2016/01/23 by Thi Thao Vu, Vu, Thi-Thao, T. T. Vu
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1601.06253
openalex publication_date 2016/01/23 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
In this thesis, the connection between recently introduced algebraic\nstructures (tridiagonal algebra, q-Onsager algebra, generalized q-Onsager\nalgebras), related representation theory (tridiagonal pair, Leonard pair,\northogonal polynomials), some properties of these algebras and the analysis of\nrelated quantum integrable models on the lattice (the XXZ open spin chain at\nroots of unity) is first reviewed. Then, the main results of the thesis are\ndescribed: (i) for the class of q-Onsager algebras associated with\n widehatsl2 and ADE type simply-laced affine Lie algebras, higher order\nanalogs of Lusztig's relations are conjectured and proven in various cases,\n(ii) for the open XXZ spin chain at roots of unity, new elements (that are\ndivided polynomials of q-Onsager generators) are introduced and some of their\nproperties are studied. These two elements together with the two basic elements\nof the q-Onsager algebra generate a new algebra, which can be understood as\nan analog of Lusztig's quantum group for the q-Onsager algebra. Some\nperspectives are presented.\n