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An introduction to loopoids

Janusz Grabowski (2016)

Commentationes Mathematicae Universitatis Carolinae

We discuss a concept of loopoid as a non-associative generalization of Brandt groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.

An obstruction to represent abelian Lie algebras by unipotent matrices.

J. C. Benjumea, F. J. Echarte, Núñez, J.,Tenorio, A. F. (2004)

Extracta Mathematicae

The aim of this paper is the study of abelian Lie algebras as subalgebras of the nilpotent Lie algebra gn associated with Lie groups of upper-triangular square matrices whose main diagonal is formed by 1. We also give an obstruction to obtain the abelian Lie algebra of dimension one unit less than the corresponding to gn as a Lie subalgebra of gn. Moreover, we give a procedure to obtain abelian Lie subalgebras of gn up to the dimension which we think it is the maximum.

Control Systems on the Orthogonal Group SO(4)

Ross M. Adams, Rory Biggs, Claudiu C. Remsing (2013)

Communications in Mathematics

We classify the left-invariant control affine systems evolving on the orthogonal group S O ( 4 ) . The equivalence relation under consideration is detached feedback equivalence. Each possible number of inputs is considered; both the homogeneous and inhomogeneous systems are covered. A complete list of class representatives is identified and controllability of each representative system is determined.

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