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About the origins of the Virasoro cocycle. (Sur les origines du cocycle de Virasoro.) (French) Zbl 1211.81092

The paper presents a short and concise historical introduction to Virasoro cocycles and other related notions. To do so, the author goes back to the origins of their appearance in physics and their theoretical formalization and link with other topics, which are revisited due to their relation. In this way, the main goal of the paper is to show the origins and motivations for the appearance of Virasoro cocycles, as a physical-mathematical object, but without being comprehensive.
In fact, the paper covers a wide range of topics where Virasoro cocycles have importance and applications. To give a theoretical definition of Virasoro algebra, the author recalls its construction starting from the Lie algebra of tangent vector fields of the circle \(S^1\).
As starting point of Virasoro cocycle, the author expounds two sources or roots: the first is based on Schur’s multipliers and belongs to the field of algebra; whereas the second is related to physics and Schwinger’s terms, corresponding to anomalies in quantum mechanics. In common, both origins share the existence of a universal cocycle, which rules central extensions of Jacobi matrices or restricted linear groups. In fact, this cocycle is not only behind the notion of Virasoro cocycle, but of other related concepts like gauge groups or Kac-Moody algebras, the latter being inseparable from Virasoro algebras.
The article expounds the relation of Virasoro algebra with other notions, like: (universal) central extensions of Lie groups and algebras (with Schur multipliers as the first case studied in the literature); Lie algebra cohomology (in particular, for Lie algebras of formal and vector fields); Witt algebra (i.e., Virasoro algebra in fields of characteristic zero); Courant algebras (in Yang-Mills fields), Kac-Moody algebras; solving soliton equations; or the theory of anomalies in quantum mechanics (taking anomalies as cohomology classes in the context of Fock spaces).

MSC:

81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B68 Virasoro and related algebras
81-03 History of quantum theory
17B56 Cohomology of Lie (super)algebras
01A60 History of mathematics in the 20th century
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E66 Analysis on and representations of infinite-dimensional Lie groups
81T50 Anomalies in quantum field theory
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