2013/03/17 by Guillaume Provencher, Yvan Saint-Aubin, Provencher, Guillaume +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1303.4102
openalex publication_date 2013/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)=limq→ qc (S^±)r/[r]!.