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\begin{document}

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\section*{primitive element}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{primitive_element_in_a_coalgebra}{Primitive element in a coalgebra}\dotfill \pageref*{primitive_element_in_a_coalgebra} \linebreak
\noindent\hyperlink{primitive_element_in_a_comodule}{Primitive element in a comodule}\dotfill \pageref*{primitive_element_in_a_comodule} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{primitives_in_a_hopf_algebra}{Primitives in a Hopf algebra}\dotfill \pageref*{primitives_in_a_hopf_algebra} \linebreak


(This article is about primitive elements in coalgebra theory, not about primitive elements for finite [[field extensions]].)

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Working over a [[commutative ring]] $R$, recall that a \emph{unital} or \emph{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 $1$.

\hypertarget{primitive_element_in_a_coalgebra}{}\subsubsection*{{Primitive element in a coalgebra}}\label{primitive_element_in_a_coalgebra}

An element $x$ in a [[coalgebra]] $C$ is \textbf{primitive} if $\Delta(x) = 1\otimes x + x\otimes 1$. This condition implies $\epsilon(x) = 0$.

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

\hypertarget{primitive_element_in_a_comodule}{}\subsubsection*{{Primitive element in a comodule}}\label{primitive_element_in_a_comodule}

Let $N$ be a [[comodule]] over $C$, with co-action map $\Psi \colon N \longrightarrow C \otimes_R N$. Then an element $n \in N$ is called a \textbf{primitive element} or \textbf{[[coinvariant]]} if $\Psi(n) = 1 \otimes n$.

The $R$-module $prim(N)$ of primitive elements of $N$ is [[natural isomorphism|naturally]] to

\begin{enumerate}%
\item the module of comodule homomorphisms out of $R$ into $N$

\begin{displaymath}
prim(N) \simeq Hom_{C}(R,N)
  \,.
\end{displaymath}

\item the [[cotensor product]] of $N$ with $A$

\begin{displaymath}
prim(N) \simeq A \Box_R N
  \,.
\end{displaymath}


\end{enumerate}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{primitives_in_a_hopf_algebra}{}\subsubsection*{{Primitives in a Hopf algebra}}\label{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

\begin{displaymath}
P: HopfAlg \to LieAlg
\end{displaymath}
(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]]:

\begin{displaymath}
U(L) = T(L)/I
\end{displaymath}
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:

\begin{prop}
\label{}\hypertarget{}{}
The functor $U: LieAlg \to BiAlg$ is [[left adjoint]] to the functor $P: BiAlg \to LieAlg$.

\end{prop}
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:

\begin{prop}
\label{}\hypertarget{}{}
For $R = k$ a [[field]] of [[characteristic zero]], the [[adjunction unit|unit]] $L \to P U(L)$ is an [[isomorphism]].

\end{prop}
An immediate consequence is that for such ground fields $k$, the functor $U: LieAlg \to BiAlg$ is [[fully faithful functor|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.

(More needs to be added.)

[[!redirects primitive element in a coalgebra]]

[[!redirects primitive elements]]



\end{document}