We work out in detail a theory of integrability on the braided covector Hopf algebra and the braided vector Hopf algebra of type An introduced by Majid. Using a braided Fourier transform very similar to the one defined by Kempf and Majid we obtain n-dimensional analogs of results by Koornwinder expressing the correspondence between products of the q2-Gaussian gq2(x_) times monomials, and products of the q2-Gaussian Gq2(∂_) times q2-Hermite polynomials under the transform. We invert the correspondence by finding a suitable inversion, different from the one of Kempf and Majid. We show that with this transforms, whenever n⩾2, the Plancherel measure will depend on the parity of the power series that we are transforming.

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