2017/02/27 by Iohara, K., Lehrer, G. I., Zhang, R. B.
#16W22 #46L37 #81R15 #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1702.08128
When the parameter q is a root of unity, the Temperley-Lieb algebra TLn(q) is non-semisimple for almost all n. Jones showed that there is a canonical symmetric bilinear form on TLn(q), whose radical Rn(q) is generated by a certain idempotent E_ℓ∈ TLℓ-1(q)⊆ TLn(q), which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. In this work, we study the quotients Qn(ℓ):=TLn(q)/Rn(q), where |q2|=ℓ, which are precisely the algebras generated by Jones' projections. We give the dimensions of their simple modules, as well as dim(Qn(ℓ)); en route we give generating functions and recursions for the dimensions of cell modules and associated combinatorics. When the order |q2|=4, we obtain an isomorphism of Qn(ℓ) with the even part of the Clifford algebra, well known to physicists through the Ising model. When |q2|=5, we obtain a sequence of algebras whose dimensions are the odd-indexed Fibonacci numbers. The general case is described explicitly.