Lie Algebras of Formal Power Series

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 obtained by using some constant multiples of its coefficients. The space of pseudodifferential 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 pseudodifferential operators, Jacobi-like forms, and modular forms for a discrete subgroup of SL(2,R).