Quantum stochastic calculus and representations of Lie superalgebras (Q1264964)

The monograph treats a generalization of the relation between bosonic and fermionic stochastic calculi, which gives rise to a \({\mathbb Z}_2\)-graded theory of multi-dimensional quantum stochastic calculus. In this theory one has a mixture of multi-dimensional bosonic and fermionic creation and annihilation processes along with \({\mathbb Z}_2\)-graded multi-dimensional gauge process. The commutation and anticommutation properties of these processes are shown to provide a time-indexed family of representations of a broad class of Lie superalgebras. The Lie superalgebra representation result is used to derive a formula for chaotic decomposition of elements of the universal enveloping superalgebra of the Lie superalgebra associated with \({\mathbb Z}_2\)-graded quantum stochastic calculus. It is suggested that this calculus may be useful in the theory of quantum groups.