A Temperley–Lieb Analogue for the BMW Algebra
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 \({\mathfrak{s}\mathfrak{l}}_{2}\) on \({(\mathbb{C}{(q)}^{2})}^{\otimes n}\). We define and study a quotient of the Birman–Wenzl–Murakami algebra, which plays an analogous role for the three-dimensional representation of quantum \({\mathfrak{s}\mathfrak{l}}_{2}\). In the course of the discussion, we prove some general results about the radical of a cellular algebra, which may be of independent interest.