#### For citation:

Volokitina E. Y. Cohomology of the Lie Algebra of Vector Fields on Some One-dimensional Orbifold. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2013, vol. 13, iss. 3, pp. 14-28. DOI: 10.18500/1816-9791-2013-13-3-14-28

# Cohomology of the Lie Algebra of Vector Fields on Some One-dimensional Orbifold

I. M. Gelfand and D. B. Fuchs have proved that the cohomology algebra of the Lie algebra of vector fields on the unit circle is isomorphic to the tensor product of the polynomial ring with one generator of degree two and the exterior algebra with one generator of degree three. In the present paper the cohomology of the Lie algebra of vector fields on the one-dimensional orbifold S1/Z2 are studied. S1/Z2 is the orbit space under the Z2 group action on the unit circle by reflection in the Ox axis. It has been proved that the cohomology algebra of the Lie algebra of vector fields on the orbifold is isomorphic to the tensor product of the exterior algebra with two generators of degree one and the polynomial ring with one generator of degree two. To prove this result author used the Gelfand–Fuchs method with some modifications

- Volokitina Е. Y. Cohomology of Lie algebra of vector fields on S1/Z2. Izv. Sarat. Univ. N.S. Ser. Math. Mech. Inform., 2012. vol. 12, iss. 1, pp. 8–15 (in Russian).
- Gelfand I. M., Fuks D. B. Cohomologies of Lie algebra of tangential vector fields of a smooth manifold. Functional Analysis and Its Applications, 1969, vol. 3, iss. 3, pp. 194–210.
- Schwartz L. The´ orie des distributiones. Paris, Hermann, 1951, 418 p.
- Gelfand I. M., Vilenkin N. Ja. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. New York, Academic Press, 1964, 384 p. (Rus. ed.: Gelfand I. M., Vilenkin N. Ja. Nekotorye primeneniia garmonicheskogo analiza. Osnashchennye gil’bertovy prostranstva. Moscow, Fizmatgiz, 1961, 472 p.)
- Gelfand I. M., Shilov G. E. Prostranstva osnovnykh i obobshchennykh funktsii [Spaces of test and generalized functions]. Moscow, Fizmatgiz, 1958, 308 p. (in Russian).
- Rham G. de. Differentsiruemye mnogoobraziia [Differentiable manifolds]. Moscow, Izdatelstvo inostrannoj literatury, 1956, 250 p. (in Russian).
- Godeman R. Algebraicheskaia topologiia i teoriia puchkov [Algebraic topology and theory of sheaves]. Moscow, Izdatelstvo inostrannoj literatury, 1961, 320 p. (in Russian). 8. Fomenko A. T., Fuks D. B. Kurs gomotopicheskoi topologii [A course in homotopy topology]. Moscow, Nauka, 1989, 528 p. (in Russian).

- 1143 reads