Exponentials of derivations

Roughly speaking, if D is a derivation of a non-associative algebra, then exp(D) is an automorphism of the algebra. Of course this can only hold under certain conditions, such as convergence of exp, etc. Even though the most well-known context of this fact is ‘Lie groups and Lie algebras’, it plays a role in various areas of mathematics, as it depends on the very basic functional equation exp(x+y)=exp(x).exp(y). Over fields of prime characteristic p the ordinary exponential series does not generally make sense because all factorials at the denominators vanish from some point on. However, substitutes for it have been important in the theory of modular Lie algebras, for a technique called ‘toral switching’, starting with a ‘truncated’ exponential in 1969. In the increasingly sophisticated generalizations which followed, the connection with the classical exponential series and its functional equation got lost to the technical details of the specialized context. We describe how those generalizations can be explained (and extended) in terms of Artin-Hasse exponentials, and more generally in terms of certain generalized Laguerre polynomials. An important point will be that weakened analogues of the functional equation of the ordinary exponential are satisfied by those generalized versions.