Quantum dissipative systems. IV. Analog of Lie algebra and Lie group

The requirement of consistent quantum description of dissipative systems leads to necessity to go beyond Lie algebra and group. In order to describe dissipative (non-Hamiltonian) systems in quantum theory we need to use non-Lie algebra (algebras for which the Jacoby identity is not satisfied) and analytic quasigroups (nonassociative generalization of analytic groups). We prove that this analog is a commutant Lie algebra (an algebra, the commutant of which is a Lie subalgebra) and a commutant associative loop (a loop, commutators of which form an associative subloop (group)). We prove that the tangent algebra of an analytic commutant associative loop (Valya loop) is a commutant Lie algebra (Valya algebra). Examples of commutant Lie algebras are considered.