Invariant differential on a group scheme

An invariant differential on a group scheme is a differential one-form preserved by every left translation. For a smooth group scheme $G$ over a field $k$, translation identifiesThus a covector at the identity extends uniquely to a left-invariant form. On the additive algebraic group the invariant form is $dx$; on the multiplicative algebraic group it is $dx/x$, since $d(ax)/(ax)=dx/x$. An abelian variety of dimension $g$ therefore has $g$ linearly independent invariant one-forms, providing the local linear data behind its global group geometry.