For which category (if any) are Lie algebras the algebras of a monad?

Question

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after I realized that I heard similar things were true for Lie algebras (i.e. that they simplify calculations for Lie groups), as well as for the matrix representations of groups.

I know that (free) monoids are the algebra for the monad on $Set$ which given a set $X$ returns the set $X^*$ of words in elements of $X$. Since Lie algebras are monoids, I figured that they might also be the algebra of some monad. In other words, I would like to know if it is possible to make the above analogy rigorous.

Do Lie algebras (over $\mathbb{R}$ or $\mathbb{C}$) form the category of algebras for any monad $T$?

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What I tried to do to find an answer (these efforts were not successful)

Ideally I would probably want to find some adjunction that generates an appropriate monad, but since I am new at this, I decided to Google instead. Here is the most relevant result I found, from page 133, edited by Paul Gregory Goerss and Stewart Priddy, of the collection of papers:

"Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory : an International Conference on Algebraic Topology, March 24-28, 2002, Northwestern University"

Specifically the result I am looking at, Proposition 1.2.16, says the following:

We assume that the ground ring $\mathbb{K}$ is a field of characteristic $p >0$. Recall that an algebra over the monad $\mathcal{S(L)}$ is a Lie algebra $\mathcal{G}$ together with a Lie bracket $[-,-]:\mathcal{G}\otimes\mathcal{G} \to \mathcal{G}$ such that $[x,y]=-[y,x]$, for all $x,y \in \mathcal{G}$. An algebra over the monad $\Lambda(\mathcal{L})$ is a Lie algebra $\mathcal{G}$ together with a Lie bracket $[-,-]: \mathcal{G} \otimes \mathcal{G} \to \mathcal{G}$ such that $[x,x]=0$. An algebra over the monad $\Gamma(\mathcal{L})$ is a restricted Lie algebra $\mathcal{G}$.

Since I don't really understand the article, I am not quite sure what all of the notation means, except that $S$ is a functor which takes operads to monads (whatever operads are). Also this might not be a result I am interested in, since it seems to only hold for fields of characteristic $p>0$, i.e. not for Lie algebras over the real or complex numbers.

Most of the other Google search results I found for "Lie algebra monad" also did not seem to answer my question. I thought this question might help: Do adjoint functors really define monads?, but I'm not sure how to interpret it -- was the result that Lie algebras cannot be/are not the algebra of any monad?

Answer 1

Yes, Lie algebras (over any commutative ring) are the algebras of a monad over $\text{Set}$. This is true more generally for any kind of algebraic object defined by some operations satisfying some equational identities. The monad is built out of the free object functor. In other words, the forgetful functor

$$\text{LieAlg} \to \text{Set}$$

has a left adjoint, sending a set to the free Lie algebra on this set, and this adjunction is monadic.

There are various gadgets that can be used to describe various kinds of algebraic objects which, in particular, can be turned into monads; operads are an example (and Lie algebras are also the algebras of an operad), and Lawvere theories are another (and Lie algebras are also the models of a Lawvere theory).

Answer 2

In addition to Qiaochu Yuan's answer, there is the following paper by Ardizzoni and Menini that describes how the category of Lie algebras over a field of characteristic zero is equivalent to a category of algebras over a monad different than the one mentioned above, see https://arxiv.org/pdf/1401.2037 (the authors actually prove a more general result for braided Lie algebras -with underlying symmetric braiding- in suitable monoidal categories).

In characteristic p, a similar result holds for restricted Lie algebras: this is proven in the article https://arxiv.org/abs/1510.02587. The introduction to the latter paper gives an overview of the similarities and of the -slight- differences between two approaches (in characteristic zero and in characteristic p).