Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

Abstract

The functional Ito formula, in the form df(Λ) = f(Λ + d Λ) −f(Λ), is formulated and proved in the context of a Lie algebra L associated with a quantum (non-commutative) stochastic calculus. Here f is an element of the universal enveloping algebra U of L, and f(Λ) + d(Λ) − f(Λ) is given a meaning using the coproduct structure of U even though the individual terms of this expression have no meaning. The Ito formula is equivalent to a chaotic expansion formula for f(Λ) which is found explicitly. 1991 Mathematics Subject Classification: primary 81S25; secondary 60H05; tertiary 18B25.

Locations

• Proceedings of the London Mathematical Society - View - PDF