Algebras, Dialgebras, and Polynomial Identities

2010 Mathematics Subject Classification: Primary 17A30. Secondary 16R10, 17-08, 17A32,This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc − bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raúl Felipe, Luiz A. Peresi, and Juana Sánchez-Ortega.∗ This paper is an expanded version of the lecture notes from the author’s talk at the Second International Workshop on Polynomial Identities, 2–6 September 2011, which took place at the Atlantic Algebra Centre, Memorial University, St. John’s, Newfoundland, Canada