Summary
The Mukunda conjecture has been proved and further appropriately extended to semi-simple symmetry groups by Guest, under the assumptions that the functions in the realized Lie algebra and furthermore the Hamiltonian are not constant on every nonempty open set of the phase space («regularity»). In this paper it is proved that any relization of a simple Lie algebra induces a regular realization of it on some nonempty open set of the phase space. Moreover, this implies that Guest’s assumptions can be weakened or can even be given up.
Riassunto
Da parte di Guest si è dimostrata l’ipotesi di Makunda e la si è estesa in maniera appropriata a gruppi di simmetria semisemplici, supponendo che sia le funzioni nell’algebra di Lie realizzata sia l’hamiltoniano non siano costanti in ogni insieme aperto non vuoto dello spazio delle fasi («regolarità»). In questo articolo si dimostra che ogni realizzazione di un’algebra semplice di Lie induce una sua realizzazione regolare in un certo insieme aperto non vuoto dello spazio delle fasi. Ciò tuttavia implica che le ipotesi di Guest possono essere indebolite o addirittura trascurate.
Реэюме
При определенных предположениях Гестом было докаэано и эатем распространено на полупростые группы симметрии, что функции в реалиэованной алгебре Ли и, следовательно, Гамильтониан не являются постоянными на любом непустом открытом множестве фаэового пространства («регулярность»). В зтой работе докаэывается, что любая реалиэация простой алгебры Ли индуцирует регулярную реалиэацию зтой алгебры на некотором непустом открытом множестве фаэового пространства. Более того, иэ зтого следует, что предположения Геста могут быть ослаблены или даже от них можно откаэаться.
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References
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F. Bunchaft:Lett. Nuovo Cimento,10, 149 (1972).
In general, realization on ℬ0(M) is less stringent, from a mathematical point of view, than realization on the Lie algebra ℬ(M) ofC ∞ differentiable functions. The kernel of the homomorphism ℬ(M) → χℋ(M), that is the constant functions, is unimportant for classical mechanics (4). For any given Hamiltonian vector field it corresponds to the arbitrariness in the zero level of the energy of a system (2).
N. Mukunda:Journ. Math. Phys.,8, 1069 (1957).
L. P. Eisenhart:Continuous Group of Transformations (New York, N.Y., 1963), p. 283.
P. B. Guest:Nuovo Cimento,61 A, 593 (1969).
See ref. (10), p. 596, footnote and recall that any extension of a semi-simple Lie algebrag is inessential (a well-known corollary of Levi-Malcev’s theorem) so that it becomes equivalent to consider realizations ofg into ℬ0(M) or into ℬ(M).
F. Brickell andR. S. Clark:Differentiable Manifolds (London, 1970), see Proposition 3.4.6 and its proof.
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Bunchaft, F. Mukunda’s conjecture and regularity on realizations of semi-simple lie algebras. Nuov Cim A 25, 110–116 (1975). https://doi.org/10.1007/BF02735613
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DOI: https://doi.org/10.1007/BF02735613