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Introduction to the Yang-Baxter Equation with Open Problems
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest, Romania
Received: 29 March 2012; in revised form: 17 April 2012 / Accepted: 18 April 2012 / Published: 26 April 2012
Abstract: The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have found solutions for the Yang-Baxter equation, obtaining qualitative results (using the axioms of various algebraic structures) or quantitative results (usually using computer calculations). However, the full classification of its solutions remains an open problem. In this paper, we present the (set-theoretical) Yang-Baxter equation, we sketch the proof of a new theorem, we state some problems, and discuss about directions for future research.
Keywords: Yang-Baxter equation; set-theoretical Yang-Baxter equation; algebra structures; Hopf algebras; quantum groups; relations on sets
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MDPI and ACS Style
Nichita, F. Introduction to the Yang-Baxter Equation with Open Problems. Axioms 2012, 1, 33-37.
Nichita F. Introduction to the Yang-Baxter Equation with Open Problems. Axioms. 2012; 1(1):33-37.
Nichita, Florin. 2012. "Introduction to the Yang-Baxter Equation with Open Problems." Axioms 1, no. 1: 33-37.