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The vector space \otimes^nC^2 upon which the XXZ Hamilonian with n spins actsbears the structure of a module over both the Temperley-Lieb algebraTL_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] (seealso 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 ofirreducible modules. We construct the projectors (idempotents of the algebra ofendomorphisms of \otimes^nC^2) onto each of these irreducible modules as linearcombinations of elements of U_qsl_2. When q=q_c is a root of unity, theTL_n-module \otimes^nC^2 (with n large enough) can be written as a direct sumof indecomposable modules that are not all irreducible. We also give theidempotents projecting onto these indecomposable modules. Their expression nowinvolve some new generators, whose action on \otimes^nC^2 is that of thedivided powers (S^\pm)^{(r)}=\lim_{q\to q_c} (S^\pm)^r/[r]!.

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