# Is there a version of Temperley-Lieb using sl(3) rather than sl(2)?

This question is a spin-off from Sammy Black's question on super Temperley-Lieb. Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the algebra of sl(2) intertwiners of d copies of the defining rep of sl(2) (and then asks about a supersymmetric version).

My question: in what ways does Sammy's story extend to other simple Lie groups, e.g. sl(3)? For example, I think that the Schur-Weyl duality still holds, so that HomUsl(3)(V⊗d) is a quotient of the group algebra of the symmetric group Sd (here V is the defining representation of sl(3)). Is it something like the Young Tableaux with three rows? And is there a diagrammatic description of this algebra?

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Cvitanovic is great for the groups themselves. But if you care about the quantum group (and this is really where TL shines) some good references are Kuperberg's work on Spiders and Scott Morrison's thesis.

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Yes... I saw it first in Stedman's work (Diagram Techniques in Group Theory, G. E. Stedman, Cambridge University Press, 1990), but it may also exist elsewhere. The basic idea is to combine symmetrization and anti-symmetrization across different sets of strands, roughly in correspondence with the Young tableaux.

The other place to look for more general diagrams for general Lie algebras is Cvitanovic (Group Theory: Birdtracks, Lie's, and Exceptional Groups, Predrag Cvitanović, Princeton University Press, 2008, http://birdtracks.eu/). The text is available online and it is extremely impressive.

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You should be careful saying "Temperley-Lieb at index d". You're better off saying "the d-strand Temperley-Lieb algebra". The problem is that Temperley-Lieb algebras are intimately related to subfactors (in fact, there's a map from the Temperley-Lieb algebra to the standard invariant of any subfactor), and there the paramater q is related to the index of the subfactor, via [N:M] = (q+q^{-1})^2.

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Oops, sorry! Yeah, I had a sneaky suspicion that the word "index" was wrong. –  Theo Johnson-Freyd Oct 30 '09 at 22:29

See the paper

Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

by Hideo Mitsuhashi.

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The short answer is yes. The map to sl(m) kills exactly the representations with more than m rows. Unfortunately, the resulting algebra has no nice general presentation I know of.

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