Journal of the Mathematical Society of Japan

Twisting the $q$-deformations of compact semisimple Lie groups


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Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q$ > 0, we can define new associativity morphisms in ${\rm Rep}(G_q)$ using a $3$-cocycle $\Phi$ on the dual of the center of $G$, thus getting a new tensor category ${\rm Rep}(G_q)^\Phi$. For a class of cocycles $\Phi$ we construct compact quantum groups $G^\tau_q$ with representation categories ${\rm Rep}(G_q)^\Phi$. The construction depends on the choice of an $r$-tuple $\tau$ of elements in the center of $G$. In the simplest case of $G=SU(2)$ and $\tau=-1$, our construction produces Woronowicz's quantum group $SU_{-q}(2)$ out of $SU_q(2)$. More generally, for $G=SU(n)$, we get quantum group realizations of the Kazhdan–Wenzl categories.

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J. Math. Soc. Japan, Volume 67, Number 2 (2015), 637-662.

First available in Project Euclid: 21 April 2015

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Zentralblatt MATH identifier

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]

compact quantum group $q$-deformation tensor category


NESHVEYEV, Sergey; YAMASHITA, Makoto. Twisting the $q$-deformations of compact semisimple Lie groups. J. Math. Soc. Japan 67 (2015), no. 2, 637--662. doi:10.2969/jmsj/06720637.

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