is a square matrix

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<annotation encoding="application/x-tex">{\displaystyle v=(v_{ij})_{i,j=1,\dots ,n}}</annotation>

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with entries in A (so that v ∈ M(n, A)) such that

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<annotation encoding="application/x-tex">{\displaystyle \forall i,j:\qquad \epsilon (v_{ij})=\delta _{ij}.}</annotation>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> van Daele, A. and Wang, S. "Universal quantum groups" Int. J. Math. (1996), 255-263.

</li>

</ol></div></div></div><div><div class="zim-footer">

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