so(3) ⊂ su(3) revisited
DOI:
https://doi.org/10.14311/AP.2024.64.0336Keywords:
lie algebra, realisation, representation, decomposition, embeddingAbstract
This paper reproduces the result of Elliot, namely that the irreducible finite dimensional representation of the Lie algebra su(3) of highest weight (m, n) is decomposed according to the embedding so(3) ⊂ su(3). First, a realisation (a representation in terms of vector fields) of the Lie algebra su(3) is constructed on a space of polynomials of three variables. The special polynomial basis of the representation space is given. In this basis, we find the highest weight vectors of the representation of the Lie subalgebra so(3) and in this way the representation space is decomposed to the direct sum of invariant subspaces. The process is illustrated by the example of the decomposition of the representation of highest weight (2, 2). As an additional result, the generating function of the decomposition is given.
Downloads
References
J. P. Elliott. Collective motion in the nuclear shell model. I. Classification schemes for states of mixed configurations. In Proceedings of the Royal Society A, vol. 245, pp. 128–145. 1958. https://doi.org/10.1098/rspa.1958.0072
V. N. Tolstoy. SU(3) symmetry for orbital angular momentum and method of extremal projection operators. Physics of Atomic Nuclei 69(6):1058–1084, 2006. https://doi.org/10.1134/S1063778806060160
M. Moshinsky, J. Patera, R. T. Sharp, P. Winternitz. Everything you always wanted to know about SU(3) ⊃ O(3). Annals of Physics 95(1):139–169, 1975. https://doi.org/10.1016/0003-4916(75)90048-2
R. M. Asherova, Y. F. Smirnov, B. N. Tolstoi. Projection operators for simple Lie groups. II. General scheme for constructing lowering operators. The groups SU(n). Theoretical and Mathematical Physics 15(1):392–401, 1973. https://doi.org/10.1007/BF01028268
R. M. Asherova, Y. F. Smirnov. On asymptotic properties of a quantum number ω in a system with SU(3) symmetry. Reports on Mathematical Physics 4(2):83–95, 1973. https://doi.org/10.1016/0034-4877(73)90015-3
T. Cerquetelli, N. Ciccoli, M. C. Nucci. Four dimensional Lie symmetry algebras and fourth order ordinary differential equations. Journal of Nonlinear Mathematical Physics 9:24–35, 2002. https://doi.org/10.2991/jnmp.2002.9.s2.3
R. M. Edelstein, K. S. Govinder, F. M. Mahomed. Solution of ordinary differential equations via nonlocal transformations. Journal of Physics A: Mathematical and General 34(6):1141–1152, 2001. https://doi.org/10.1088/0305-4470/34/6/306
M. Molati, F. M. Mahomed, C. Wafo Soh. A group classification of a system of partial differential equations modeling flow in collapsible tubes. Journal of Nonlinear Mathematical Physics 16:179–208, 2009. https://doi.org/10.1142/S1402925109000406
O. O. Vaneeva, R. O. Popovych, C. Sophocleous. Extended symmetry analysis of two-dimensional degenerate Burgers equation. Journal of Geometry and Physics 169:104336, 2021. https://doi.org/10.1016/j.geomphys.2021.104336
G. I. Kruchkovich. Lektsii po gruppam dvizhenii. Vyp. 1. [In Russian; Lectures on groups of motions. No. 1]. Erevan University, Erevan, 1977.
W. M. Boothby. A transitivity problem from control theory. Journal of Differential Equations 17(2):296–307, 1975. https://doi.org/10.1016/0022-0396(75)90045-5
A. Bourlioux, C. Cyr-Gagnon, P. Winternitz. Difference schemes with point symmetries and their numerical tests. Journal of Physics A: Mathematical and General 39(22):6877–6896, 2006. https://doi.org/10.1088/0305-4470/39/22/006
R. O. Popovych, V. M. Boyko, M. O. Nesterenko, M. W. Lutfullin. Realizations of real low-dimensional Lie algebras. Journal of Physics A: Mathematical and General 36(26):7337–7360, 2003. https://doi.org/10.1088/0305-4470/36/26/309
A. Morozov, M. Reva, N. Tselousov, Y. Zenkevich. Polynomial representations of classical Lie algebras and flag varieties. Physics Letters B 831:137193, 2022. https://doi.org/10.1016/j.physletb.2022.137193
M. Havlíček, W. Lassner. Canonical realizations of the Lie algebras gl(n,R) and sl(n,R). I. Formulae and classification. Reports on Mathematical Physics 8(3):391–399, 1975. https://doi.org/10.1016/0034-4877(75)90081-6
J. E. Humphreys. Introduction to Lie algebras and representation theory, vol. 9 of Graduate Texts in Mathematics. Springer-Verlag New York, USA, 1972. https://doi.org/10.1007/978-1-4612-6398-2
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Čestmír Burdík, Severin Pošta, Erik Rapp
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
Accepted 2024-05-02
Published 2024-09-08
