Existence of (not necessarily positive) integrals on the Hopf *-algebras 𝔄_t^c

Determine whether there exists any (not necessarily positive) integral on the Hopf *-algebras 𝔄_t^c arising from the Jimbo-type deformation in Proposition 4.8a. Specifically, decide whether there is a nonzero linear functional Ο• on 𝔄_t^c satisfying the Hopf algebra invariance property, given that Proposition 2.13d rules out the existence of positive integrals.

Background

The paper constructs Hopf *-algebras 𝔄_tc as deformations of the enveloping algebra of su(2) and shows in Proposition 2.13d that there cannot be any positive integral on these Hopf *-algebras.

However, the nonexistence proof for positive integrals does not address the possibility of non-positive integrals. Establishing whether any (possibly non-positive) integral exists would clarify the status of 𝔄_tc within the broader framework of algebraic quantum groups, where the presence of integrals is central.

References

A few properties are still not completely clear. First we have the existence of the integral on the Hopf $*$-algebras $\mathfrak A_tc$ that we have in Proposition \ref{prop:4.8a}. As we found in Proposition \ref{prop:2.13d}, it is relatively easy to show that there can not be a positive integral. But the proof of that fact can not be used to prove the non-existing of a (possibly non-positive) integral.

— The discrete quantum group $su_q(2)$ and its dual  (2603.29701 - Daele, 31 Mar 2026) in Section 7 (Conclusion and further research)