Abstract
The universal central extension V of the Lie algebra W of vector fields on the circle with finite Fourier series is called by physicists the Virasoro algebra. However, W was known in characteristic p as the Witt algebra, in characteristic 0 as the infinite-dimensional Witt algebra, and Gelfand and Fuks [3] determined the second cohomology group of W with trivial coefficients, thereby describing V. The algebra V and an associated superalgebra play a fundamental role in the study of elementary particles [6].
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References
J. Dixmier, Enveloping Algebras. North-Holland, 1977.
B. L. Peigln and D. B. Fuks, Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra, Funct. Anal. Appl. 16(1982), no. 2, 47–63; English translation 114-126.
I. M. Gelfand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle, Funct. Anal. Appl. 2(1968), no. 4, 92–93; English translation 342-343.
V. G. Kac, Some problems on infinite dimensional Lie algebras and their representations, pp. 117-126 in Lie Algebras and Related Topics, Springer Lecture Notes no. 933, 1982.
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A. Meurman and L. J. Santharoubane, Cohomology and Harish-chandra modules over the Virasoro algebra. Preprint, MSRI.
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Kaplansky, I., Santharoubane, L.J. (1985). Harish-Chandra Modules Over the Virasoro Algebra. In: Kac, V. (eds) Infinite Dimensional Groups with Applications. Mathematical Sciences Research Institute Publications, vol 4. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1104-4_8
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DOI: https://doi.org/10.1007/978-1-4612-1104-4_8
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