Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures

These are the lecture notes from my short course of the same title at the CIMPA Research School on Associative and Nonassociative Algebras and Dialgebras: Theory and Algorithms - In Honour of Jean-Louis Loday (1946-2012), held at CIMAT, Guanajuato, Mexico, February 17 to March 2, 2013. The underlying motivation is to apply the theory of noncommutative Grobner bases in free associative algebras to the construction of universal associative envelopes for nonassociative structures defined by multilinear operations. Trilinear operations were classified by the author and Peresi in 2007. In her Ph.D. thesis of 2012, Elgendy studied the universal associative envelopes of nonassociative triple systems obtained by applying these trilinear operations to the 2-dimensional simple associative triple system. In these notes I use computer algebra to extend some aspects of her work to the 4-dimensional and 6-dimensional simple associative triple systems.