For a certain reason, I am interested in the Lie algebra generated by the generators of the Temperley-Lieb algebra, in its natural Catalan- dimensional representation. In other words given the T-L algebra with 2n strands, I am looking at a C_n x C_n matrix algebra quotient, where C_n is the nth Catalan number. Then in this quotient, the 2n-1 Temperley-Lieb generators also generate a Lie algebra. I only need 8 strands for the problem that I am studying. Thus it's a question in 14 x 14 matrices.
I wrote a SAGE/Python program to compute the matrices of the Temperley- Lieb generators, and then I gave them to GAP from SAGE. It isn't all that fast, because GAP does not use the fact that the matrices are sparse, but it works. Sort of. The problem is that the Temperley- Lieb algebra has a parameter d, and I want to know the answer for all d. My code can give me the answer for a specific d working over QQ. But if I want the answer for all d, I should be working over the polynomial ring QQ[d]. Note that it is not hard to rephrase the question as a module question in R^(14^2), where R is the ground ring. I can write down the Lie adjoint action of the Temperley-Lieb generators on 14 x 14 matrices expressed as vectors. I can generate an invariant submodule, and I am then interested in the Smith Normal Form of this submodule, or equivalently the isomorphism type of the quotient module. I have to think a bit about keeping the number of generators of the module under control, but maybe there is a way. My impression is that both GAP and SAGE can easily understand the question when it is posed over a field, but not over a univariate polynomial ring as I actually want. Am I wrong and is there a reasonable way to do this? --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
