nLab derivation on a group

Derivations:

For convenience we assume below that $M$ is a $G$-module, it does not in general have to be abelian and it suffices to have it a $G$-group.

Suppose $G$ is a group and $M$ a $G$-module and let $\delta : G \to M$ be a derivation in the sense that $\delta(g_1g_2) = \delta(g_1) +g_1\delta(g_2)$ for all $g_1, g_2 \in G$ (a crossed homomorphism – notice that it’s not $\delta(g_1)g_2 + g_1\delta(g_2)$ as for the other notion of derivation.)

For calculations, the following lemma is very valuable, although very simple to prove.

Lemma

If $\delta : G \to M$ is a derivation, then

1. $\delta(1_G) = 0$;

2. $\delta(g^{-1}) = -g^{-1}\delta(g)$ for all $g \in G$;

3. for any $g \in G$ and $n\geq 1$,

   $$\delta(g^n) = (\sum^{n-1}_{k=0}g^k)\delta(g).$$

Proof

As was said, these are easy to prove.

$\delta(g) = \delta(1) + 1\delta(g)$, so $\delta(1)= 0$, and hence (1); then

$$\delta(1) = \delta(g^{-1}g) = \delta(g^{-1}) + g^{-1}\delta(g)$$

to get (2), and finally induction to get (3).

Remarks and examples:

• There is a mapping from $G$ to its augmentation ideal, $I(G)$, defined by $d_G(g)= g-e_G$. This is the universal derivation towards $G$-modules.

• The Fox derivatives are examples of derivations. It is worth noting that this lemma allows a simplification of the conditions given there (as noted there).

Example calculation using the Fox derivative w.r.t a generator.

Let $X = \{u,v\}$, with $r \equiv u v u v^{-1} u^{-1} v^{-1} \in F = F(u,v),$ then

$$\frac{\partial r}{\partial u} = 1 + u v - u v u v^{-1} u^{-1},$$

$$\frac{\partial r}{\partial v} = u - u v u v^{-1} - u v u v^{-1} u^{-1} v^{-1}.$$

This relation, $r$, is the typical braid group relation, here in $Br_3$.

Relative derivations

These are a useful relative form of derivation. The notion is often avoided as it can easily be reduced to the more standard form above by restricting the module structure along $\varphi$.

Let $\varphi : H \rightarrow G$ be a homomorphism of groups. A $\varphi$-derivation from a group to a module,

$$\partial : H \rightarrow M,$$

from $H$ to a left $\mathbb{Z}[G]$-module, $M$, is a mapping from $H$ to $M$, which satisfies the equation

$$\partial (h_1 h_2 ) = \partial (h_1 ) + \varphi (h_1)\partial (h_2 )$$

for all $h_1$, $h_2 \in H$.

There is a universal such $\varphi$-derivation, $d_\varphi:H\to D_\varphi$. The codomain of this is variously called the derived module of $\varphi$ (e.g. by Crowell) or the $\varphi$-differential module by Morishita.

The set of $\varphi$-derivations is often written $Der_\varphi(H,M)$, or simply $Der_\varphi(M)$.

Related concepts

• crossed homomorphism

References

For the original version of derived module, see

• R. H. Crowell, The derived module of a homomorphism, Advances in Math., 5, (1971), 210–238 (doi:10.1016/0001-8708(71)90016-8)

Textbook account:

• Kenneth Brown, p. 45 of: Cohomology of Groups, Graduate Texts in Mathematics, 87, Springer 1982 (doi:10.1007/978-1-4684-9327-6)

Application in arithmetic topology:

• Masanori Morishita, Knots and Primes: An Introduction to Arithmetic Topology, 2012 (web)