PASCAL programs for identification of Lie algebras. III: Levi decomposition and canonical basis. These are the last parts of a program package based on PASCAL for the identification of a Lie algebra L (given by its structure constants) by computer means. The program LEVI (5100 lines) calculates by a recursive algorithm the semisimple part S(L), of L, together with the radical R(L) (determined by RADICAL, a program presented earlier) there is then realized a Levi decomposition of L. The program CANONIK (10300 lines) uses the parts RADICAL, SPLIT (also presented earlier) and LEVI to present a given Lie algebra with a standardized basis and gives possibilities for its comparison with other Lie algebras. Examples of Lie algebras with canonical bases of dim=10 are given, for the theory of applied algorithms see also Zbl 0668.17004. With respect to the examples and informations the whole package presented by the authors seems to be very effective and useful.

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  1. Kanatchikov, Igor V.: Schrödinger wave functional in quantum Yang-Mills theory from precanonical quantization (2018)
  2. Fisher, David J.; Gray, Robert J.; Hydon, Peter E.: Automorphisms of real Lie algebras of dimension five or less (2013)
  3. Šnobl, L.; Winternitz, P.: Solvable Lie algebras with Borel nilradicals (2012)
  4. Nikolayevsky, Y.: Einstein solvmanifolds and the pre-Einstein derivation (2011)
  5. Hrivnák, Jiří; Novotný, Petr: Twisted cocycles of Lie algebras and corresponding invariant functions (2009)
  6. Schwarz, Fritz: Algorithmic Lie theory for solving ordinary differential equations (2008)
  7. Šnobl, L.; Winternitz, P.: A class of solvable Lie algebras and their Casimir invariants (2005)
  8. Ndogmo, J. C.: Invariants of a semi-direct sum of Lie algebras (2004)
  9. Winternitz, Pavel: Lie groups, singularities and solutions of nonlinear partial differential equations (2003)
  10. Agaoka, Yoshio: An algorithm to determine the isomorphism classes of 4-dimensional complex Lie algebras (2002)
  11. Bracken, P.; Grundland, A. M.: Symmetry properties and explicit solutions of the generalized Weierstrass system (2001)
  12. Clarkson, Peter A.; Winternitz, Pavel: Symmetry reduction and exact solutions of nonlinear partial differential equations (1999)
  13. de Graaf, W. A.: Using Cartan subalgebras to calculate nilradicals and Levi subalgebras of Lie algebras (1999)
  14. MacCallum, M. A. H.: On the classification of the real four-dimensional Lie algebras (1999)
  15. de Graaf, W. A.: An algorithm for the decomposition of semisimple Lie algebras (1997)
  16. de Graaf, W. A.: Constructing faithful matrix representations of Lie algebras (1997)
  17. De Graaf, W. A.; Ivanyos, G.; Küronya, A.; Rónyai, L.: Computing Levi decompositions in Lie algebras (1997)
  18. Ait Abdelmalek, M.; Leng, X.; Patera, J.; Winternitz, P.: Grading refinements in the contractions of Lie algebras and their invariants (1996)
  19. De Graaf, Willem; Ivanyos, Gábor; Rónyai, Lajos: Computing Cartan subalgebras of Lie algebras (1996)
  20. Reid, G. J.; Lisle, I. G.; Boulton, A.; Wittkopf, A. D.: Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs (1992)

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