Canonical realizations of classical lie algebras |
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Authors: | P Exner M Havlí?ek W Lassner |
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Institution: | (1) Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O.Box 79, Moscow, U.S.S.R.;(2) Nuclear Center at the Faculty of Mathematics and Physics of the Charles University, Prague |
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Abstract: | We construct sets of canonical realizations for all classical Lie algebras (A
n
,B
n
,C
n
,D
n
). These realizations depend ond parameters,d=1, 2, 3,...,n; all Casimir operators are realized by multiples of identity. For most of the real forms of these algebras we give sets of realizations which are, moreover, in well-defined sense skew-Hermitian. Further we study extremal cases of the presented realizations. The realizations with minimal numbers of canonical pairs are discussed from the point of view of general results concerning minimal realizations. On the other hand, a connection is found between our maximal realizations ofA
n
and the Gel'fand-Kirillov Conjecture.The authors would like to thank Prof. A.Uhlmann for his kind interest in this work. They are very grateful to Prof. A. A.Kirillov and Prof. D. P.Zhelobenko for helpful discussions and to Prof. J.Dixmier for his informative letter concerning the problem mentioned in Sect. 5.One of the authors (W. L.) thanks Prof. I.Úlehla for the hospitality at the Nuclear Center of the Charles University, Praha. |
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Keywords: | |
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