On the automorphism group of a linear algebraic monoid

Abstract:

Let $S$ be a connected regular monoid with zero. It is shown that an automorphism of $S$ is inner if and only if it sends each idempotent of $S$ to a conjugate idempotent. In the language of semigroup theory, the automorphism group of $S$ maps homomorphically into the automorphism group of the finite lattice of $\mathcal {G}$-classes of $S$, and the kernel of this homomorphism is the group of inner automorphisms of $S$. In particular, if the $\mathcal {G}$-classes of $S$ are linearly ordered, then every automorphism of $S$ is inner.