## Distributive laws between the Three Graces

### Martin Markl – 20 December 2017

(joint work with Murray Bremner)

Experience teaches us that the most common classes of algebras are the Three Graces - associative, commutative associative, and Lie - together with other that combine these in a specific way. The most prominent example of a combined structure are Poisson algebras which are combinations of Lie and commutative associative algebras by means of a quadratic homogeneous distributive law.

Our aim was to investigate whether the commonly known combinations of the Three Graces are the only possible ones via such a distributive law. The answer turned out, rather surprisingly, no.

In my talk I will present results of our on-going work including examples of "exotic" structures combining the Three Graces.

Our research was facilitated by advances in computer-assisted mathematics, and in particular the computer algebra system Maple worksheets written by the first author expressly for this project extended hand calculations of the second author dating from some 20 years ago.

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