Sine laws on semigroups with an anti-automorphism: A left-translation approach
Dang Vo Phuc
TL;DR
The paper addresses extending sine-law functional equations on semigroups to involutive anti-automorphisms $(\sigma^2=\mathrm{id})$ by replacing the failing right-regular action with a left-translation framework. It introduces the conjugation identity $J R(\sigma(y)) J = L(y)$ (with $J h = h \circ \sigma$) to recover a well-behaved representation, enabling a Levi-Civita style closure in the anti-homomorphic setting. The main contributions are a) establishing $L$-invariance of the solution space $V=\mathrm{span}\{f,g\}$ and a finite affine-linear matrix form for $L(y)|_V$, and b) proving an analogue of the generalized sine law: the equation implies a simpler $xy$-addition law and specific transformation rules $f\circ\sigma$ and $g\circ\sigma$, with a dichotomy on parameters. Concrete examples from $GL_n(F)$, $S_n$, and $SO(3)$ illustrate the construction and invariant subspaces. This work extends the structural theory of trigonometric functional equations to anti-homomorphic settings and highlights left translations as a robust tool for maintaining representation-theoretic control.
Abstract
Stetkær's matrix method is a useful tool for analyzing functional equations on semigroups involving a homomorphism $σ$. However, this method fails when $σ$ is an anti-automorphism because the underlying right-regular representation reverses composition order. To resolve this, we introduce a new approach based on a key conjugation identity. Let $J$ denote the operator of composition with $σ$; then the identity $J R(σ(y)) J = L(y)$ provides the foundation for our method. This identity restores a well-behaved representation via left translations, making the matrix method applicable again. This left-translation approach is illustrated with several concrete examples from matrix groups and symmetric groups. Using this approach, we extend Stetkær's main structural theorem for the generalized sine law on semigroups to the anti-automorphic setting. For linearly independent solutions, we show that the equation implies a simpler addition law and that the solutions obey the same transformation rules ($f\circσ=βf$, etc.) as in the homomorphic case.