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

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".

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 Not only enveloping algebra can be contructed there, but also the appropriate internal Hopf analogues of GL(n) (unpublished work of mine) and the internal Weyl algebras (work of a student of mine).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.

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.