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 $df({\bf \Lambda})=f({\bf \Lambda} +d{\bf \Lambda})-f({\bf \Lambda} )$ associated with a quantum (non-commutative) stochastic calculus. Here is an element of the universal enveloping algebra is given a meaning using the coproduct structure of even though the individual terms of this expression have no meaning. The Ito formula is equivalent to a chaotic expansion formula for which is found explicitly. 1991 Mathematics Subject Classification: primary 81S25; secondary 60H05; tertiary 18B25.