Quasigroup homogeneous spaces and linear representations. (English) Zbl 0994.20054

The aim of the present paper is to describe the structure of the ring \((\pi(\mathbb{C} G),+,E_P)\) and its representation \[ \rho_{P\setminus Q}\colon(\pi(\mathbb{C} G),+,E_P)\to\text{End}_\mathbb{C}\mathbb{C} P\setminus Q.\tag{1} \] In the final section the author examines sufficient conditions for commutativity of the image of (1), implying mutual commutativity of the various transition matrices \(R_{P\setminus Q}(q)=A^+_PR(q)A_P\) in the corresponding iterated function system.


20N05 Loops, quasigroups
Full Text: DOI


[1] Barnsley, M.F., Fractals everywhere, (1988), Academic Press San Diego · Zbl 0691.58001
[2] Boullion, T.L.; Odell, P.L., Generalized inverse matrices, (1971), Wiley New York
[3] Chein, O., Quasigroups and loops: theory and applications, (1990), Heldermann Berlin · Zbl 0719.20036
[4] Divinsky, N.J., Rings and radicals, (1965), Univ. of Toronto Press Toronto · Zbl 0138.26303
[5] Johnson, K.W.; Smith, J.D.H., Characters of finite quasigroups, European J. combin., 5, 43-50, (1984) · Zbl 0537.20042
[6] Penrose, R., A generalized inverse for matrices, Proc. Cambridge philos. soc., 51, 406-413, (1955) · Zbl 0065.24603
[7] Pierce, R.S., Associative algebras, (1982), Springer-Verlag New York · Zbl 0497.16001
[8] Smith, J.D.H., Centralizer rings of multiplication groups on quasigroups, Math. proc. Cambridge philos. soc., 79, 427-431, (1976) · Zbl 0335.20035
[9] Smith, J.D.H., Representation theory of infinite groups and finite quasigroups, (1986), Univ. of Montreal Montreal · Zbl 0599.20110
[10] Smith, J.D.H., Quasigroup actions: Markov chains, pseudoinverses, and linear representations, Southeast Asian bull. math., 23, 719-729, (1999) · Zbl 0944.20059
[11] Smith, J.D.H.; Romanowska, A.B., Post-modern algebra, (1999), Wiley New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.