How are the universal enveloping algebra and PBW-theorem for differential graded Lie algebras defined? Differential graded Lie algebras are defined as graded vector spaces $L=\bigoplus_{i \in \mathbb{Z}}$ over a field of characteristic zero equipped with a bilinear bracket $[-,-] \colon L_i \otimes L_j \to L_{i+j}$ satisfying the graded antisymmetric and graded Jacobi identity together with a differential $d \colon L_{i} \to L_{j}$ which satisfies the graded Leibniz rule. I want to know how the universal enveloping algebra and PBW-theorem are defined for differential graded Lie algebras? Introducing related papers will be highly appreciated.
 A: I will ignore the differential for a moment, and only deal with the grading for now.
The degree of a homogeneous element $x$ will be denoted by $|x|$.

*

*For any two graded vector spaces $V$ and $W$, the tensor product $V ⊗ W$ inherits a grading from $V$ and $W$ given by
$$
  |v ⊗ w| = |v| + |w| \,.
$$
In other words, we have $(V ⊗ W)_i = \bigoplus_{j + k = i} V_j ⊗ W_k$ for every degree $i$.
It follows that for every natural exponent $n ≥ 0$ the tensor power $V^{⊗n}$ inherits a grading from $V$ given by
$$
  |v_1 ⊗ \dotsm ⊗ v_n|
  =
  |v_1| + \dotsb + |v_n| \,.
$$
Therefore, the tensor algebra $\mathrm{T}(V) = \bigoplus_{n ≥ 0} V^{⊗ n}$ inherits a grading from $V$.
This grading makes $\mathrm{T}(V)$ into a graded algebra.
This ‘graded tensor algebra’ has the expected universal property:
wherever $A$ is graded algebra, every homomorphism of graded vector spaces from $V$ to $A$ extends uniquely to a homomorphism of graded algebras from $\mathrm{T}(V)$ to $A$.


*Every graded algebra $A$ can be made into a graded Lie algebra via the graded commutator
$$
  [a, b] = ab - (-1)^{|a| |b|} ba \,.
$$


*Let $L$ be a graded Lie algebra.
The universal enveloping algebra of $L$ is a graded Lie algebra $\mathrm{U}(L)$ together with a homomorphism of graded Lie algebras $i \colon L \to \mathrm{U}(L)$ satisfying the following universal property:
for every graded algebra $A$ and every homomorphism of graded Lie algebras $f \colon L \to A$ there exists a unique homomorphism of graded algebras $φ \colon \mathrm{U}(L) \to A$ with $φ ∘ i = f$.


*The universal enveloping algebra of a graded Lie algebra $L$ can be constructed in the usual way.
We consider the tensor algebra $\mathrm{T}(L)$ and the ideal $I$ of $\mathrm{T}(L)$ generated the elements of the form $[x, y]_{\mathrm{T}(L)} - [x, y]_L$ with $x, y ∈ L$.
In other words, the ideal $I$ is generated by elements of the form $x ⊗ y - (-1)^{|x| |y|} y ⊗ x - [x, y]_L$ where $x$ and $y$ are homogeneous elements of $L$.
The ideal $I$ is generated by homogeneous elements and is therefore homogeneous.
The quotient algebra $\mathrm{U}(L) := \mathrm{T}(L) / I$ therefore inherits a grading from $\mathrm{T}(L)$.
The composite
$$
  i \colon L \to \mathrm{T}(L) \to \mathrm{U}(L)
$$
is the required homomorphism of graded Lie algebras.


*Let $(x_λ)_{λ ∈ Λ}$ be a vector space basis of $L$ consisting of homogeneous elements, and where $(Λ, ≤)$ linearly ordered.
The universal enveloping algebra $\mathrm{U}(L)$ has a vector space basis consisting of the ordered monomials
$$
  x_{λ_1}^{n_1} \dotsm x_{λ_r}^{n_r}
$$
with $r ≥ 0$, $λ_1 < \dotsb < λ_r$, $n_i ≥ 1$, and $n_i = 1$ if $x_{λ_i}$ is of odd degree.
The above graded PBW-theorem can be found as part of Theorem 21.1 in Rational Homotopy Theory by Felix, Halperin and Thomas.
Suppose now that we are not only dealing with gradings, but also with differentials.
I assume that these gradings are supposed to be degree $-1$, i.e., that $|d(x)| = |x| - 1$ for every homogeneous element $x$.
We can then check that all of the above constructions inherit suitable differentials:

*

*For any two differential graded vector spaces $V$ and $W$, their tensor products $V ⊗ W$ inherits a differential from $V$ and $W$ given by
$$
  d(v ⊗ w) = d(v) ⊗ w + (-1)^{|v|} v ⊗ d(w) \,.
$$
This differentail makes the graded vector space $V ⊗ W$ into a differential graded vector space.
It follows that for every natural exponent $n ≥ 0$ the graded vector space $V^{⊗ n}$ becomes a differential graded vector space with differential
$$
  d(v_1 ⊗ \dotsb ⊗ v_n)
  =
  \sum_{i = 0}^n
  (-1)^{|v_1| + \dotsb + |v_i|}
  v_1 ⊗ \dotsb ⊗ d(v_i) ⊗ \dotsb v_n \,.
$$
The resulting differential on the tensor algebra $\mathrm{T}(V) = \bigoplus_{n ≥ 0} V^{⊕ n}$ makes it into a differential graded algebra.


*We have constructed the graded universal enveloping algebra $\mathrm{U}(L)$ as a quotient $\mathrm{T}(L) / I$ for a specific ideal $I$. This ideal satisfies $d(I) ⊆ I$, whence the differential of $\mathrm{T}(L)$ induces a differential on $\mathrm{U}(L)$, making it into a differential graded algebra.


*Every differential graded algebra becomes a differentail graded Lie algebra via the graded commutator.


*The homomorphism of graded Lie algebras $i$ from $L$ to $\mathrm{U}(L)$ is a homomorphism of differential graded Lie algebras, and it satisfies the expected universal property:
for every differential graded algebra $A$ and every homomorphism of differential graded Lie algebras $f$ from $L$ to $A$, there exists a unique homomorphism of differential graded algebras $φ$ from $\mathrm{U}(L)$ to $A$ with $φ ∘ i = f$.


*The PBW-theorem remains unchanged since it makes no reference to the differential.
Some possible sources:

*

*Chapter 21 in Rational Homotopy Theory by Felix, Halperin, Thomas (Graduate Texts in Mathematics, number 205).

*Appendix B in Rational Homotopy Theory by Quillen.

*I once gave a seminar talk about a related topic; the handout of this talk can be found online at https://gitlab.com/cionx/topology-seminar-hopf-algebras-ss19/-/raw/master/handout.pdf.

