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\begin{document}

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\section*{differential graded Hopf algebra}

\begin{quote}%
under construction


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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{graded_hopf_algebras}{Graded Hopf algebras}\dotfill \pageref*{graded_hopf_algebras} \linebreak
\noindent\hyperlink{derivations_on_hopf_algebras}{Derivations on Hopf algebras}\dotfill \pageref*{derivations_on_hopf_algebras} \linebreak
\noindent\hyperlink{shuffle_product_on_}{Shuffle product on $T(V)$}\dotfill \pageref*{shuffle_product_on_} \linebreak
\noindent\hyperlink{gradedcommutative_hopf_algebra_structure_on_}{Graded-commutative Hopf algebra structure on $T(V)$}\dotfill \pageref*{gradedcommutative_hopf_algebra_structure_on_} \linebreak
\noindent\hyperlink{gradedcocommutative_hopf_algebra_structure_on_}{Graded-cocommutative Hopf algebra structure on $T(V)$}\dotfill \pageref*{gradedcocommutative_hopf_algebra_structure_on_} \linebreak
\noindent\hyperlink{the_enveloping_algebra_of_a_lie_algebra_}{The enveloping algebra of a Lie algebra, $U(L)$.}\dotfill \pageref*{the_enveloping_algebra_of_a_lie_algebra_} \linebreak
\noindent\hyperlink{the_lie_algebra_of_primitives_}{The Lie algebra of primitives, $P$}\dotfill \pageref*{the_lie_algebra_of_primitives_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{graded_hopf_algebras}{}\subsubsection*{{Graded Hopf algebras}}\label{graded_hopf_algebras}

A \emph{$\mathbb{Z}$-graded Hopf algebra} (pre-gha) is a $\mathbb{Z}$-graded vector space, which, for that grading, is both a $\mathbb{Z}$-graded algebra, $(A,\mu)$, with unity, $\eta : K \to A$, and a $\mathbb{Z}$-graded coalgebra $(A, \Delta, \varepsilon)$ such that:

\begin{itemize}%
\item $\eta : K \to A$ is a morphism of $\mathbb{Z}$-graded coalgebras;
\item $\varepsilon  : A \to K$ is a morphism of $\mathbb{Z}$-graded algebras;
\item $\mu : A \otimes A \to A$ is a morphism of $\mathbb{Z}$-graded coalgebras

\end{itemize}
\textbf{Remark}

We can replace the third condition by:

\begin{itemize}%
\item $\Delta : A \to A \otimes A$ is a morphism of $\mathbb{Z}$-graded algebras.

\end{itemize}
Of course, wherever possible, we will abbreviate $(A,\Delta,\mu,\epsilon,\eta)$ to $A$.

A [[homomorphism]] of pre-ghas is a linear map of degree zero compatible with both the algebra and coalgebra structures. We may write $pre GHA$ for the resulting category.

If $A$ and $A'$ are two pre-ghas, $A\otimes A'$ is a pre-gha for the algebra and coalgebra structures already defined.

\hypertarget{derivations_on_hopf_algebras}{}\subsubsection*{{Derivations on Hopf algebras}}\label{derivations_on_hopf_algebras}

Let $A$ be a pre-gha. A \emph{Hopf algebra derivation} of $A$ of degree $p\in \mathbb{Z}$ is a linear mapping $\theta \in Hom_p(A,A)$, defining both an algebra and a coalgebra derivation.

A differential $\partial$ of pre-ghas is a Hopf algebra derivation of degree -1 such that $\partial\circ \partial = 0$. The pair $(A,\partial)$ is called a \emph{differential $\mathbb{Z}$-graded Hopf algebra} (pre-dgha). Its homology $H(A,\partial)$ is also a pre-gha. A morphism of pre-dghas is a morphism, at the same time, of pre-ghas and pre-dgvs. This gives a category $pre DGHA$.

A pre-gha $(A,\Delta,\mu,\epsilon,\eta)$ is \emph{commutative} if $(A,\mu)$ is commutative and is \emph{cocommutative} if $(A,\Delta,\varepsilon)$ is cocommutative.

This gives categories $pre CDGHA$ and $pre CoDGHA$ respectively.

A \emph{cocommutative} (resp. \emph{commutative}) \emph{dgha} is an object of $pre CoDGHA$ (resp. $pre CDGHA$, which has a lower (resp. upper) grading.

A cocommutative (resp. commutative) dgha $A$ is \emph{$n$-connected} if $\bar{A}_p = 0$ (resp $\bar{A}^p = 0$) for $p\leq n$.

\hypertarget{shuffle_product_on_}{}\subsubsection*{{Shuffle product on $T(V)$}}\label{shuffle_product_on_}

Let $V$ be a pre-gvs. The gvs $T(V)$ is a pre-cga for the [[shuffle]] product defined by

\begin{displaymath}
(v_1\otimes \ldots \otimes v_p)\star (v_{p+1}\otimes\ldots \otimes v_n) = \sum_\sigma \varepsilon(\sigma)v_{\sigma^{-1}(1)}\otimes\ldots \otimes v_{\sigma^{-1}(n)},
\end{displaymath}
where the sum is over all $(p,n-p)$ [[shuffles]], $\varepsilon(\sigma)$ is the Koszul sign of $\sigma$ and the elements $v_i$ of $V$ are all homogeneous.

\hypertarget{gradedcommutative_hopf_algebra_structure_on_}{}\subsubsection*{{Graded-commutative Hopf algebra structure on $T(V)$}}\label{gradedcommutative_hopf_algebra_structure_on_}

The underlying algebra structure is $T(V)$ with the shuffle product. The reduced diagonal is given by

\begin{displaymath}
\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n).
\end{displaymath}
\hypertarget{gradedcocommutative_hopf_algebra_structure_on_}{}\subsubsection*{{Graded-cocommutative Hopf algebra structure on $T(V)$}}\label{gradedcocommutative_hopf_algebra_structure_on_}

The underlying algebra structure this time is $T(V)$ with the usual product

\begin{displaymath}
(v_1\otimes \ldots \otimes v_p)\cdot(v_{p+1}\otimes \ldots \otimes v_n) = v_1\otimes \ldots \otimes v_p\otimes v_{p+1}\otimes \ldots \otimes v_n,
\end{displaymath}
but with the reduced diagonal given by

\begin{displaymath}
\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1}\sum_\sigma \varepsilon(\sigma) (v_{\sigma(1)}\otimes \ldots \otimes v_{\sigma(p)})\otimes(v_{{\sigma(p+1)}}\otimes \ldots \otimes v_{\sigma(n)}),
\end{displaymath}
where the sum is over all $p$ and all $(p,n-p)$-shuffles and, as usual, $\varepsilon(\sigma)$ is the Koszul sign.

The diagonal $\Delta$ is thus defined by the conditions

\begin{itemize}%
\item $\Delta v = v\otimes 1 + 1\otimes v$ if $v \in V$;


\item $\Delta$ is a morphism of $\mathbb{Z}$-graded algebras.



\end{itemize}
A commutative and cocommutative $\mathbb{Z}$-graded Hopf algebra structure on $\bigwedge V$ is obtained by using the algebra and coalgebra structures defined in [[differential graded algebra]] and [[differential graded coalgebra]]. respectively.

\hypertarget{the_enveloping_algebra_of_a_lie_algebra_}{}\subsubsection*{{The enveloping algebra of a Lie algebra, $U(L)$.}}\label{the_enveloping_algebra_of_a_lie_algebra_}

Let $L$ be a pre-gla, $U(L)$, is the quotient algebra of the tensor algebra $T(L)$ by the two sided ideal generated by the elements

\begin{displaymath}
x\otimes y - (-1)^{|y||x|}y\otimes x - [x,y], \quad x,y,\in L.
\end{displaymath}
The diagonal $\Delta : L \to L\times L$, with $\Delta(x) = (x,x)$ defines a homomorphism of pre-gas,

\begin{displaymath}
U(\Delta) : U(L)\to U(L\times L) \cong U(L)\otimes U(L),
\end{displaymath}
which makes $U(L)$ a pre-gha which is cocommutative and conilpotent.

If $L$ is a free Lie algebra on $V$, then the enveloping algebra is the tensor algebra: $U\mathbb{L}(V) \cong T(V)$.

Let $(L,\partial)$ be a pre-dgla, the differential $\partial$ extends to an algebra differential on $T(L)$. With the quotient differential, $U(L)$ becomes a cocommutative pre-dgha, which will be denoted $U(L,\partial)$.

The differential $\partial$ determines a differential, also denoted $\partial$, on the cocommutative pre-gca $\bigwedge' L$, (for which gca see [[differential graded coalgebra]]). It satisfies:

\begin{displaymath}
\bigwedge' H(L,\partial) \cong H(\bigwedge' L,\partial).
\end{displaymath}
Let $i : L \to U(L)$ be the linear mapping $L\to T(L) \to U(L)$, then define $e: \bigwedge' L \to U(L)$ by

\begin{displaymath}
e(x_1\wedge \ldots x_n) = \frac{1}{n!}\sum_\sigma \varepsilon(\sigma)i(x_{\sigma(1)})\ldots i(x_{\sigma(n)}),
\end{displaymath}
where the sum is over all permutations and $\varepsilon(\sigma)$ is the Koszul sign.

\begin{theorem}
\label{}\hypertarget{}{}
(Poincar\'e{}-Birkhoff-Witt)(cf. \textbf{Quillen})

The mapping $e$ is an isomorphism of pre-dgcas.

\end{theorem}
\begin{corollary}
\label{}\hypertarget{}{}
$i : L \to U(L)$ defines an isomorphism between $L$ and the space of primitives of $U(L)$.

\end{corollary}
\begin{corollary}
\label{}\hypertarget{}{}
The natural map $UH(L,\partial)\to H(U(L,\partial)$ is an isomorphism of cocommutative pre-ghas.

\end{corollary}
\hypertarget{the_lie_algebra_of_primitives_}{}\subsubsection*{{The Lie algebra of primitives, $P$}}\label{the_lie_algebra_of_primitives_}

Let $(A,\partial)$ be a cocommutative pre-dgha. The vector space $P(A)$ of primitive elements (for the coalgebra structure, cf. [[differential graded coalgebra]]), is not stable under the multiplication, however the commutator $[\alpha,\beta]$ of two elements of $P(A)$ is again in $P(A)$. This defines a pre-gla structure on $P(A)$ and we can put the induced differential on it to obtain $P(A,\partial)$.

The inclusion $P(A)\to A$ extends to a morphism of cocommutative pre-dghas $\sigma: UP(A)\to A.$

\begin{theorem}
\label{}\hypertarget{}{}
If $A$ is conilpotent, $\sigma$ is an isomorphism.

\end{theorem}
The above theorem and earlier corollary show that $U$ and $P$ are inverse equivalences between the category, $pre DGLA$ and that of cocommutative, conilpotent pre-dghas.

\textbf{Remark}

The enveloping algebra of a free Lie algebra $\mathbb{L}(V)$ coincides with the tensor algebra, $T(V)$. It is conilpotent from which one gets $PT(V) = \mathbb{L}(V)$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hopf algebra]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item D. Tanr\'e{}, Homotopie rationnelle: Mod\`e{}les de Chen, Quillen, Sullivan, Lecture Notes in Maths No. 1025, Springer, 1983.


\item [[Daniel Quillen]], \emph{Rational Homotopy Theory}, Ann. of Math., (2) 90 (1969), 205-295.



\end{itemize}
Discussion of [[bar-cobar construction]] for dg-Hopf algebras:

\begin{itemize}%
\item [[Benoit Fresse]], \emph{The universal Hopf operads of the bar construction} (\href{https://arxiv.org/abs/math/0701245}{arXiv:math/0701245})


\item [[Murray Gerstenhaber]], [[Alexander Voronov]], Section 3.2 of: \emph{Homotopy G-algebras and moduli space operad}, Internat. Math. Research Notices (1995) 141-153 (\href{https://arxiv.org/abs/hep-th/9409063}{arXiv:hep-th/9409063})


\item Justin Young, \emph{Brace Bar-Cobar Duality} (\href{https://arxiv.org/abs/1309.2820}{arXiv:1309.2820})



\end{itemize}
[[!redirects differential graded Hopf algebras]]

[[!redirects dg-Hopf algebra]] [[!redirects dg-Hopf algebras]]



\end{document}