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We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras over a finite field are unitary for the action of the braid groups.

Image of the braid groups inside the finite Temperley–Lieb algebras