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Journal of the London Mathematical Society 2002 65(2):361-380; doi:10.1112/S0024610701002976
© 2002 by London Mathematical Society
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© The London Mathematical Society

Calculus In Enveloping Algebras

R. L. Hudson

Department of Computing and Mathematics, Nottingham Trent University Burton Street, Nottingham NG1 4BU

Received 4 July 2000. Revision received 13 August 2001.

Motivated by, but independent of, some recent work in quantum stochastic calculus, a theory of differential and integral calculus is developed which is intrinsic to the universal enveloping algebra of a Lie algebra whose Lie bracket is obtained by taking commutators in an associative algebra. The differential map satisfies a generalisation of Leibniz' formula called the Leibniz–Itô formula, which involves the associative multiplication. There is an analogue of the Taylor–Maclaurin expansion. Through passing to formal power series, a theory of product integrals is developed; such integrals are characterised by a group-like property with respect to the coproduct.


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