Lie Algebras of Formal Power Series

Abstract

Pseudodifferential operators are formal Laurent series in the formal inverse -1 of the derivative operator whose coefficients are holomorphic functions. Given a pseudodifferential operator, the corresponding formal power series can be ob tained by using some constant multiples of its coefficients. The space of pseu dodifferential operators is a noncommutative algebra over C and therefore has a natural structure of a Lie algebra. We determine the corresponding Lie algebra structure on the space of formal power series and study some of its properties. We also discuss these results in connection with automorphic pseudodifferen tial operators, Jacobi-like forms, and modular forms for a discrete subgroup of SL(2, R).