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Many additions and changes to Leibniz algebra. The purpose is to outline that the (co)homology and abelian and even nonabelian extensions of Leibniz algebras follow the same pattern as Lie algebras. One of the historical motivations was that the Lie algebra homology of matrices which lead Tsygan to the discovery of the (the parallel discovery by Connes was just a stroke of genius without an apparent calculational need) cyclic homology. Now, if one does the Leibniz homology instead then one is supposedly lead the same way toward the Leibniz homology (for me there are other motivations for Leibniz algebras, including the business of double derivations relevant for the study of integrable systems).
Matija and I have a proposal how to proceed toward candidates for Leibniz groups, that is an integration theory. But the proposal is going indirectly through an algebraic geometry of Lie algebras in Loday-Pirashvili category. Maybe Urs will come up with another path if it drags his interest.
So with a notion of Chevalley-Eilenberg algebra for Leinniz-algebras, one should do the following:
Dmitry Roytenberg has that notion of hemistrict Lie 2-algebra modeled on Leibniz algebras. There should be a notion of “hemistrict” $L_\infty$-algebra.
Ordinary $L_\infty$-algebras (of finite type) are, as we know, dually the same as semifree graded-commutative dgas, their CE-algebras.
So a natural question is: what characterizes CE-algebras of “hemistrict” $L_\infty$-algebras, Leibniz type.
It seems the answer should be: dg-algebras whose underlying graded-commutative algebra is not quite free, but has a nontrivial binary piece. Would be good to relate this to whatever the people cited in the Leinbiz algebra enty tought about.
I am not sure now any more. I mean I did not write the answer to your main question as I am not sure. I mean there is the CE chain complex anc CE cochain complex in Leibniz algebra case, which is evident from the formulas. Mainly we have tensor product instead of exterior power. Now, I am not sure about the algebra structure which you enthusiatically asked about. Now if my noise about Koszul operad makes any sense than it should be a algebra over a dual Leibniz operad, understood as a dg-operad, that Koszul dual is studied in the Loday’s world. But need to get closer to the subject, I have spent most of my time in Loday-Pirashvili category instead of the original category of Leibniz algebras where I am not yet comfortable as in the LP category.
The dual Leibniz algebra (in $Vec$) is a nonassociative algebra with a product satisfying
$(xy)z = x(zy + yz).$Could you enter the link (I will go for a bus in few minutes) to the appropriate Roytenberg’s paper somewhere in the entry Leibniz algebra ? I have added a section on dual Leibniz algebra.
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