Generalized Schur-Weyl duality and symmetries of integrable spin chains
by Nenad Manojlovic (GFMUL / Universidade do Algarve, Portugal)
Important properties of quantum integrable systems are related with their symmetry algebra and are defined by a bigger algebra which gives the main relations underlining integrability, the so-called RLL-relations. In the case of isotropic Heisenberg chain of spin 1/2 (XXX-model) the symmetry algebra is Lie algebra sl2, the Hamiltonian is an element of the group algebra ℂ[ 𝔖N] of the symmetric group 𝔖N. The fundamental relations of the auxiliary L-matrix entries generate an infinite dimensional quantum algebra -- the Yangian Y(sl2). The actions of sl2 and 𝔖N on the space of states H = ⊗N1ℂ2 are mutually commuting (the Schur-Weyl duality). Extension of this scheme to a particular case of the Hecke algebra -- the Temperley-Lieb algebra, instead of the symmetric group and corresponding new quantum algebras we proposed previously. Here we consider a further generalization -- the case of the Birman-Wenzl-Murakami algebra. In many cases the two algebras give multiplicity free decomposition of the space of states into irreducible representations of these algebras. As a consequence one gets the structure of the multiplets and degeneracy of the spectra of the Hamiltonians.