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.
See also Alexander polynomial Free group Ring (mathematics) Integral domain Derivation (differential algebra)
References Brown, Kenneth S. (1972). Cohomology of Groups. Graduate Texts in Mathematics. Vol. 87. Springer Verlag. ISBN 0-387-90688-6. MR 0672956. Fox, Ralph (May 1953). "Free Differential Calculus, I: Derivation in the Free Group Ring". Annals of Mathematics. 57 (3): 547–560. doi:10.2307/1969736. JSTOR 1969736. MR 0053938. Fox, Ralph (March 1954). "Free Differential Calculus, II: The Isomorphism Problem of Groups". Annals of Mathematics. 59 (2): 196–210. doi:10.2307/1969686. JSTOR 1969686. MR 0062125. Fox, Ralph (November 1956). "Free Differential Calculus, III: Subgroups". Annals of Mathematics. 64 (2): 407–419. doi:10.2307/1969592. JSTOR 1969592. MR 0095876. Chen, Kuo-Tsai; Ralph Fox; Roger Lyndon (July 1958). "Free Differential Calculus, IV: The Quotient Groups of the Lower Central Series". Annals of Mathematics. 68 (1): 81–95. doi:10.2307/1970044. JSTOR 1970044. MR 0102539. Fox, Ralph (May 1960). "Free Differential Calculus, V: The Alexander Matrices Re-Examined". Annals of Mathematics. 71 (3): 408–422. doi:10.2307/1969936. JSTOR 1969936. MR 0111781.