The idempotents of the TLₙ-modules ⊗ⁿC² in terms of elements of Uqsl₂

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The vector space ⊗ⁿC² upon which the XXZ Hamilonian with n spins acts bears the structure of a module over both the Temperley-Lieb algebra TLₙ(β=q+1/q) and the quantum algebra Uqsl₂. The decomposition of ⊗ⁿC² as a Uqsl₂-module was first described by Rosso [23], Lusztig [15] and Pasquier and Saleur [20] and that as a TLₙ-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 TLₙ-module ⊗ⁿC² is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of ⊗ⁿC²) onto each of these irreducible modules as linear combinations of elements of Uqsl₂. When q=qc is a root of unity, the TLₙ-module ⊗ⁿC² (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 ⊗ⁿC² is that of the divided powers (S±)⁽ʳ⁾=→ qc (S±)ʳ/[r]!.