Basic constructions for graded algebras. I'm reading about the Weil algebra of a Lie group and it involves some constructions I'm not very familiar with, for instance the "free graded-commutative graded algebra on $a_1...a_n$ with degrees $deg(a_i)$." Does anyone know a good source for basic graded-algebra constructions like this?  Anything i find is either too basic or too advanced to do these constructions in detail. Thanks!
 A: I would start by reading the first part of Kassel's Quantum Groups.
Let us give some examples involving graded constructions.
We focus on graded vector spaces over a field; the module theory is similar.
It seems you are interested in the graded symmetric algebra over a graded vector space.
In general, let $V:=\{V_i\}_{i\in\mathbb Z}$ be a $\mathbb Z$-graded vector field over $\mathbb R$: every homogeneous component $V_i$ is a vector space over $\mathbb R$.
Elements $a_i\in V_i$ are said of being homogeneous of degree $|a_i|:=i$. The free graded algebra over $V$ is the tensor algebra $T(V)$, which, as a set, is given by
$$T(V)=\mathbb R\oplus V\oplus V^{\otimes 2}\oplus V^{\otimes 3}\oplus\dots $$
Its homogeneous component $T^i:= T(V)^i$ of degree $i$ is then
$$T^i:=\bigoplus_{j\geq 0}(V^{\otimes j})^i, $$
and $(V^{\otimes j})^i:=\oplus_{i_1+...+i_j=i} V^{i_1}\otimes\cdots\otimes V^{i_j}.$
 Note that the index $j$ is not a grading: it is called the "weight". The associative product is given by concatenation of tensors and the unit is the multiplicative unit $1\in \mathbb R$.
Let $(R)$ be the two sided graded ideal in $T(V)$ generated by all the expressions of the form 
$$v\otimes w -(-1)^{|v_i||w_j|}w\otimes v, $$
for all $v$ and $w$ in $V$ homogeneous of degrees $|i|$ and $|j|$. In other words,
$$(R)=T(V)\otimes R\otimes T(V) $$
with $R:=\operatorname{span}_{\mathbb R}(v\otimes w -(-1)^{|v_i||w_j|}w\otimes v)$.
Note that $R\subset V^{\otimes 2}$. We denote by $(R)_j$, $j\geq 2$ the component(s) of the ideal $(R)$ in $V^{\otimes j}$, i.e.
$$(R)_2=R, $$
$$(R)_3=V\otimes R\oplus R\otimes V, $$
and so on. Once again, the index $j$ denotes the "weight", and not the grading.
We are done: the quotient 
$$S(V):=T(V)/ (R)=\mathbb R\oplus V\oplus V^{\otimes 2}/R\oplus V^{\otimes 3}/(R)_3 \oplus\dots  $$
is the graded symmetric algebra over $V$.
The case with a finite number of letters $a_1,...a_n$ in the OP can be reduced to the case of a finite dimensional graded vector space $V$ (with all homogeneous components $V_i$ finite dimensional, and in finite number different from $\{0\}$). 
