Lie algebras under constraints and non-bijective canonical transformations

Abstract

The concept of a Lie algebra under constraints is developed in connection with the theory of non-bijective canonical transformations. A finite-dimensional vector space M, carrying a faithful linear representation of a Lie algebra L, is mapped into a lower-dimensional space, M in such a manner that a subalgebra L0 of L is mapped into D(L0)=0. The Lie algebra L under the constraint D(L0)=0 is the largest subalgebra L1 of L that can be represented faithfully on M. If L0 is semisimple, then L1 is shown to be the centraliser cent LL0. If L is semisimple and L0 is a one-dimensional diagonal subalgebra of a Cartan subalgebra of L, then L1 is shown to be the factor algebra cent LL0/L0. The later two results are applied to non-bijective canonical transformations generalising the Kustaanheimo-Stiefel transformation.