Hopf algebra structure on the universal enveloping algebra of a Leibniz algebra?

Question

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 $End(M) \cong M \otimes M^*$ 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 $L$: $$ (x,m) \mapsto [x,m] \textrm{ and } (m,x) \mapsto [m,x], \forall m \in M \textrm{ and } x \in L $$

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 $T(L\oplus L)$ of $L \oplus L$. Let $l_x = (x,0)$ and $r_x = (0,x) \in L \oplus L$. 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): $$ r_{[x,y]} - r_y r_x - l_x r_y \\ l_x r_y - r_y l_x - r_{[x,y]} \\ l_x l_y - l_y l_x - l_{[x,y]} $$ for all $x, y \in L$, and where I have omitted the $\otimes$'s.

Notice that adding the first two, we can substitute one of them by the simpler $$ r_y (l_x + r_x) = 0 $$

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

Answer 1

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".

Answer 2

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.

Answer 3

Let $L$ be the abelian Leibniz algebra of dimension $n$. Then $U(L)$ is the polinomial algebra $k[X_1, \cdots, X_n, Y_1, \cdots, Y_n]$ subject to the relations: $$ X_i X_j = X_j X_i, \quad X_j Y_i = Y_i X_j = - Y_j Y_i $$ for all $i$, $j$. Then $U(L)$ is not a Hopf algebra (whith the usual structures, i.e. $X_i$ and $Y_i$ are primitives) since the above two-sided ideal is not a coideal.