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The logarithm of the derivative operator and higher spin algebras of \(W_{\infty{}}\) type. (English) Zbl 0765.35049
Summary: The authors use the notion of the logarithm of the derivative operator to describe \(W_ \infty\) type algebras as central extensions of the algebra of differential operators. They also provide closed formulae for the truncations of \(W_{1+\infty}\) to higher spin algebras with \(s\geq M\), for all \(M\geq 2\). The results are extended to matrix valued differential operators, introducing a logarithmic generalization of the Maurer-Cartan cocycle.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
58J70 Invariance and symmetry properties for PDEs on manifolds
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
35S05 Pseudodifferential operators as generalizations of partial differential operators
17B68 Virasoro and related algebras
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