The Temperley-Lieb algebra has the same generators as the $S\_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen in [the Wikipedia article](http://en.wikipedia.org/wiki/Temperley-Lieb_algebra).



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> Is there a standard analogue for other Weyl groups? Perhaps some specialization of the Hecke algebra?
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There is a standard basis for the Temperley-Lieb algebra, which can be indexed by 321-avoiding permutations: pick a reduced word for $v$ in $S\_n$'s generators, and interpret the generators inside T-L instead. This association is well-defined, because the 321-avoiders are also the "fully commutative" elements, meaning that each has only one reduced word up to commutation relations.


This subset of $S\_n$ is also the set of "lambda-cominuscule" elements (meaning, the $T$-action on the Bruhat cell $BvB/B$ includes the dilation action), and is an ideal in left/right weak Bruhat order. In Weyl groups other than $S\_n$ the lambda-cominuscule condition is strictly stronger than "fully commutative".



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> Assuming the first question, do the lambda-cominuscules, or some other subset of W, similarly give a basis?
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Of course references would be most appreciated.



Temperley-Lieb algebras for other Weyl groups?
First I don't think you have stated the defining relations correctly. [AK: you were quite right, and I've edited the question.]


It sounds as though you should start with the work of Richard Green.
The following papers seem relevant.


<http://arxiv.org/abs/q-alg/9712018>


<http://arxiv.org/abs/math/0102003>


<http://arxiv.org/abs/math/0108076>


