This article is cited in 6 papers

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

V. E. Tarasov

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University

Abstract: 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.

Received: 30.04.1996

DOI: 10.4213/tmf962

 English version: Theoretical and Mathematical Physics, 1997, 110:2, 168–178

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