Turning the Quantum Group Invariant XXZ Spin-Chain Hermitian: A Conjecture on the Invariant Product - Mathematical PhysicsReport as inadecuate




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Abstract: This is a continuation of a previous joint work with Robert Weston on thequantum group invariant XXZ spin-chain math-ph-0703085. The previous resultson quasi-Hermiticity of this integrable model are briefly reviewed and thenconnected with a new construction of an inner product with respect to which theHamiltonian and the representation of the Temperley-Lieb algebra becomeHermitian. The approach is purely algebraic, one starts with the definition ofa positive functional over the Temperley-Lieb algebra whose values can becomputed graphically. Employing the Gel-fand-Naimark-Segal GNS constructionfor C*-algebras a self-adjoint representation of the Temperley-Lieb algebra isconstructed when the deformation parameter q lies in a special section of theunit circle. The main conjecture of the paper is the unitary equivalence ofthis GNS representation with the representation obtained in the previous paperemploying the ideas of PT-symmetry and quasi-Hermiticity. An explicit exampleis presented.



Author: Christian Korff

Source: https://arxiv.org/



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