Closed elliptic structures on compact semisimple Lie groups

In this work, we prove that, under a topological condition, the cohomology associated with left-invariant elliptic structures on compact semisimple Lie groups can be computed using only left-invariant forms. This reduces the analytical problem to a purely algebraic one, while also providing a generalization of the classic works of Chevalley and Eilenberg [Trans. Amer. Math. Soc. 63 (1948), 85–124] on the de Rham cohomology of compact Lie groups and of Pittie [Proc. Indian Acad. Sci. Math. Sci. 98 (1988), 117–152] on the Dolbeault cohomology of compact semisimple Lie groups to the context of elliptic structures. We use spectral sequences as our primary tool, which facilitates the construction of an isomorphism between the left-invariant differential complex and the usual differential complex.