African Diaspora Journal of Mathematics

KV-Cohomology and Differential Geometry of Affinely Flat Manifolds. Information Geometry

P. M. Byande, F. Ngakeu, M. Nguiffo Boyom, and R. Wolak

Full-text: Open access

Abstract

This paper is devoted to the socalled twisted cohomology of Koszul-Vinberg algebras. We discuss relationships between the twisted cohomology of Koszul-Vinberg algebras and Chevalley-Eilenberg cohomology of the commutator algebra of these algebras. We also discuss some geometry applications of these relationships. For instance we obtain some homological criteria for hyberbolicity and for completeness of locally flat manifolds. We also discuss some topics which are related to twisted cohomology. In particular, we use some techniques of information geometry to discuss canonical representations of locally flat connections.

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 14, Number 2 (2012), 197-226.

Dates
First available in Project Euclid: 31 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1375293546

Mathematical Reviews number (MathSciNet)
MR3093244

Zentralblatt MATH identifier
1297.53020

Subjects
Primary: 18Gxx: Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx] 17D99: None of the above, but in this section 53B15: Other connections 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords
$KV$-algebras twisted $KV$-modules twisted $KV$-cohomology twisted $KV$-homology Spectral sequence locally flat manifolds

Citation

Nguiffo Boyom, M.; Ngakeu, F.; Byande, P. M.; Wolak, R. KV-Cohomology and Differential Geometry of Affinely Flat Manifolds. Information Geometry. Afr. Diaspora J. Math. (N.S.) 14 (2012), no. 2, 197--226. https://projecteuclid.org/euclid.adjm/1375293546


Export citation

References

  • S-I Amari and H Nagaoka, Methods of information geometry, Translations of Mathematical Monographs, AMS-OXFORD University Press, vol 191.
  • O. Baues and W. M. Goldman, Is the deformation space of complete affine structures on the 2-torus smooth. Contemporary Mathematics, Vol 389 (2005) 69-89.
  • F. Chapoton and M. Livernet, Pre-Lie algebras and the root tree operads, Internat. Math. Res. 8 (2001), 395-408.
  • Chevalley Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Amer. Math. Society 63 (1948), 85-124.
  • P. Delanoé, Remarques sur les variétés localement hessiennes, Osaka J. Math. 26 (1989), 65-69.
  • A. Dzhumadil'daev, Cohomologies and Deformations of Right-Symmetric Algebras, Journal of mathematical sciences, vol. 93, No.6, (1999), 836-876.
  • M. Gerstenhaber, On deformations of Rings and Algebras, Ann. of Math. Vol. 79, No 1, (1964), 59-103.
  • M. Gromov, In a Search for a Structure, Part 1: Entropy. Proc ECM6, Krakow 2012....
  • G. Hochschild, On the Cohomology Groups of an Associative Algebra, Annals of Mathematics, vol. 46, No 1, (1945) 58-67.
  • G. Hochschild and J-P. Serre, Cohomology of Lie algebras, Annals of Mathematics, Vol. 57, No. 3, (1953)591-603.
  • S. Kobayashi and K. Nomizu, Foundations of Differential Geometry,vol I, 1963 and vol II, 1969, Interscience Publishers.
  • M. Kontsevich, Deformation Quantization of Poisson manifolds, Lett. Math. Phys. Vol 66, No 3, (2003), 157-216.
  • J-L. Koszul, Déformation des connexions localement plates, Ann. Inst. Fourier, No 18, (1968)103-114.
  • J-L Koszul, Homologie des complexes de formes différentielles d'ordre supérieur, Ann. Scient. Ec. Norm. Sup. Vol. 7 (1974),139-153.
  • J-L. Koszul, Homologie et cohomologie des algébres de Lie, Bull. Soc. Math. France, 78 (1950), 1-63.
  • J. McCleary, A User's Guide to Spectral Sequences. Cambridge University Press, 2001.
  • M. Nguiffo Boyom, Algèbres à associateurs symétriques et algèbres de Lie reductibles, thèse de doctorat, T. cycle, Univ. Grenoble, 1968.
  • M. Nguiffo Boyom, Sur les structures affines homotopes à zéro des groupes de Lie, J. Differential Geom. 31:3 (1990), 859-911.
  • M. Nguiffo Boyom, Structures localement plates isotropes des groupes de Lie, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 91-131.
  • M. Nguiffo Boyom, Structures localement plates dans certaines variétés symplectiques, Math. Scand. 76:1 (1995), 61-84.
  • M. Nguiffo Boyom, Cohomology of Koszul-Vinberg algebroids and Poisson manifolds $I$, Banach Center Publ. 54, Polish Acad. Sci., Warsaw, 2001,99-110.
  • M. Nguiffo Boyom and R. A. Wolak, Affine structures and KV-cohomology, J. Geom. Phys. 42:4 (2002), 307-317.
  • M. Nguiffo Boyom and R. Wolak, Local structure of Koszul-Vinberg and of Lie algebroids, Bull. Sci. Math. 128:6 (2004),467-479.
  • M. Nguiffo Boyom, KV-cohomology of Koszul-Vinberg algebroids and Poisson manifolds, Internat. J. Math. 16:9 (2005), 1033-1061.
  • M. Nguiffo Boyom, The Cohomology of Koszul-Vinberg Algebras, Pacific Journal of Mathematics, Vol.225, No. 1, (2006), 119-153.
  • M. Nguiffo Boyom, Some Lagrangian Invariants of Symplectic Manifolds, Geometry and Topology of Manifolds, Banach Center Publications, vol. 76 (2007), 515-525.
  • M. Nguiffo Boyom, Réductions Kahlériennes dans les groupes de Lie Résolubles et Applications, Osaka J. Math. 47 (2010), 237-283.
  • M. Nguiffo Boyom, Convexité local dans l'ensemble des connxions de Chensov localement plate. Workshop Information geometry and optimal transport, IHP 2010.....
  • M. Nguiffo Boyom and P. M. Byande, The KV cohomology in information geometry. Matrix information geometry, 69-92, Springer, Heidelberg 2013.
  • M. Nguiffo Boyom, Convexit'e locale des connexions de Chentsov localement plates.GDR MSPCS Symposium Institut Henri Poincaré. Workshop WWW. Ceremade dauphine.
  • M. Nguiffo Boyom and F. Ngakeu, Cohomology and Homology of Abelian Groups Graded Koszul-Vinberg Algebras, Manuscript 2008.
  • A. Nijenhuis, Sur une classe de propriétés communes à quelques types différents d'algèbres,Enseignement Math. (2) 14 (1968), 225-277.
  • H. Shima, The Geometry of Hessian Structures, World Scientific, 2007.
  • A. Tsemo, Dynamique des variétés affines, J. London Math. Soc. 63,(2001), 469-486.