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Yemon Choi
Yemon Choi
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Clebsch Gordan coeffiecentscoefficients of compact quantum groups

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Jim Humphreys
Jim Humphreys
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Consider a compact quantum group G$G$. Let a, b$a, b$ and c$c$ be irreducible unitary corepresentations and assume that c$c$ is contained in a \otimes b$a \otimes b$. Let U$U$ be the intertwiner from the representation Hilbert space of c$c$ to the one of a \otimes b$a \otimes b$ and assume that U$U$ is a partial isometry. By Schur hisSchur's lemma U$U$ is determined up to a phase factor.

Question: Can we assume that U has real (Clebsch-Gordan) coefficients?

If not, is there a good class of examples for which this is true? Or is it true for free orthogonal/unitary QG's?

Question: Can we assume that $U$ has real (Clebsch-Gordan) coefficients?
If not, is there a good class of examples for which this is true? Or is it true for free orthogonal/unitary QG's?

Consider a compact quantum group G. Let a, b and c be irreducible unitary corepresentations and assume that c is contained in a \otimes b. Let U be the intertwiner from the representation Hilbert space of c to the one of a \otimes b and assume that U is a partial isometry. By Schur his lemma U is determined up to a phase factor.

Question: Can we assume that U has real (Clebsch-Gordan) coefficients?

If not, is there a good class of examples for which this is true? Or is it true for free orthogonal/unitary QG's?

Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation Hilbert space of $c$ to the one of $a \otimes b$ and assume that $U$ is a partial isometry. By Schur's lemma $U$ is determined up to a phase factor.

Question: Can we assume that $U$ has real (Clebsch-Gordan) coefficients?
If not, is there a good class of examples for which this is true? Or is it true for free orthogonal/unitary QG's?

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user61080
user61080
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Clebsch Gordan coeffiecents of compact quantum groups

Consider a compact quantum group G. Let a, b and c be irreducible unitary corepresentations and assume that c is contained in a \otimes b. Let U be the intertwiner from the representation Hilbert space of c to the one of a \otimes b and assume that U is a partial isometry. By Schur his lemma U is determined up to a phase factor.

Question: Can we assume that U has real (Clebsch-Gordan) coefficients?

If not, is there a good class of examples for which this is true? Or is it true for free orthogonal/unitary QG's?