2005/03/02 by Anastasia Doikou, Paul Martin, Paul P. Martin
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic structures and combinatorial models #Bethe ansatz #Crossing #Hamiltonian (control theory) #Hecke algebra #Homotopy and Cohomology in Algebraic Topology #Integrable system #Irreducible representation #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum group #Quantum mechanics #Representation theory #S-matrix #hep-th
paper · pdf · doi:10.1088/1742-5468/2006/06/p06004
43 pages, multiple figures
arxiv created 2005/03/02 · openalex publication_date 2006/06/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’ representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang–Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley–Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the boundary Yang–Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin’s commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary—despite differing from this model in having reference states.) Using these multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.