Journal of Lie Theory 35 (2025), No. 3, 455--506
Copyright Heldermann Verlag 2025
Central Extensions, Derivations, and Automorphisms of Semi-Direct Sums of the Witt Algebra with its Intermediate Series Modules
Lucas Buzaglo
Department of Mathematics, University of California at San Diego, La Jolla, U.S.A.
lbuzaglo@ucsd.edu
Girish S. Vishwa
Maxwell Institute and School of Mathematics, The University of Edinburgh, Scotland, United Kingdom
G.S.Vishwa@sms.ed.ac.uk
[Abstract-pdf]
Lie algebras formed via semi-direct sums of the Witt algebra $\mathcal{W} = {\rm Der}(\mathbb{C}[t,t^{-1}])$ and its modules have become increasingly prominent in both physics and mathematics in recent years. In this paper, we complete the study of (Leibniz) central extensions, derivations and automorphisms of the Lie algebras formed from the semi-direct sum of the Witt algebra and its indecomposable intermediate series modules (that is, graded modules with one-dimensional graded components). Our techniques exploit the internal grading of the Witt algebra, which can be applied to a wider class of graded Lie algebras.
Keywords: Witt algebra, Virasoro algebra, intermediate series module, tensor density module, central extension, derivation, automorphism, Lie algebra cohomology, Leibniz cohomology.
MSC: 17B40, 17B56; 17B65, 17B68.
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