I hate to ask such a naive question, but here goes. Suppose $A$ and $B$ are rational Lie algebras, i.e. rational vector spaces together with a bracket. Then, $A\otimes\_{\mathbb{Q}} B$ is a rational vector space. Can it be endowed with a Lie bracket in such a way that the category of rational Lie algebras is a monoidal category? I came up with this question while reading Quillen's Rational Homotopy Theory and Schwede-Shipley's "Equivalences of Monoidal Model Categories", which mentions Quillen's work, but stops short of claiming any monoidal properties.



Is the category of rational Lie algebras monoidal?
Andrew Salch wrote me a nice email about this question, and with his permission, I'm turning it into an answer. Any mistakes are my own. I've made this answer CW, so I don't get any points for Andrew's work. The following is Andrew's email, lightly edited.


"The Milnor-Moore theorem tells you that, if $k$ is a field of characteristic zero, then the category of Lie algebras over $k$ is equivalent to the category of co-commutative primitively generated Hopf algebras over $k$: from Lie algebras to Hopf algebras, you have the universal enveloping algebra functor $U$, and from a Hopf algebra $A$ you can take its Lie algebra $PA$ of primitives, and if the ground field has characteristic zero, then $P$ and $U$ give you the Milnor-Moore equivalence of categories.


You have a nice extension theory in both settings: for Lie algebras it's very classical, and extension theory for Hopf algebras was developed in a series of papers by Bill Singer. The functor $U$ sends an extension of Lie algebras to an extension of Hopf algebras and the functor $P$ sends an extension (in Singer's sense) of co-commutative primitively generated Hopf algebras to an extension of Lie algebras. (Actually, I don't remember ever seeing that last fact in Singer's papers, but I think it's part of the motivation for what he was doing. I guess you better not take my word for it, and actually work out a proof if you're going to use this.)


Given two Lie algebras $g$ and $h$, you can form the trivial extension of $g$ by $h$, which is just the Cartesian product $g \times h$, like you'd expect. The trivial extension of the Hopf algebra $Ug$ by the Hopf algebra $Uh$ is just the tensor product $Ug \otimes\_k Uh$.


So the ordinary Cartesian product gives you a monoidal structure on the category of Lie algebras over $k$, and on applying $U$ to translate all the Lie algebras into Hopf algebras, you get the tensor product (over $k$) of the Hopf algebras. This isn't so far from what you were asking about--it's not a tensor product of Lie algebras, but it's a monoidal product on Lie algebras that induces the tensor product on their universal enveloping algebras--so maybe it's useful."


