Publication:
Computer-Aided Analysis of Solvable Rigid Lie Algebras with a Given Eigenvalue Spectrum

Loading...
Thumbnail Image
Full text at PDC
Publication Date
2022-08-30
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
MDPI
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
With the help of symbolic computer packages, the study of the cohomological rigidity of real solvable Lie algebras of rank one with a maximal torus of derivations t and the eigenvalue spectrum spec(t)=(1,k,k+1,⋯,n+k−2) initiated in a previous work is continued for arbitrary values k≥2, obtaining new hierarchies of solvable rigid Lie algebras.
Description
UCM subjects
Unesco subjects
Keywords
Citation
1. Frölicher, A.; Nijenhuis, A. A theorem on stability of complex structures. Proc. Natl. Acad. Sci. USA 1957, 3, 239–241. 2. Kodaira, K.; Spencer, D.C. On deformations of complex analytic structures I. Ann. Math. 1958, 67, 328–401. 3. Carles, R. Sur certaines classes d’orbites ouvertes dans les variétés d’algèbres de Lie. C. R. Acad. Sci. Paris 1981, 293, 545–547. 4. Carles, R. Déformations et éléments nilpotents dans les schémas définis par les identités de Jacobi. C. R. Acad. Sci. Paris 1991, 312, 671–674. 5. Carles, R.; Márquez García, M.C. Different methods for the study of obstructions in the schemes of Jacobi. Ann. Inst. Fourier 2011, 61, 453–490. 6. Richardson, R.W. On the rigidity of semi-direct products of Lie algebras. Pac. J. Math. 1967, 22, 329–344. 7. Page, S.S. A characterization of rigid algebras. J. Lond. Math. Soc. 1970, 2, 237–240. 8. Chevalley, C.; Eilenberg, S. Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 1948, 63, 85–124. 9. Tolpygo, A.K. On the cohomology of parabolic Lie algebras. Mat. Zamet. 1972, 12, 251–255. 10. Rauch, G. Effacement et déformation. Ann. Inst. Fourier 1972, 22, 239–269. 11. Rim, D.S. Deformation of transitive Lie algebras. Ann. Math. 1966, 83, 339–357. 12. Tôgô, S. Outer derivations of Lie algebras. Trans. Amer. Math. Soc. 1967, 128, 264–276. 13. Bratzlavsky, F. Sur les algèbres admettant un tore d’automorphismes donné. J. Algebra 1974, 30, 305–316. 14. Favre, G. Système des poids sur une algèbre de Lie nilpotente. Manuscr. Math. 1973, 9, 53–90. 15. Ancochea, J.M.; Goze, M. Algorithme de construction des algèbres de Lie rigides. Publ. Math. Univ. Paris VII 1989, 31, 277–298. 16. Goze, M. Critères cohomologiques pour la rigidité de lois algébriques. Bull. Soc. Math. Belg. 1991, 43, 33–42. 17. Ancochea, J.M.; Goze, M. Le rang du système linéaire des racines d’une algèbre de Lie rigide résoluble complexe. Commun. Algebra 1992, 20, 875–887. 18. Nelson, E. Internal set theory: A new approach to nonstandard analysis. Bull. Amer. Math. Soc. 1977, 83, 1165–1198. 19. Goze, M.; Ancochea, J.M. Algèbres de Lie rigides. Indag. Math. 1985, 47, 397–415. 20. Goze, M.; Ancochea, J.M. Algèbres de Lie rigides dont le nilradical est filiforme. C. R. Acad. Sci. Paris 1991, 312, 21–24. 21. Goze, M.; Ancochea, J.M. On the classification of rigid Lie algebras. J. Algebra 2001, 245, 68–91. 22. Carles, R. Sur certaines classes d’algèbres de Lie rigides. Math. Ann. 1985, 272, 477–488. 23. Ancochea, J.M.; Campoamor-Stursberg, R. Classification of solvable real rigid Lie algebras with a nilradical of dimension n ≤ 6. Linear Algebra Appl. 2015, 451, 54–75. 24. Bouarroudj, S.; Navarro, R.M. Cohomologically rigid solvable Lie superalgebras with model filiform and model nilpotent nilradical. Commun. Algebra 2021, 49, 5061–5072. 25. Goze, M.; Hakimjanov, Y. Sur les algèbres de Lie nilpotentes admettant un tore de dérivations. Manuscr. Math. 1994, 84, 115–124. 26. Ancochea, J.M.; Campoamor-Stursberg, R. Cohomologically rigid solvable real Lie algebras with a nilradical of arbitrary characteristic sequence. Linear Algebra Appl. 2016, 488, 135–147. 27. Ancochea, J.M.; Campoamor-Stursberg, R. Rigidity-preserving and cohomology-decreasing extensions of solvable rigid Lie algebras. Linear Multilinear Algebra 2017, 66, 525–539. 28. Bérubé, D.; de Montigny, M. The computer calculation of graded contractions of Lie algebras and their representations. Comput. Phys. Commun. 1993, 76, 389–410. 29. Grozman, P.; Leites, D. MATHEMATICA aided study of Lie algebras and their cohomology. From supergravity to ballbearings and magnetic hydrodynamics. Trans. Eng. Sci. 1997, 15, 185–192. 30. Ancochea, J.M.; Campoamor-Stursberg, R.; Oviaño García, F. New examples of rank one solvable real rigid Lie algebras possessing a nonvanishing Chevalley cohomology. Appl. Math. Comput. 2018, 339, 431–440. 31. Carles, R. Sur la structure des algèbres de Lie rigides. Ann. Inst. Fourier 1984, 34, 65–82. 32. Carles, R. Un exemple d’algèbres de Lie résolubles rigides, au deuxième groupe de cohomologie non nul et pour lesquelles l’application quadratique de D.S. Rim est injective. C. R. Acad. Sci. Paris 1985, 300, 467–469. 33. Carles, R. Sur la cohomologie d’une nouvelle classe d’algèbres de Lie qui généralisent les sous-algèbres de Borel. J. Algebra 1993, 154, 310–334. 34. Carles, R.; Petit, T. Versal deformations and versality in central extensions of Jacobi schemes. Transform. Groups 2009, 14, 287–317. 35. Campoamor-Stursberg, R.; Oviaño, F. Algorithmic construction of solvable rigid Lie algebras determined by generating functions. Linear Multilinear Algebra 2022, 70, 280–296. 36. Campoamor-Stursberg, R.; Oviaño, F. Some features of rank one real solvable cohomologically rigid Lie algebras with a nilradical contracting onto the model filiform Lie algebra Qn. Axioms 2019, 8, 10. 37. Mal’cev, A.I. Solvable Lie algebras. Izv. Akad. Nauk SSSR 1945, 9, 329–356. 38. Šnobl, L.; Winternitz, P. Classification and Identification of Lie Algebras; CRC Monograph Series; American Mathematical Society: Providence, RI, USA, 2014; Volume 33. 39. Fialowski, A. Deformations and contractions of algebraic structures. Proc. Steklov Inst. Math. 2014, 286, 240–252. 40. Rauch, G. Variations d’algébres de Lie résolubles. C. R. Acad. Sci. Paris 1969, 269, 285–288. 41. Murray, F.J. Perturbation theory and Lie algebras. J. Math. Phys. 1962, 3, 89–105. 42. Gerstenhaber, M. On the deformations of rings and algebras. Ann. Math. 1964, 79, 59–103. 43. Nijenhuis, A.; Richardson, R.W. Cohomology and deformations of algebraic structures. Bull. Amer. Math. Soc. 1964, 70, 406–411. 44. Boyer, C.P. Deformations of Lie algebras and groups and their applications. Rev. Mex. Fis. 1974, 23, 99–122. 45. Grunewald, F.; O’Halloran, J. Deformations of Lie algebras. J. Algebra 1993, 162, 210–224. 46. Hochschild, G.; Serre, J.P. Cohomology of Lie algebras. Annals Math. 1953, 57, 591–603.
Collections