Computing the Haar state of $\mathcal{O}(S{L}_q(3))$ using value preserving (anti)homomorphisms

In this paper, we introduce two (anti)homomorphisms that preserve the Haar state values of monomials. Together with the modular automorphism, the three (anti)homomorphisms are used in our new algorithm to compute the Haar states of monomials on $\mathcal{O}(S{L}_q(3))$. Comparing with the algorithm proposed in the author’s previous work \cite{lu2023}, the new algorithm reduces the linear relations used in the computation to a half.