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arXiv:hep-th/9311051

Universality of the universal R-matrix and applications to quantum integrable systems

Anjan Kundu

Published Nov 9, 1993• 0 citations
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Abstract

Results obtained by us are overviewed from a general set up. The universal $R$-matrix is exploited to obtain various important relations and structures involved in quantum group algebra, which are used subsequently for generating different classes of quantum integrable systems through a systematic scheme. This recovers known models as well as discovers a series of new ones.

DOI
10.48550/arxiv.hep-th/9311051

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Results obtained by us are overviewed from a general set up. The universal RRR-matrix is exploited to obtain various important relations and structures involved in quantum group algebra, which are used subsequently for generating different classes of quantum integrable systems through a systematic scheme. This recovers known models as well as discovers a series of new ones.
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Various aspects of the theory of quantum integrable systems are reviewed. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures related to the Yang-Baxter equations. Physical meaning of abstract mathematical notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided algebra, Hopf algebra etc. and the role played by them in integrable systems are highlighted. Systematic construction of quantum integrable lattice as well as field models and their exact excitation spectra are presented through examples. The coordinate and algebraic formulations of the Bethe ansatz are illustrated with comparison, along with the description of nested and functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians from the Lax operators are demonstrated on concrete models. The exposition of this review is kept in a fairly elementary level with emphasis on the physical contents.
A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given RRR matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known discrete and field models a new class of inhomogeneous and impurity models is obtained.
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