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

Publication date: 22 February 2021

Abstract: Let

q = | q | e^{i p i h e t a},, h e t a i n (- 1, 1],

be a nonzero complex number such that

| q | e q 1

and consider the compact quantum group

U_{q} (2)

. For

h e t a o t i n m a t h b b Q s e t m i n u s 0, 1

, we obtain the

K

-theory of the

C^{*}

-algebra

C (U_{q} (2))

. We construct a spectral triple on

U_{q} (2)

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

K

-homology class of the associated Fredholm module is shown to be nontrivial.

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