Quantum Integrability of Non-Ultralocal Models Through Baxterization of Quantized Braided Algebra

Abstract

A scheme suitable for describing quantum non-ultralocal models, including supersymmetric ones, is proposed. Braided algebras are generalized to be used through Baxterization for constructing braided quantum Yang—Baxter equations. Supersymmetric and some known non-ultralocal models are derived in the framework of this approach. As further applications of the scheme, the construction of new quantum integrable non-ultralocal models, like mKdV and anyonic supersymmetric models including deformed anyonic superalgebra, is outlined.

Ultralocal solutions for quantum integrable non-ultralocal models

“…The BYBE (2) provides similar Hopf algebra structure to the integrable NUL models, though braiding relation (1) induces now braided algebra property by modifying the multiplication rule [10,11].…”

“…Nevertheless there is now a considerable number of NUL systems, namely, nonabelian Toda chain [2], the quantum mapping [3], model related to the Coulomb gas picture of CFT [4], current in WZWN [5], integrable model on moduli space [6] and the quantum mKdV [7,8], for which the quantum integrability is established and the braided extensions of YBE have been formulated [9,10,11]. We shall call such NUL models genuine quantum integrable models.…”

Unifying approaches in integrable systems: Quantum and statistical, ultralocal and non-ultralocal

“…In spite of such braided extension of the multiplication rule, the associated coproduct structure of the underlying Hopf algebra, crucial for transition to the global QYBE, must be preserved. Such a braided extension of the Hopf algebra [49,50] was implemented in formulating the integrability theory of nonultralocal models through an unified approach [51]. The basic idea is to complement the commutation rule for the Lax operators at the same site with their braiding property at different lattice sites.…”

“…Detail discussion of this problem and the classification of the Z-matrices allowing factorization is given in [51]. Investigations of some nonultralocal systems from a different angle were done in [52].…”

“…Some other nonultralocal models known in the literature need introduction of braiding beyond NN, basic formulation of which can be found in [50,51]. Examples of such models having same braiding between any two different sites are i.)…”