Polynomial Lie Algebras Research Articles

Polynomial lie algebras and associated pseudogroup structures in composite quantum models

Polynomial Lie (super)algebras ℷ pd are introduced via G i -invariant polynomial Jordan maps in quantum composite models with Hamiltonians H having invariance groups G i . The algebras ℷ pd have polynomial structure functions in commutation relations, they are related to pseudogroup structures by exp V, V ϵ ℷ pd , and describe dynamical symmetry of models under study. Physical applications of algebras ℷ pd in quantum optics and in composite field theories are briefly discussed.

Commuting Polynomials in Quantum Canonical Operators and Realizations of Lie Algebras

It is pointed out that the problem of realizing Lie algebras through polynomials in quantum canonical operators is not equivalent to its classical counterpart because the polynomial Lie algebras taken with respect to the classical and quantum Lie brackets are not isomorphic. Yet there are still many results which are common to both. To show this, the properties of commuting polynomials in quantum canonical operators are analyzed. This makes possible an extension from the classical to the quantum domain of a number of theorems on realizations of semisimple Lie algebras. At the same time it is stressed that differences can arise in the classical and quantum solutions, and some of these are described.