#
Distributive laws between the Three Graces

##
Murray Bremner and Martin Markl

By the Three Graces we refer, following J.-L. Loday, to the algebraic
operads Ass, Com, and Lie, each generated by a single binary
operation; algebras over these operads are respectively associative,
commutative associative, and Lie. We classify all distributive laws
(in the categorical sense of Beck) between these three operads. Some
of our results depend on the computer algebra system Maple,
especially its packages `LinearAlgebra`

and
`Groebner`

.

Keywords:
Algebraic operads, distributive laws, Koszul duality, associative
algebras, commutative associative algebras, Lie algebras, Poisson
algebras, linear algebra over polynomial rings, Gr\"obner bases for
polynomial ideals, computer algebra

2010 MSC:
Primary 18D50. Secondary 13P10, 16R10, 16S10, 16S37, 16W10, 17B60, 17B63,
18-04, 68W30.

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 41, pp 1317-1342.

Published 2019-12-05.

http://www.tac.mta.ca/tac/volumes/34/41/34-41.pdf

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