Fox derivative

In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.

Definition

If G {\displaystyle G} is a free group with identity element e {\displaystyle e} and generators g i {\displaystyle g_{i}}, then the Fox derivative with respect to g i {\displaystyle g_{i}} is a function from G {\displaystyle G} into the integral group ring Z G {\displaystyle \mathbb {Z} G} which is denoted ∂ ∂ g i {\displaystyle {\frac {\partial }{\partial g_{i}}}}, and obeys the following axioms:

∂ ∂ g i ( g j ) = δ i j {\displaystyle {\frac {\partial }{\partial g_{i}}}(g_{j})=\delta _{ij}}, where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta
∂ ∂ g i ( e ) = 0 {\displaystyle {\frac {\partial }{\partial g_{i}}}(e)=0}
∂ ∂ g i ( u v ) = ∂ ∂ g i ( u ) + u ∂ ∂ g i ( v ) {\displaystyle {\frac {\partial }{\partial g_{i}}}(uv)={\frac {\partial }{\partial g_{i}}}(u)+u{\frac {\partial }{\partial g_{i}}}(v)} for any elements u {\displaystyle u} and v {\displaystyle v} of G {\displaystyle G}.

The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product rule. As a consequence of the axioms, we have the following formula for inverses

∂ ∂ g i ( u − 1 ) = − u − 1 ∂ ∂ g i ( u ) {\displaystyle {\frac {\partial }{\partial g_{i}}}(u^{-1})=-u^{-1}{\frac {\partial }{\partial g_{i}}}(u)} for any element u {\displaystyle u} of G {\displaystyle G}.

Applications

The Fox derivative has applications in group cohomology, knot theory, and covering space theory, among other areas of mathematics.

Fox derivative

Definition

Applications

See also