  We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra.
Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar
algebra motivated by Kuperberg's A_2-spider. This A_2-planar algebra contains a
subfamily of vector spaces which will capture the double complex structure
pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell
system, including both the periodicity three coming from the A_2-Temperley-Lieb
algebra as well as the periodicity two coming from the subfactor basic
construction. We use an A_2-planar algebra to obtain a description of the
(Jones) planar algebra for the Wenzl subfactor in terms of generators and
relations.
