# nLab primitive element

### Context

#### Algebra

higher algebra

universal algebra

## Idea

Working over a commutative ring $R$, recall that a unital or coaugmented coalgebra is an $R$-coalgebra $(C, \Delta: C \to C \otimes_R C, \epsilon: C \to R)$ equipped with a coalgebra map $u: R \to C$. Abusing notation, denote $u(1) \in C$ by $1$; then an element $x\in C$ is primitive if $\Delta(x) = 1\otimes x + x\otimes 1$. This condition implies $\epsilon(x) = 0$.

This notion generalizes straightforwardly to unital corings over $R$.

## Primitives in a Hopf algebra

A straightforward calculation shows that the module of primitive elements in a Hopf algebra $H$ (or even in a bialgebra $H$) is a Lie subalgebra of the underlying Lie algebra of $H$ (whose bracket is the algebra commutator). Thus, taking primitive elements yields a functor

$P: HopfAlg \to LieAlg$

(and of course we have more generally a functor $P: BiAlg \to LieAlg$ which is an extension along the full inclusion $HopfAlg \to BiAlg$).

For a Lie algebra $L$, let $U(L)$ be its universal enveloping algebra:

$U(L) = T(L)/I$

where $I$ is the two-sided ideal generated by elements of the form $x y - y x - [x, y]$ where $x, y \in L$. This carries a bialgebra structure whose comultiplication $\delta: U(L) \to U(L) \otimes U(L)$ is uniquely determined by the rule $\delta(x) = 1 \otimes x + x \otimes 1$ for $x \in L$. Since this says $x \in L$ is primitive, the counit $\epsilon: U(L) \to R$ is forced to be the algebra map such $\epsilon(x) = 0$ for all $x \in L$, and also the Hopf antipode is uniquely determined: $\sigma(x) = -x$ for $x \in L$.

The following proposition is entirely straightfoward:

###### Proposition

The functor $U: LieAlg \to BiAlg$ is left adjoint to the functor $P: BiAlg \to LieAlg$.

This result is essentially tautologous and holds for any commutative ring of arbitrary characteristic. (This despite the fact that the $U(L)$ as defined above is not as well-behaved in nonzero characteristic as one might like; e.g. the PBW theorem fails.) More information on this adjunction may require more restrictive hypotheses:

###### Proposition

For $R = k$ a field of characteristic zero, the unit $L \to P U(L)$ is an isomorphism.

An immediate consequence is that for such ground fields $k$, the functor $U: LieAlg \to BiAlg$ is fully faithful. Of course, $U$ lands in the full subcategory of cocommutative Hopf algebras, which is exactly the category of group objects in the cartesian monoidal category of cocommutative coalgebras.

The Milnor-Moore theorem gives further information: for Hopf algebras over a field of characteristic zero, the counit $U P(H) \to H$ is a monomorphism, and an isomorphism in case $H$ satisfies a suitable conilpotency condition.