Central extensions, derivations, and automorphisms of semi-direct sums of the Witt algebra with its intermediate series modules
Lucas Buzaglo, Girish S. Vishwa
TL;DR
Using the internal grading of the Witt algebra, the paper completely classifies central extensions, derivations, and automorphisms for semi-direct sums of the Witt algebra with indecomposable intermediate series modules $A(\\lambda)$, $B(\\lambda)$ and the irreducible bounded module $\\widetilde{I}$. It provides explicit Lie and Leibniz 2-cocycles, describes automorphism groups as InnAut\\(\\mathcal{W}_X(\\lambda)\\) semi-direct product with explicit outer subgroups $G_X(\\lambda)$, and determines derivation spaces with clear decompositions into inner parts and finite-dimensional outer contributions, together with a universal Virasoro-type extension in relevant cases. A key technical tool is the internal grading, which reduces cohomology of $\\mathcal{W}_X(\\lambda)$ to degree-zero computations and yields a unified approach applicable to broader graded Witt-type algebras. The results advance the deformation and symmetry analysis of Witt-related semi-direct sums and have potential applications in mathematical physics where such algebras arise.
Abstract
Lie algebras formed via semi-direct sums of the Witt algebra $\text{Der}(\mathbb{C}[t,t^{-1}])$ 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.