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In this article we consider conditions under which projection operators inmultiplicity free semi-simple tensor categories satisfy Temperley-Lieb likerelations. This is then used as a stepping stone to prove sufficient conditionsfor obtaining a representation of the Birman-Murakami-Wenzl algebra from abraided multiplicity free semi-simple tensor category. The results are found byutalising the data of the categories. There is considerable overlap with theresults found in arXiv:1607.08908, where proofs are shown by manipulatingdiagrams.

Temperley-Lieb and Birman-Murakami-Wenzl like relations from multiplicity free semi-simple tensor system