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Is there a cohomology theory for magmas? Or cohomology theories for any class of non-associative algebras (other than Lie and maybe Jordan)?

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    $\begingroup$ Isn't there a cohomology for algebras over any operad? $\endgroup$
    – Fernando Muro
    Commented Nov 10, 2013 at 20:51

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Fernando is right, magmatic algebras are algebras over the magma operad. This operad is free, it is generated by a bilinear operation.

We can define André-Quillen cohomology of magmatic algebras: the situation is very nice because the magma operad is Koszul.

If you like homotopical algebra you can produce a closed model category of simplicial or differential graded magmatic algebras (of $R$-modules) and produce André-Quillen homology as the derived functor of indecomposable elements and so forth and so on...

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    $\begingroup$ Thnaks to all - I should have thought of that - must be getting rusty $\endgroup$
    – Jim Stasheff
    Commented Nov 17, 2013 at 21:25
  • $\begingroup$ but Andre-Quillen means going simplicial - because R could be over charateristic p? $\endgroup$
    – Jim Stasheff
    Commented Nov 17, 2013 at 21:26
  • $\begingroup$ There is no problem with the characteristic because the operad is free, you can work in the dg setting. $\endgroup$
    – David C
    Commented Nov 18, 2013 at 7:17
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Eilenberg S., Mac Lane S., Algebraic cohomology groups and loops, Duke Math. J. 14 (1947), 435-463.

Johnson K.W., Leedham-Green C.R., Loop cohomology, Czech. Math. J. 40 (1990), 182-194.

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  • $\begingroup$ They asked for magmas, not necc. quasigroups... $\endgroup$
    – Daniel Donnelly
    Commented Feb 9, 2022 at 0:06

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