Generalized Operator Yang-Baxter Equations, Integrable ODEs and Nonassociative Algebras

Abstract

A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter.

A unified algebraic approach to the classical Yang–Baxter equation

“…The formula (3.17) was also given in [31] and a similar construction for Novikov algebras (leftsymmetric algebras with commutative right multiplication operators) was given with r satisfying some additional conditions in [40]. We would like to point out that the above construction cannot get all left-symmetric algebras.…”

“…Although some scattered results are known in certain references (cf. [9,[28][29][30][31], etc. ), we give a systematic study on the relations between left-symmetric algebras and CYBE in this paper.…”

Improved q-exponential and q-trigonometric functions

“…The quantum calculus (q-calculus) is an old, classical branch of mathematics, which can be traced back to Euler and Gauss [1,2] with important contributions of Jackson a century ago [3,4]. In recent years there are many new developments and applications of the q-calculus in mathematical physics, especially concerning special functions [5][6][7] and quantum mechanics [8][9][10][11][12][13][14].…”

On the degenerations of solvable Leibniz algebras

“…Largely because of their importance to string theory, quantum field theory and other branches of fundamental research in mathematical physics, noncommutative analogs of many classical constructions have received much attention in the past few years [8,10].…”