The Haar state of \mathcal{O}(SL_q(3)) on a monomial basis

The quantum group \mathcal{O}(SL_q(n)) is a non-(co)commutative Hopf algebra dual to the Jimbo-Drinfeld quantum group $\mathcal{U}_q(sl(n))$. It is known that O(SLq(3)) becomes a pre-Hilbert space with the Haar state as the inner product. The matrix coefficients of the Peter-Weyl decomposition of \mathcal{O}(SL_q(n)) are orthogonal with respect to the inner product. However, there is no orthonormal basis on this pre-Hilbert space due to a lack of efficient methods to evaluate the Haar state values of monomials on $\mathcal{O}(SL_q(n)). In this talk, we investigate the Haar state on \mathcal{O}(SL_q(3)). We will define a monomial basis that spans the linear subspace consisting of elements with non-zero Haar state values. Then, we give the Haar state values of those monomials in the basis as rational polynomials in variable q. We will also discuss the method to achieve an orthonormal basis which is proposed by Noumi, Yamada, and Mimachi.

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