Generalized Cartan Calculus in general dimension

Understand

We develop the generalized Cartan Calculus for the groups $G=$SL$(2,\mathbb{R})\times\mathbb{R}^+$, SL$(5,\mathbb{R})$ and SO(5,5).

• They are the underlying algebraic structures of $d=9,7,6$ exceptional field theory, respectively.
• These algebraic identities are needed for the "tensor hierarchy" structure in exceptional field theory.
• The validity of Poincar\'e lemmas in this new differential geometry is also discussed.