TEMPERLEY-LIEB ALGEBRAS FOR NON-PLANAR STATISTICAL MECHANICS — THE PARTITION ALGEBRA CONSTRUCTION

We give the definition of the Partition Algebra Pn(Q). This is a new generalisation of the Temperley–Lieb algebra for Q-state n-site Potts models, underpinning their transfer matrix formulation on arbitrary transverse lattices. In Pn(Q) subalgebras appropriate for building the transfer matrices for all transverse lattice shapes (e.g. cubic) occur. For the Partition algebra manifests either a semi-simple generic structure or is one of a discrete set of exceptional cases. We determine the Q-generic and Q-independent structure and representation theory. In all cases (except Q = 0) simple modules are indexed by the integers j ≤ n and by the partitions λ ˫ j. Physically they may be associated, at least for sufficiently small j, to 2j 'spin' correlation functions. We exhibit a subalgebra isomorphic to the Brauer algebra.