nLab Fox pairing

Idea

The notion of Fox pairings in group theory/algebra was introduced by Massuyeau & Turaev 2013, who used them to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces.

Definition

By a Fox pairing in an augmented algebra, $A$, over a commutative ring $K$, we mean a $K$-bilinear map $\eta : A \times A \to A$, which is a left Fox derivative with respect to the first variable and a right Fox derivative with respect to the second variable.

Properties

Proposition

Given a Fox pairing $\eta$, the following product formulas hold:

$$\eta(a_1 a_2, b) = \eta(a_1, b) aug(a_2) + a_1\eta(a_2, b)$$

for all $a_1, a_2, b\in A$; and

$$\eta(a, b_1b_2) = \eta(a, b_1)b_2 + aug(b_1)\eta(a, b_2)$$

for all $a, b_1, b_2 \in A$.

References

• Gwénaël Massuyeau, Vladimir Turaev: Fox pairings and generalized Dehn twists, Annales de l’Institut Fourier 63 5 (2013) 2403-2456 [doi:10.5802/wbln.38, arXiv:1109.5248]