“Dilute” Birman–Wenzl–Murakami Algebra

Explicit expressions for three series of R matrices which are related to a “dilute” generalisation of the Birman–Wenzl–Murakami algebra are presented. Of those, one series is equivalent to the quantum R matrices of the D (2) n+1 generalised Toda systems whereas the remaining two series appear to be new. A “dilute” generalisation of the Birman–Wenzl–Murakami (BWM) algebra [1, 2] has recently been introduced [3, 4]. It appears [3] as a generalised braid–monoid algebra [5] related to certain exactly solvable lattice models of two-dimensional statistical mechanics. Alternatively, it can be regarded as a particularly simple case of a two-colour braid–monoid algebra [4] where one colour is represented trivially (in the sense that the corresponding representation of the subalgebra generated by the elements of this colour is one-dimensional). In Ref. [3], it was shown that representations of this algebra can be “Baxterised” [6], i.e., one can find a general expression for a local Yang–Baxter operator X j(u) [5] (u denoting the spectral parameter) in terms of the generators of the dilute BWM algebra for which the Yang–Baxter relations X j(u)X j+1(u+ v)X j(v) = X j+1(v)X j(u+ v)X j+1(u) X j(u)X k(v) = X k(v)X j(u) for |j − k| > 1 (1) follow algebraically. This implies that every suitable representation of the dilute BWM algebra defines a solvable lattice model. As an example, one series of R matrices of this kind has been given in Ref. [3] which were shown to be equivalent to the D (2) n+1 vertex models [7, 8]. In this letter, we present the explicit form of three such series of R matrices (where the series mentioned above is included for completeness). In this case, the Yang–Baxter operator X j(u) acts on a tensor space V ⊗ V ⊗ . . .⊗ V (where V ∼= C d+1 with some integer d) as X j(u) = I ⊗ I ⊗ . . .⊗ I ⊗ Ř(u)⊗ I ⊗ . . .⊗ I ⊗ I (2)