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$%M. H. Lee, Lie algebras of formal power series, Rev. Mat. Complut. 20 (2007), no. 2, 463–481.%$
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 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,
).
Key words: Lie algebras, pseudodifferential operators, Jacobi-like forms, modular forms.
2000 Mathematics Subject Classification: 17B60, 11F50, 11F11, 35S05.