The idempotents of the TL_n-modules \otimes^nC^2 in terms of elements of U_qsl_2

TL;DR

This paper constructs explicit idempotents projecting onto irreducible and indecomposable modules of the TL_n-module ^nC^2, expressed in terms of U_qsl_2 elements, including at roots of unity where modules are indecomposable.

Contribution

It provides explicit formulas for idempotents in the TL_n-module ^nC^2 using U_qsl_2 elements, extending to roots of unity where modules are indecomposable.

Findings

Explicit idempotents for generic q as linear combinations of U_qsl_2 elements.

Idempotents for roots of unity involving divided powers and new generators.

Decomposition of ^nC^2 into irreducible and indecomposable modules.

Abstract

The vector space \otimes^nC^2 upon which the XXZ Hamilonian with n spins acts bears the structure of a module over both the Temperley-Lieb algebra TL_n(\beta=q+1/q) and the quantum algebra U_qsl_2. The decomposition of \otimes^nC^2 as a U_qsl_2-module was first described by Rosso [23], Lusztig [15] and Pasquier and Saleur [20] and that as a TL_n-module by Martin [17] (see also Read and Saleur [21] and Gainutdinov and Vasseur [9]). For q generic, i.e. not a root of unity, the TL_n-module \otimes^nC^2 is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \otimes^nC^2) onto each of these irreducible modules as linear combinations of elements of U_qsl_2. When q=q_c is a root of unity, the TL_n-module \otimes^nC^2 (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involve some new generators, whose action on \otimes^nC^2 is that of the divided powers (S^\pm)^{(r)}=\lim_{q\to q_c} (S^\pm)^r/[r]!.