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The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in natural modules

Volume 123 / 2021

Jill Ecker, Martin Schlichenmaier Banach Center Publications 123 (2021), 141-174 MSC: Primary 17B56; Secondary 17B68, 17B65, 17B66, 14D15, 81R10, 81T40. DOI: 10.4064/bc123-6

Abstract

The main aim of this contribution is to compute the low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in the adjoint and the trivial module. The last section includes results for the general tensor densities modules, presented without proof. One of our main results is that the third algebraic cohomology of the Witt algebra with values in the adjoint module vanishes, while it is one-dimensional for the Virasoro algebra. The first and the second algebraic cohomology of the Witt and the Virasoro algebra with values in tensor densities modules vanish for almost all modules. In the case they do not vanish, we give explicit expressions for the generating cocycles. In our work, we consider algebraic cohomology and not only the sub-complex of continuous cohomology, meaning we do not put any continuity constraints on the cochains. Consequently, our results are independent of any choice of an underlying topology, and valid for any concrete realizations of the considered Lie algebras.

Authors

  • Jill EckerUniversity of Luxembourg
    Faculty of Science, Technology and Medicine
    Maison du Nombre, 6, Avenue de la Fonte
    L-4364 Esch-sur-Alzette, Luxembourg
    e-mail
  • Martin SchlichenmaierUniversity of Luxembourg
    Faculty of Science, Technology and Medicine
    Maison du Nombre, 6, Avenue de la Fonte
    L-4364 Esch-sur-Alzette, Luxembourg
    e-mail

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