Contraction Admissible Pairs of Complex Six-Dimensional Nilpotent Lie Algebras
All possible pairs of complex six-dimensional nilpotent Lie algebras are considered and necessary contraction conditions are verified. The complete set of the Lie algebra couples that do not admit contraction is obtained.
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Abstract All possible pairs of complex six-dimensional nilpotent Lie algebras are considered and necessary contraction conditions are verified. The complete set of the Lie algebra couples that do not admit contraction is obtained.
1 Introduction Notion of contraction originates from the work of I. Segal [1] and later in the work [2] of E. Inonu and E. Wigner it was shown that different physical theories are connected by the contractions of their underlying symmetry algebras. There exist two parallel branches of scientists studying contractions in modern science: the first branch is “algebraical”, they mainly study the varieties of Lie algebras by means of deformations and orbit closures (degenerations); another one is “physical”, it deals with the limit processes between different theories and with the applications of contractions and deformations to physics, e.g. quantization, uncoupling of the coupled systems, etc. Both problems of description of all possible contractions of a fixed Lie algebra or description of all contractions of Lie algebras of a fixed dimension are rather complicated (e.g., up to now the complete classification of contractions is known only for dimensions not greater then four [3], or for some subsets). Nevertheless, some sets of Lie algebras are closed with respect to contractions and can be studied independently. One of the closed sets is the set of nilpotent Lie algebras of a fixed dimension. In this paper we will deal with complex six-dimensional nilpotent Lie algebras. Note, that complex six-dimensional nilpotent Lie algebras have been already considered M. Nesterenko Department of Applied Research, Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kyiv-4, Ukraine e-mail: [email protected] S. Posta (B) Faculty of Nuclear Sciences and Physical Engineering, Department of Mathematics, Czech Technical University in Prague, 13 Trojanova Str., 120 00 Prague, Czech Republic e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2016 V. Dobrev (ed.), Lie Theory and Its Applications in Physics, Springer Proceedings in Mathematics & Statistics 191, DOI 10.1007/978-981-10-2636-2_41
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by Seeley [4], but this paper contains a number of misprints and mistakes, in particular, for the algebras 12346C and 1246 from classification by Seeley the Jacobi identity is not valid, and there is no contraction from the algebra 246C to the algebra 13 + 13 that is indicated in [4]. So we would like to correct and enhance the result as far as it forms a base for the proof of the statement that all contractions of nilpotent Lie algebras of dimensions up to six are equivalent to the generalized Inonu–Wigner contractions. The effective method that allows one to study all inequivalent contractions of some closed family of Lie algebras consists of two main steps: (i) to exclude all the pairs of Lie algebras that do not admit any contraction; (ii) to construct explicitly the contraction matrix for the rest of the pairs. We will focus on the step
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