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Calculus In Enveloping Algebras
Author(s) -
Hudson R. L.
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610701002976
Subject(s) - mathematics , universal enveloping algebra , associative property , lie algebra , algebra over a field , graded lie algebra , formal power series , pure mathematics , associative algebra , differential (mechanical device) , differential calculus , taylor series , power series , multiplication (music) , cellular algebra , algebra representation , mathematical analysis , combinatorics , engineering , aerospace engineering
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.