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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.
added this pointer on relation to the embedding tensor and tensor hierarchies in gauged supergravity:
I have touched the layout of the entry.
Added this as an earlier reference where the concept of Leibniz algebras appears (just not by that name):
added pointer to this review:
added more references relating Leibniz algebras to dg-Lie algebras and to tensor hierarchies:
Sylvain Lavau, Jakob Palmkvist, Infinity-enhancing of Leibniz algebras (arXiv:1907.05752)
Sylvain Lavau, Jim Stasheff, $L_\infty$-algebra extensions of Leibniz algebras (arXiv:2003.07838)
and I have added pointer to
which, I gather, is the actual reason that the concept is attributed to Loday alone, instead of to Loday & Pirashvili as the usual pointer to Loday & Pirashvili 93 would suggest.
I have completed the previously missing last words in the title of the reference
and added the comment that this is about realizing Leibniz algebras as Lie algebra objects
(both somewhat crucial for knowing what that reference has to do on this page…)
I have turned the previous first section “Motivation” into an “Idea”-section that starts out with stating the simple idea:
A Leibniz algebra is like a Lie algebra, but without the condition that the product, often still written as a bracket $[-,-]$, is skew-symmetric. The Jacobi identity however is retained as a condition in its form as the derivation-property of the product over itself. In view of the analogous product law of differentiation (also a derivation-property) attributed to Gottfried Leibniz, this is then called the Leibniz identity which gives Leibniz algebras their modern name (Loday 93, Loday-Pirashvili 93) even though the concept itself is older (Blokh 65).
After that I rephrased the content of the previous “Motivation”-section as follows:
Leibniz algebras were motivated in Cuvier 91, Loday-Pirashvili 93 as generalizing the relation between Lie algebra cohomology and cyclic homology (Loday-Quillen 84) to one between Leibniz cohomology and Hochschild homology: Where the nilpotency of the differential in the Chevalley-Eilenberg algebras that compute Lie algebra cohomology is equivalent to the Jacobi identity in the corresponding Lie algebra, Leibniz cohomology is defined on non-skew symmetric dg-algebras where now it is the generalization of the Jacobi identity in form of the Leibniz rule (eq:LeibnizRule) which still guarantees the nilpotency of the differential.
and then I added this line afterwards:
More recently, Leibniz algebras have been argued to clarify the nature of the embedding tensor and the resulting tensor hierarchies in gauged supergravity (Lavau 17).
I have made explicit the cross-links with Lie algebra object and added pointer to the reference.
The respective paragraph now reads this way:
As internal Lie algebras
Leibniz algebras are equivalenlty Lie algebra objects in the Loday–Pirashvili tensor category of linear maps with (exotic) “infinitesimal tensor product” (Loday-Pirashvili 98)
Question:
If there is this article from 1991
how is it the concept is attributed to Loday’s article(s) from 1993, two years later?
(I haven’t managed to find the actual document of Cuvier’s article. If anyone knows of an electronic copy, let’s add a pointer to the entry/)
Added an Examples-section, and added there statement and proof (here) that a Lie module equipped with the “embedding tensor” becomes a Leibniz algebra
Thanks. I have added cross-link with Lie’s third theorem.
But the definition of Lie racks etc. should really be in a separate entry of that name.
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