Equivariant spectral triple for the compact quantum group $U_q(2)$ for complex deformation parameters

Abstract: Let $q=|q|e^{i\pi\theta}$ be a nonzero complex number such that $|q|\neq 1$, and consider the compact quantum group $U_q(2)$. In this talk, we discuss a complete list of inequivalent irreducible representations of $U_q(2)$ and its Peter-Weyl decomposition. Then, for $\theta\notin\mathbb{Q}\setminus\{0,1\}$, we discuss the $K$-theory of the underlying $C^*$-algebra $C(U_q(2))$, and a spectral triple which is equivariant under its own comultiplication action. The spectral triple obtained here is even, $4^+$-summable, non-degenerate, and the Dirac operator acts on two copies of the $L^2$-space of $U_q(2)$. The Chern character of the associated Fredholm module is nontrivial.

operator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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