Properties of the monad for the LieAlgebra adjunction

Question

The Lie Algebra is indeed an algebra for a monad. We see some background here. As such, the category of Lie Algebras has an adjunction into Set. This means that there is a monad, $(L,\mu, \eta)$ on Set which is generated by the adjunction. Can we see a concrete example of:

1. $L(S)$, for $S \in Set$
2. $\mu : L \cdot L \rightarrow L$
3. $\eta : 1_{Set} \rightarrow L$

Does the adjunction also generate a comonad on Set? If so, can we see concrete examples of the co-multiplication and co-unit?

I have a pet physical theory that states that the category of physical data for a system should admit a (co)monad structure and the theory itself is a factorization of the (co)monad. I don't have any proof of the general theorem. In this case, the category of the data for the theory is Set. LieAlg is a theory of physics. Set admits bimonads, and I am expecting the functor $L$ to admit a bimonad structure.

Answer 1

1. $L(S)$ is the underlying set of the free Lie algebra over set $S$:
   We take formal linear combinations of formal Lie brackets of elements of $S$, then we quotient out the Lie algebra identities.
2. Well, $LL(S)$ is then the free Lie algebra built on $L(S)$, we can express its elements by putting in parentheses in the formal expressions, e.g. with $a,b,c\in S$, the following formal expression is in $LL(S)$:
   $[(2[a,b]-[a,c]),\ (3a)]\ $ as a simple Lie bracket of two elements of $L(S)$.
   Then $\mu$ just opens up the parentheses.
3. Guess what, $\eta$ will send $a\in S\,$ to $\,a\in L(S)$.
4. Yes, by duality, every adjunction also determines a comonad on the other category.
   Note that here the free functor $F:Set\to LieAlg\,$ is left adjoint to the underlying set functor $U:LieAlg\to Set$, and that we have $L=UF$.
   Then the comonad structure will be induced on the functor $FU:LieAlg\to LieAlg$.
   Based on this, you can describe its counit and comultiplication.

Note also, as Derek commented, this is a very general pattern, applicable to all kinds of algebraic structures defined by equations, and even beyond..