First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $\mathbb{F}{S}_{n}$ of the symmetric group ${S}_{n}$ over a field $\mathbb{F}$ of characteristic 0 (or $p>n$). The goal is to obtain a constructive version of the isomorphism $\psi :{\u2a01}_{\lambda}{M}_{{d}_{\lambda}}\left(\mathbb{F}\right)\u27f6\mathbb{F}{S}_{n}$ where $\lambda $ is a partition of $n$ and ${d}_{\lambda}$ counts the standard tableaux of shape $\lambda $....

We first discuss the construction by Pérez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra...

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...

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