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Semidirect product of groupoids and associated algebras

Pysiak L., Eckstein M., Heller M., Sasin W.

Demonstratio Mathematica

2014

Vol. 47, nr 2

289--299

One of the pressing problems in mathematical physics is to find a generalized Poincaré symmetry that could be applied to nonflat space-times. As a step in this direction, we define the semidirect product of groupoids Γ0 x Γ1 and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid Γ1 and prove that it is isomorphic to the convolutive algebra of the groupoid Γ0 x Γ1. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. An important example is the Poincaré groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.

On representation of a semidirect product of cyclic groups by a 2-state time-varying mealy automaton

Woryna A.

Zeszyty Naukowe. Matematyka - Fizyka / Politechnika Śląska

2004

z. 91

343-354

We present thesubgroup K - [W, W].Z of the wreath product W = Zlx0Z relative to an n-element set X0 as a group generated by a 2-state time-varying Mealy automaton. The action of K on the sets of finite and infinite words over the changing alphabet we study.

W splocie W = Z I x0 Z względem n-elementowego zbioru X 0 rozważamy podgrupę K = [W, W]. Z jako grupę generowaną przez zmienny w czasie automat Mealy'ego o dwóch stanach. Badamy działanie grupy K na zbiorach słów skończonych i nieskończonych nad zmiennym alfabetem.