|On a functional equation involving group actions|
Aequationes mathematicae 77, 25-32, 2009.
Abstract: During the forty-first ISFE in Noszvaj, Hungary, G. Guzik posed a problem on a functional equation involving group actions which arose in a generalization of Bargman theory occurring in Quantum Mechanics. (Cf. 18. Problem and Remark in "Report of Meeting", Aequationes Mathematicae, Vol. 67 (2004) 312-313.)
Let (G, ⋅) be a group which is acting on a set X and let (K, +) be an abelian group. Describe all functions f: G × G × X→ K satisfying
for all g1, g2, g3∈ G and x∈ X.
f(g1, g2, x)+f(g1g2, g3, x)=f(g2, g3, g1-1x)+f(g1, g2g3, x)..
This problem was solved in a particular case by B. Ebanks. (Cf. 19. Remark in "Report of Meeting", Aequationes Mathematicae, Vol. 67 (2004) p. 313.) We present the general solution of this problem.
|GDPR||On a functional equation involving group actions|