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A Leibniz algebra L may be thought of as a noncommutative generalisation of a Lie algebra. One drops the requirement that the bracket be alternating and substitutes the Jacobi identity for the Leibniz identity

[x,[y,z]] = [[x,y],z] + [y,[x,z]]   for all x,y,z in L

Remark. What is being defined above is a left Leibniz algebra. There is also the notion of a right Leibniz algebra where the Leibniz identity now says that it is the right multiplication [-,x] which is a derivation, instead of the left multiplication [x,-] as in the equation above.

Since the bracket is no longer alternating, left- and right-multiplications are no longer related simply by a sign as in the case of Lie algebras, and this means that representations in general admit two actions of L: one on the left and one on the right, satisfying some identities which are explained, for example, in a paper of Loday (who seems to have introduced the concept) and Pirashvili (Math. Ann. 296 (1993) pp. 139–158). In that paper they also define the universal enveloping algebra U(L) of a Leibniz algebra L and show that the there is a categorical equivalence between representations of L and left modules over U(L). (Right modules of U(L) correspond to the notion of corepresentation.) Also notice that in that paper they work with right Leibniz algebras, so everything there is the mirror image to what I'm saying here. One difference with the case of a Lie algebra is that U(L) is a quotient of the tensor algebra of L\oplus L, to take into account the two actions of L on a representation.

My question is whether there is a Hopf algebra structure on U(L).

My interest in this question is that in some recent work on the deformation theory of n-Leibniz algebras, I studied cohomology with values in a representation M of a Leibniz algebra L and also with values on End(M). The action of L on End(M) follows from the formalism and one can check that it is indeed a representation, but it does not follow in any obvious way from the action of L on M. In Lie theory, we are used to the fact that if M is a (finite-dimensional) representation of a Lie algebra G, then we have an isomorphism End(M) = M \otimes M^* as representations of G, where to determine the action of G on M \otimes M^* we use the Hopf algebra structure on U(G). Hence my question.

EDIT: I am adding more details about U(L), as requested in the comment below by Theo Johnson-Freyd.

To motivate it, let us first define a representation M of a (left) Leibniz algebra L to be a vector space admitting two actions of L:

  (x,m) \mapsto [x,m]  and  (m,x) \mapsto [m,x] for all m in M and x in L

satisfying three identities, which are obtained from the Leibniz identity above by replacing x,y,z in turn by m; that is,

           [m,[x,y]] = [[m,x],y] + [x,[m,y]]

           [x,[m,y]] = [[x,m],y] + [m,[x,y]]

           [x,[y,m]] = [[x,y],m] + [y,[x,m]]

To define U(L) we start with the tensor algebra T(L+L) of L \oplus L. Let lx = (x,0) and rx = (0,x) in L \oplus L. Then U(L) is the quotient of T(L+L) by the two-sided ideal generated by the following elements (which can be read off from the conditions defining a representation):

r[x,y] - ry rx - lx ry

lx ry - ry lx - r[x,y]

lx ly - ly lx - l[x,y]

for all x and y in L, and where I have omitted the \otimes's.

Notice that adding the first two, we can substitute one of them by the simpler

ry (lx + rx) = 0

I don't know what the coalgebra structure is, though. That's part of the original question.

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I'm curious: where do Leibniz algebras show up in mathematics? –  S. Carnahan Oct 29 '09 at 19:37
They were introduced by Loday in his book on Cyclic Cohomology, but that's not where I have met them. The most recent place I have met them is in the study of n-Lie algebras, which was prompted by some recent developments in the gauge/gravity correspondence. Essentially, metric n-Lie algebras (and more generally metric n-Leibniz algebras) are used in formulating some three-dimensional superconformal field theories with desirable properties. This started with work of Bagger and Lambert (check the hep-th arXiv if you are interested) in 2006/7, with a minor explosion of activity in 2008/9. –  José Figueroa-O'Farrill Oct 30 '09 at 2:30
Continuing with the above comment... n-Lie algebras are n-ary generalizations of Lie algebras, with which they agree for n=2. They appeared originally for n=3 in work of Nambu on generalised hamiltonian dynamics. There's a canonical metric 3-Lie algebra (albeit infinite-dimensional) attached to every oriented compact 3-manifold, for example. Underlying every n-Lie (or more generally n-Leibniz) algebra is a Leibniz algebra structure on its n-th tensor power. If you forgive pointing to a paper of mine, you can read about this in arxiv.org/abs/0903.4871 and references therein. –  José Figueroa-O'Farrill Oct 30 '09 at 2:35
Would it take too much space for you to describe/define the space U(L) and its algebra and coalgebra structures? –  Theo Johnson-Freyd Oct 31 '09 at 23:44
More space than allowed in a comment, but I will add it to the main body of the question. I hope that's alright. –  José Figueroa-O'Farrill Nov 1 '09 at 6:16

2 Answers 2

up vote 6 down vote accepted

Do you know the paper of Loday and Pirashvili? They discuss what, in their opinion, should replace the notion of a Hopf algebra in Leibniz setting, "Hopf algebras in the category of linear maps".

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Thank you for this reference. Although it might take some time to digest it, it seems that this might be the answer to the question I <em>should</em> have asked. –  José Figueroa-O'Farrill Nov 4 '09 at 3:16
It is perhaps worth pointing out that the link currently in the answer is no longer valid. The full reference info, for anyone who wants it, is: Jean-Louis Loday, Teimuraz Pirashvili, The tensor category of linear maps, Georg. Math. J. vol. 5, n.3 (1998) 263–276. –  zibadawa timmy Dec 24 '14 at 21:16
@zibadawatimmy: thank you! I fixed the link, it did change apparently. –  Vladimir Dotsenko Dec 25 '14 at 10:37

There is more than one version of what should be the universal enveloping algebra of a Leibniz algebra. One version is a usual associative algebra, and another (in my opinion better one) is an internal Hopf algebra in the Loday-Pirashvili tensor category (article mentioned above and some follow up articles). Both have the same categories of modules. The internal geometry in LP has a better chance at natural description of various issues. Not only enveloping algebra can be contructed there, but also the appropriate internal Hopf analogues of GL(n) (unpublished work of mine) and the internal Weyl algebras (work of a student of mine). I believe in a certain program of obtaining a theory of Leibniz groups along these lines.

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