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The Temperley–Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the usual action of quantum  \mathfraks\mathfrakl2{\mathfrak{s}\mathfrak{l}}_{2} on  (\mathbbC(q)2)Ä</font>n{(\mathbb{C}{(q)}^{2})}^{\otimes n} . We define and study a quotient of the Birman–Wenzl–Murakami algebra, which plays an analogous role for the three-dimensional representation of quantum  \mathfraks\mathfrakl2{\mathfrak{s}\mathfrak{l}}_{2} . In the course of the discussion, we prove some general results about the radical of a cellular algebra, which may be of independent interest.

A Temperley–Lieb Analogue for the BMW Algebra