Lie Symmetries of the Canonical Geodesic Equations for Four-Dimensional Lie Groups

For each of the four-dimensional indecomposable Lie algebras the geodesic equations of the associated canonical Lie group connection are given. In each case a basis for the associated Lie algebra of symmetries is constructed and the corresponding Lie brackets are written down

Minimal Matrix Representations for Six-Dimensional Nilpotent Lie Algebras

This paper is concerned with finding minimal dimension linear representations for six-dimensional real, indecomposable nilpotent Lie algebras. It is known that all such Lie algebras can be represented in gl(6, R). After discussing the classification of the 24 such Lie algebras, it is shown that only one algebra can be represented in gl(4, R). A Theorem is then presented that shows that 13 of the algebras can be represented in gl(5, R). The special case of filiform Lie algebras is considered, of which there are five, and it is shown that each of them can be represented in gl(6, R) and not gl(5, R). Of the remaining five algebras, four of them can be represented minimally in gl(5, R). That leaves one difficult case that is treated in detail in an Appendix

Symmetry Algebras of the Canonical Lie Group Geodesic Equations in Dimension Three

For each of the two and three-dimensional indecomposable Lie algebras the geodesic equations of the associated canonical Lie group connection are given. In each case a basis for the associated Lie algebra of symmetries is constructed and the corresponding Lie brackets are written down

Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A_{5,7}^{abc} to A_{18}^a. For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized

On computing joint invariants of vector fields

A constructive version of the Frobenius integrability theorem -- that can be programmed effectively -- is given. This is used in computing invariants of groups of low ranks and recover examples from a recent paper of Boyko, Patera and Popoyvich \cite{BPP}

Symmetries of the Canonical Geodesic Equations of Five-Dimensional Nilpotent Lie Algebras

In this paper, symmetries of the canonical geodesic equations of indecomposable nilpotent Lie groups of dimension five are constructed. For each case, the associated system of geodesics is provided. In addition, a basis for the associated Lie algebra of symmetries as well as the corresponding non-zero Lie brackets are listed and classified. This is a joint work with Ryad Ghanam and Gerard Thompson

How Human Resource Outsourcing Affects Organizational Learning in the Knowledge Economy

Adaptability and knowledge management, key elements of organizational learning, are critical to organizational success as a result of a fundamental shift towards a knowledge economy. HR outsourcing and the growth in contingent work can result in a significant loss in learning capital through a breakdown in the psychological contract. We explore how to preserve HR\u27s strategic role in facilitating organizational learning in the new outsourcing and offshoring context. The problem is compounded if outsourcing is introduced for cost control rather than strategic re-focusing reasons. We suggest that managers can positively influence the relationship between outsourcing and organizational learning through internal marketing tactics and enriched psychological contracts