Recall that we have the concept of tensor algebra
Theorem Let \(A\) be associated algebra, \(V\) vector space, \(f:V\to A\) linear. Then there exists \(\overline{f}:T(V)\to A\) algebra homomorphism such that \(\overline{f}\circ i=f\) where \(i:V\to T(V)\) is the canonical inclusion.
In terms of functors, this is
\[\forall A. \mathrm{Hom}_{\mathbf{Vect}}(V,A) = \mathrm{Hom}_{\mathbf{Alg}}(T(V),A).\]
In other words, the functor \(h^V_\mathbf{Vect} : \mathbf{Alg}\to {\bf Set}\) is represented by \(T(V)\), \(h^V_\mathbf{Vect} = h^{T(V)}\), here \(=\) means a natural isomorphism is specified.
Example: let \(\dim V=1\) with a basis element \(v\neq 0\). We can just define the linear map \(v\mapsto t \in \mathbb{C}[t]\), the tensor algebra is just the polynomial algebra in that \(V^{\otimes k}\mapsto t^k\) and \(T(V) \cong \mathbb{C}[t]\).
Definition A monomial in \(T(V)\) is an element of the form \(v_1\otimes \dots\otimes v_k\) for some \(v_1,\dots, v_k\in V\) for some \(1\le i,j\le k\).
The symmetric algebra on \(V\) is
\[S(V) = T(V) / \langle x\otimes y - y\otimes x | x,y\in V\rangle.\]
Example This is a commutative algebra. Note that there is a map \(V\to T(V)\to S(V)\).
Similar to tensor algebra the commutative algebra \(S(V)\) is also a universal solution
\[h^V_\mathbf{Vect} = h^{S(V)} : \mathbf{Comm}\to \mathbf{Set}.\]
Let \(V=L\) be a Lie algebra, we define the universal enveloping algebra \(\mathcal{U}(L)\) by
\[\mathcal{U}(L) := T(L)/ \langle x\otimes y - y\otimes x - [x,y]|x,y\in L\rangle.\]
Warning: \(x\otimes y - y\otimes x \in T_2(L)\) but \([x,y]\in T_1(L)= L\).
Theorem (Universal Property for \(\mathcal{U}(L)\)) Let \(A\) be an associative algebra over \(\mathbb{C}\) and let \(f:L\to [A]\) a Lie morphism, then it uniquely factor through \(L\to \mathcal{U}(L)\).
\[h^L_\mathbf{Lie} = h^{\mathcal{U}(L)} : \mathbf{Alg} \to {\bf Set}.\]
Theorem (PBW). Let \(e_1,\dots,e_n\) be a basis of \(L\), then \(\{e^\alpha = \prod_i e_i^{\alpha_i} : \alpha\ge 0\}\) is a basis of \(\mathcal{U}(L)\).
Corollary. \(L\to \mathcal{U}(L)\) is an injective map.
Outline of proof. Using an argument for vector spaces, let \(V\) be a finite dimensional vector space over \(\mathbb{C}\) with basis \(e_{1,\dots,n}\). Let \(B\) be the monomials in \(e_1,\dots,e_n \subset T(V)\).
Fix a total order on \(B\) with
\(u\ge 1\) for all \(u\in B\).
\(u\ge v \Rightarrow wu\ge wv, \forall w\in B\).
\(\{u\in B: u\le v\}\) is finite \(\forall b\in B\).
Example
We say the length of \(u=e_{i_1}\dots e_{i_k}\in B\) is \(k\), write as \(\mathrm{len}(u)=k\). And we can specify the length-lex order, which satisfy the above requirements.
Definition. Let \(x\in T(V)\) write \(x\) in terms of \(B\). The leading term of \(x\) written as \(\overline{x}\) is the maximal term with respect to \(\le\) with non-zero coefficient.
Example
Set \(e_1<e_2<e_3\), then we can order like \[ 1 < e_1 < e_2e_1 < e_3e_1 < e_3e_2 \]
Definition
Let \(x,y\in T(V)\), if \(\exists u,v,w\in B\) with \(\overline{x}=uv, \overline{y}=vw\), then the syzygy of \(x\) and \(y\) is \(x\vee y = xw-uy\).