A $\mathbb{Z}$-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:
We can replace the third condition by:
Of course, wherever possible, we will abbreviate $(A,\Delta,\mu,\epsilon,\eta)$ to $A$.
A morphism 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.
Let $A$ be a pre-gha. A 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 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 commutative if $(A,\mu)$ is commutative and is cocommutative if $(A,\Delta,\varepsilon)$ is cocommutative.
This gives categories $pre CDGHA$ and $pre CoDGHA$ respectively.
A cocommutative (resp. commutative) dgha} is an object of $pre CoDGHA$ (resp. $pre CDGHA$, which has a lower (resp. upper) grading.
A cocommutative (resp. commutative) dgha $A$ is $n$-connected if $\bar{A}_p = 0$ (resp $\bar{A}^p = 0$) for $p\leq n$.
Let $V$ be a pre-gvs. The gvs $T(V)$ is a pre-cga for the shuffle product defined by
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.
The underlying algebra structure is $T(V)$ with the shuffle product. The reduced diagonal is given by
The underlying algebra structure this time is $T(V)$ with the usual product
but with the reduced diagonal given by
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
$\Delta v = v\otimes 1 + 1\otimes v$ if $v \in V$;
$\Delta$ is a morphism of $\mathbb{Z}$-graded algebras.
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.
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
The diagonal $\Delta : L \to L\times L$, with $\Delta(x) = (x,x)$ defines a homomorphism of pre-gas,
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:
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
where the sum is over all permutations and $\varepsilon(\sigma)$ is the Koszul sign.
(Poincaré-Birkhoff-Witt)(cf. Quillen)
The mapping $e$ is an isomorphism of pre-dgcas.
$i : L \to U(L)$ defines an isomorphism between $L$ and the space of primitives of $U(L)$.
The natural map $UH(L,\partial)\to H(U(L,\partial)$ is an isomorphism of cocommutative pre-ghas.
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 $\sigma: UP(A)\to A.$
If $A$ is conilpotent, $\sigma$ is an isomorphism.
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.
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 $PT(V) = \mathbb{L}(V)$.
The source used for this lexicon was
D. Tanré, Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Maths No. 1025, Springer, 1983.
Much of the material there was based on Quillen’s paper:
D. Quillen, Rational Homotopy Theory, Ann. of Math., (2) 90 (1969), 205-295.