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 Mat. Zametki, 2005, Volume 77, Issue 1, Pages 67–79 (Mi mz2470)

Cohomology of solvable lie algebras and solvmanifolds

D. V. Millionshchikov

M. V. Lomonosov Moscow State University

Abstract: The cohomology $H^*_{\lambda\omega}(G/\Gamma,\mathbb C)$ of the de Rham complex $\Lambda^*(G/\Gamma)\otimes\mathbb C$ of a compact solvmanifold $G/\Gamma$ with deformed differential $d_{\lambda\omega}=d+\lambda\omega$, where $\omega$ is a closed 1-form, is studied. Such cohomologies naturally arise in Morse–Novikov theory. It is shown that, for any completely solvable Lie group $G$ containing a cocompact lattice $\Gamma\subset G$, the cohomology $H^*_{\lambda\omega}(G/\Gamma,\mathbb C)$ is isomorphic to the cohomology $H^*_{\lambda\omega}(\mathfrak g)$ of the tangent Lie algebra $\mathfrak g$ of the group $G$ with coefficients in the one-dimensional representation $\rho_{\lambda\omega}\colon\mathfrak g\to\mathbb K$ defined by $\rho_{\lambda\omega}(\xi)=\lambda\omega(\xi)$. Moreover, the cohomology $H^*_{\lambda\omega}(G/\Gamma,\mathbb C)$ is nontrivial if and only if $-\lambda[\omega]$ belongs to a finite subset $\widetilde\Omega_{\mathfrak g}$ of $H^1(G/\Gamma,\mathbb C)$ defined in terms of the Lie algebra $\mathfrak g$.

DOI: https://doi.org/10.4213/mzm2470

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English version:
Mathematical Notes, 2005, 77:1, 61–71

Bibliographic databases:

UDC: 515.1

Citation: D. V. Millionshchikov, “Cohomology of solvable lie algebras and solvmanifolds”, Mat. Zametki, 77:1 (2005), 67–79; Math. Notes, 77:1 (2005), 61–71

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz2470
• https://doi.org/10.4213/mzm2470
• http://mi.mathnet.ru/eng/mz/v77/i1/p67

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This publication is cited in the following articles:
1. Millionschikov D.V., “Multi-Valued Functionals, One-Forms and Deformed de Rham Complex”, Topology in Molecular Biology, Biological and Medical Physics, Biomedical Engineering, ed. MOnastyrsky M., Springer-Verlag Berlin, 2007, 189–208
2. Van Le H., Vanzura J., “Cohomology Theories on Locally Conformal Symplectic Manifolds”, Asian J. Math., 19:1 (2015), 45–82
3. Cagliero L., Tirao P., “Total cohomology of solvable Lie algebras and linear deformations”, Trans. Am. Math. Soc., 368:5 (2016), 3341–3358
4. Angella D., Otiman A., Tardini N., “Cohomologies of Locally Conformally Symplectic Manifolds and Solvmanifolds”, Ann. Glob. Anal. Geom., 53:1 (2018), 67–96
5. Andrada A., Origlia M., “Lattices in Almost Abelian Lie Groups With Locally Conformal Kahler Or Symplectic Structures”, Manuscr. Math., 155:3-4 (2018), 389–417
6. Otiman A., “Morse-Novikov Cohomology of Locally Conformally Kahler Surfaces”, Math. Z., 289:1-2 (2018), 605–628
7. Sawai H., “Examples of Solvmanifolds Without Lck Structures”, Complex Manifolds, 5:1 (2018), 103–110
8. Origlia M., “Locally Conformal Symplectic Structures on Lie Algebras of Type i and Their Solvmanifolds”, Forum Math., 31:3 (2019), 563–578
9. Andrada A., Origlia M., “Locally Conformally Kahler Solvmanifolds: a Survey”, Complex Manifolds, 6:1 (2019), 65–87
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