Abstract
We consider the question of existence of symplectic and Kahler structures on compact homogeneous spaces of solvable triangular Lie groups. The aim of the article is to clarify the situation with examples in this area. We prove that it is impossible to complete the construction of examples in the well-known article by Benson and Gordon on the structure of compact solvmanifolds with Kahler structure. We do this by proving the absence of lattices (and thereby a compact form) in the Lie groups of the above-mentioned article. We construct a new (similar) example for which, unlike the above examples, a compact form exists. We consider one class of solvable Lie groups, namely the class of almost abelian groups, and obtain for this class a characterization of those Lie groups for which the cohomologies of their compact solvmanifolds are isomorphic to the cohomologies of the corresponding Lie algebras. Until recently, such isomorphism has been known only for one specific class of Lie groups, namely the class of triangular groups. We give examples of new (almost abelian) Lie groups with such isomorphism.
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References
Hucklberry A., “Homogeneous pseudo-Kählerian manifolds,” Note Math., 10, No. 2, 337–342 (1990).
Hasegawa K., “Minimal models of nilmanifolds,” Proc. Amer. Math. Soc., 106, No. 1, 65–71 (1989).
Benson C. and Gordon C., “Kähler structures on compact solvmanifolds,” Proc. Amer. Math. Soc., 108, No. 4, 971–979 (1990).
Tralle A., “A note on solvable Lie groups without lattices and the Felix-Thomas models of fibrations,” arXiv: math.DG / 0009105 11 Sep. 2000.
Oprea J. and Tralle A., “Koszul-Sullivan models and the cohomology of certain solvmanifolds,” Ann. Glob. Anal. and Geom., 15, 347–360 (1997).
Vinberg È. B., Gorbatsevich V. V., and Onishchik A. L., “The structure of Lie groups and Lie algebras,” in: Contemporary Problems of Mathematics. Fundamental Trends. Vol. 41 [in Russian], VINITI, Moscow, 1990, pp. 5–258. (Itogi Nauki i Tekhniki.)
Hattori A., “Spectral sequence in the de Rham cohomology of fibre bundles,” J. Fac. Sci. Univ. Tokyo, 8, 289–331 (1960).
Saito M., “Sur certains groupes de Lie resolubles,” Sci. Pap. Coll. Gen. Ed. Univ. Tokyo, 7, I: No. 1, 1–11; II: No. 2, 157-168 (1957).
Raghunathan M. S., Discrete Subgroups of Lie Groups [Russian translation], Mir, Moscow (1977).
Gorbatsevich V. V., “On the lattices in Lie groups of type (E) and (R),” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 6, 56–63 (1975).
Tralle A. and Andrzejewski W., “On solvmanifolds and a conjecture of Benson and Gordon from the Hamiltonian viewpoint,” J. Lie Theory, 8, No. 2, 279–292 (1998).
Morozov V. V., “Classification of nilpotent Lie algebras of sixth order,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 4, 161–171 (1958).
Cox D., Little J. and O'Shea D., Ideals, Varieties, and Algorithms [Russian translation], Mir, Moscow (2000).
Gómez J., Jiménez-Merchán A., and Khakimdjanov Y., “Symplectic structures on the filiform Lie algebras,” J. Pure Appl. Algebra, 156, No. 1, 15–31 (2001).
Hasegawa K., “A class of compact Kählerian solvmanifolds and a general conjecture,” Geom. Dedicata, 78, 253–258 (1999).
Auslander L., “An exposition of the structure of solvmamifolds,” Bull. Amer. Math. Soc., 79, No. 2, 227–261 (1973).
Nomizu K., “On the cohomology of compact homogeneous spaces of nilpotent Lie groups,” Ann. Math., 59, 531–538 (1954).
Chebotarëv N. G., Fundamentals of the Galois Theory [in Russian], ONTI-GTTI, Moscow; Leningrad (1934).
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Gorbatsevich, V.V. Symplectic Structures and Cohomologies on Some Solvmanifolds. Siberian Mathematical Journal 44, 260–274 (2003). https://doi.org/10.1023/A:1022980703672
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DOI: https://doi.org/10.1023/A:1022980703672