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CALCULUS IN ENVELOPING ALGEBRAS

Published online by Cambridge University Press:  24 March 2003

R. L. HUDSON
R. L. HUDSON
Affiliation:
Department of Computing and Mathematics, Nottingham Trent University, Burton Street, Nottingham NG1 4BU
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Abstract

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.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 65 , Issue 2 , April 2002 , pp. 361 - 380
Copyright
The London Mathematical Society, 2002

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