Abelian fourfold of Mumford-type and Kuga-Satake varieties

Abstract

Let X be a complex abelian fourfold of Mumford-type and let V = H 1( X, ▪). The complex Mumford-Tate group of X is isogenous to SL(2) 3. We recover information about the Hodge structure of X using representations of the Lie algebras sl (2) 3 and sp (8) acting on V ⊗ ▪ ▪. Using these techniques we show that there is a Kuga-Satake variety A associated to X in such a way that A is isogenous to X 32.