Geodesic
Classification of four-dimensional transitive local Lie groups of transformations of the space $R\sp 4$ and their two-point invariants
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2008), pp. 29-42

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In the theory of physical structures the classification of metric functions (both on a single set and on two ones) plays an important role. A metric function represents a two-point invariant of a certain local Lie transformation group. Moreover, one can uniquely restore this group with the help of the invariance condition. According to this theorem, in order to find all metric functions, it suffices to construct the complete classification of local Lie transformation groups. In this paper we classify Lie algebras of simply transitive local Lie groups of local transformations of a four-dimensional space, and then we define metric functions. The obtained results admit application in physics, in particular, in thermodynamics.
Keywords: a Lie algebra, a simply transitive transformation group, basis operators, a metric function, a physical structure. 
@article{IVM_2008_6_a3,
     author = {V. A. Kyrov},
     title = {Classification of four-dimensional transitive local {Lie} groups of transformations of the space $R\sp 4$ and their two-point invariants},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {29--42},
     publisher = {mathdoc},
     number = {6},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2008_6_a3/}
}
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V. A. Kyrov. Classification of four-dimensional transitive local Lie groups of transformations of the space $R\sp 4$ and their two-point invariants. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2008), pp. 29-42. http://geodesic.mathdoc.fr/item/IVM_2008_6_a3/