Publication:
Some features of rank one real solvable cohomologically rigid lie algebras with a nilradical contracting onto the model filiform lie algebra Qn

Loading...
Thumbnail Image
Full text at PDC
Publication Date
2019
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
MDPI
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec(t) = (1, k, k +1, ..., n + k-3, n +2k -3) for k [mayor o igual] 2 are analyzed, with special emphasis on the resulting Lie algebras for which the second Chevalley cohomology space vanishes. From the detailed inspection of the values k [menor o igual] 5, some series of cohomologically rigid algebras for arbitrary values of k are determined.
Description
Keywords
Citation
1. Carles, R. Sur certaines classes d’orbites ouvertes dans les variétés d’algèbres de Lie. C. R. Acad. Sci. Paris 1981, 293 , 545–547. 2. Segal, I.E. A class of operators which are determined by groups. Duke Math. J. 1951, 18, 221–265. 3. Koszul, J.L. Homologie et cohomologie des algèbres de Lie. Bull. Soc. Math. Fr. 1950, 78, 65–127. 4. Fialowski, A. Deformations and contractions of algebraic structures. Proc. Steklov Inst. Math. 2014, 286, 240–252. 5. Gerstenhaber, M. On the deformations of rings and algebras. Ann. Math. 1964, 79, 59–103. 6. Verona, A. Introducere in Coomologia Algebrelor Lie; Editura Academiei Rep. Soc.: Bucharest, Romania, 1974. 7. Zusmanovich, P. A converse to the second Whitehead Lemma. J. Lie Theory 2008, 18, 295–299. 8. Carles, R. Déformations dans les schémas définis par les identités de Jacobi. Ann. Math. Blaise Pascal 1996, 3, 33–62. 9. 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. 10. Favre, G. Système des poids sur une algèbre de Lie nilpotente. Manuscr. Math. 1973, 9, 53–90. 11. 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. 12. Ancochea, J.M.; Campoamor-Stursberg, R. Rigidity-preserving and cohomology-decreasing extensions of solvable rigid Lie algebras. Linear Multilinear Algebra 2017, 66, 525–539. 13. Carles, R. Sur certaines classes d’algèbres de Lie rigides. Math. Ann. 1985, 272, 477–488. 14. 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. 15. Goze, M.; Ancochea, J.M. Algèbres de Lie rigides dont le nilradical est filiforme. C. R. Acad. Sci. Paris 1991, 312, 21–24. 16. Goze, M.; Ancochea, J.M. On the classification of rigid Lie algebras. J. Algebra 2001, 245, 68–91. [CrossRef] 17. Nijenhuis, A.; Richardson, R.W. Deformations of Lie algebra structures. J. Math. Mech. 1967, 17, 89–105. 18. Page, S.S. A characterization of rigid algebras. J. Lond. Math. Soc. 1970, 2, 237–240. 19. Rauch, G. Effacement et déformation. Ann. Inst. Fourier 1972, 22, 239–269. 20. Richardson, R.W. On the rigidity of semi-direct products of Lie algebras. Pac. J. Math. 1967, 22, 339–344. 21. Goze, M.; Ancochea, J.M. On the nonrationality of rigid Lie algebras. Proc. Am. Math. Soc. 1999, 127, 2611–2618. 22. Vergne, M. Cohomologie des algèbres de Lie nilpotentes. Application á l’étude de la variété des algébres de Lie nilpotentes. Bull. Soc. Math. France 1970, 98, 81–116. 23. Ancochea, J.M.; Goze, M. Algorithme de construction des algèbres de Lie rigides. Publ. Math. Univ. Paris VII 1989, 31, 285–306. 24. 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. 25. Mal’cev, A.I. Solvable Lie algebras. Izv. Akad. Nauk SSSR 1945, 9, 329–356. 26. Šnobl, L.; Winternitz, P. Classification and Identification of Lie Algebras; CRC Monograph Series; American Mathematical Society: Providence, RI, USA, 2014; Volume 33. 27. Carles, R. Sur la structure des algèbres de Lie rigides. Ann. Inst. Fourier 1984, 34, 65–82. [CrossRef] 28. Azcárraga, J.A.; Izquierdo, J.M. Lie Groups, Lie Algebras, Cohomology and Some Applications to Physics; Cambridge Univeisity Press: Cambridge, UK, 1995. 29. Campoamor-Stursberg, R. A comment concerning cohomology and invariants of Lie algebras with respect to contractions and deformations. Phys. Lett. A 2007, 362, 360–367. 30. Fialowski, A.; de Montigny, M. On deformations and contractions of Lie algebras. J. Phys. A Math. Gen. 2005, 38, 6335–6349. 31. Weimar-Woods, E. Contractions, generalized Inönü-Wigner contractions and deformations of finite-dimensional Lie algebras. Rev. Math. Phys. 2000, 12, 1505–1529. 32. Tolpigo, A.K. On the cohomology of parabolic Lie algebras. Mat. Zamet. 1972 12, 251–255. 33. Ancochea, J.M.; Campoamor-Stursberg, R.; Garcia Vergnolle, L.; Goze, M. Algèbres résolubles réelles algébriquement rigides. Monatsh. Math. 2007, 152, 187–195. 34. Ancochea, J.M.; Campoamor-Stursberg, R.; Garcia Vergnolle, L. Les algèbres de Lie résolubles rigides réelles ne sont pas nécessairement complètement résolubles. Linear Algebra Appl. 2006, 418, 657–664. 35. 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. 36. Bratzlavsky, F. Sur les algèbres admettant un tore d’automorphismes donné. J. Algebra 1974, 30, 305–316. 37. 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 2985, 300, 467–469. 38. Rim, D.S. Deformation of transitive Lie algebras. Ann. Math. 1966, 83, 339–357. 39. Dozias, J. Sur les dérivations des algèbres de Lie. C. R. Acad. Sci. Paris 1964, 259, 2748–2750. 40. Goze, M.; Hakimjanov, Yu. Sur les algèbres de Lie nilpotentes admettant un tore de dérivations. Manuscr. Math. 1994, 84, 115–124. 41. Grozman, P.; Leites, D. SuperLie and Problems (to Be) Solved with it, Preprint MPI-2003-39, MPIM-Bonn. Available online: www.mpim-bonn.mpg.de (accessed on 15 November 2018). 42. 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. 43. 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.
Collections