×

zbMATH — the first resource for mathematics

Cyclic homology of differential operators, the Virasoro algebra and a \(q\)-analogue. (English) Zbl 0761.17020
In the algebraic setting the Virasoro-Bott cocycle, which defines the universal central extension of the Lie algebra of polynomial vector fields on the unit circle, can be obtained as the restriction of a 2-cocycle defined on the Lie algebra of algebraic differential operators.
In this paper this result is reproved using cyclic homology theory. The advantage of this method is that it can be directly generalized to a \(q\)-deformed setting, the algebra of \(q\)-difference operators on \(\mathbb C[x,x^{-1}]\). This leads to a one-parameter deformation of the Hochschild 1-cocycle, which corresponds to the Virasoro-Bott cocycle. Moreover in the case where \(q\) is a root of unity there arises an infinite family of linearly independent Hochschild 1-cocycles. All these cocycles are computed explicitly.
Reviewer: A. Čap (Wien)

MSC:
17B55 Homological methods in Lie (super)algebras
17B68 Virasoro and related algebras
17B66 Lie algebras of vector fields and related (super) algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bourbaki, N.: Algèbre, Chap. 10. Algèbre homologique. Paris: Masson 1980 · Zbl 0455.18010
[2] Brylinski, J.-L.: Some examples of Hochschild and cyclic homology. Lecture Notes in Mathematics, vol 1271, pp. 33–72. Berlin, Heidelberg, New York: Springer 1987
[3] Brylinski, J.-L., Getzler, E.: The homology of algebras of pseudo-differential symbols and the non-commutative residue.K-Theory1, 385–403 (1987) · Zbl 0646.58026 · doi:10.1007/BF00539624
[4] Getzler, E.: Cyclic homology and the Beilinson-Manin-Schechtman central extension. Proc. A.M.S.104, 729–734 (1988) · Zbl 0692.17007
[5] Kac, V.G., Peterson, D.H.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981) · Zbl 0469.22016 · doi:10.1073/pnas.78.6.3308
[6] Takhtadjian, L.A.: Noncommutative homology of quantum tori. Funkts. Anal. Pril.23, 75–76 (1989); English translation: Funct. Anal. Appl.23, 147–149 (1989)
[7] Wodzicki, M.: Cyclic homology of differential operators. Duke Math. J.54, 641–647 (1987) · Zbl 0635.18010 · doi:10.1215/S0012-7094-87-05426-3
[8] Beilinson, A.A., Schechtman, V.V.: Determinant bundles and Virasoro algebras. Commun. Math. Phys.118, 651–701 (1988) · Zbl 0665.17010 · doi:10.1007/BF01221114
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.