The Commutator Functor from R -Algebras to Lie Algebras

Let $R$$R$ be a ring and $S$$S$ be an $R$$R$-algebra, i.e., there exists a map $f:R\to S$$f: R \to S$ such that $f(R)\subseteq Z(S)$$f(R) \subseteq Z(S)$, where $Z(S)$$Z(S)$ is the center of $S$$S$.

Define the commutator as follows: for any $x,y\in S$$x, y \in S$, $[x,y]=xy-yx$$[x, y] = xy - yx$.

It is easy to see that it satisfies the following properties:

• Bilinearity, i.e., $[ax+b{x}^{\prime },y]=a[x,y]+b[x^{\prime },y]$$[ax + bx', y] = a[x, y] + b[x', y]$ and similarly for $[x,ay+b{y}^{\prime }]$$[x, ay + by']$.

• The Alternating property, i.e., $[x,x]=0$$[x, x] = 0$, hence $[x,y]=[x,y+x]=[x-(y+x),y+x]=[-y,y+x]=-[y,x]$$[x, y] = [x, y + x] = [x - (y + x), y + x] = [-y, y + x] = -[y, x]$.

• The Jacobi identity, i.e., $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$$[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$.

Hence, the commutator gives us a functor $\mathcal{Com}:R\text{-alg}\to {\mathrm{LieAlg}}_R$$\mathscr{Com}: R\text{-alg} \to \mathrm{LieAlg}_R$.

For any $R$$R$-algebra $S$$S$, $\mathcal{Com}(S):=(S,[-,-])$$\mathscr{Com}(S) := (S, [-,-])$. For an $R$$R$-algebra homomorphism $f$$f$, $\mathcal{Com}(f)(x):=f(x)$$\mathscr{Com}(f)(x) := f(x)$, and it is easy to see that:

$$\begin{array}{}\text{(1)}& f[x,y]=[f(x),f(y)].\end{array}$$