Geodesic
Quantum dissipative systems. IV. Analog of Lie algebra and Lie group
Teoretičeskaâ i matematičeskaâ fizika, Tome 110 (1997) no. 2, pp. 214-227 
V. E. Tarasov. Quantum dissipative systems. IV. Analog of Lie algebra and Lie group. Teoretičeskaâ i matematičeskaâ fizika, Tome 110 (1997) no. 2, pp. 214-227. http://geodesic.mathdoc.fr/item/TMF_1997_110_2_a1/
@article{TMF_1997_110_2_a1,
     author = {V. E. Tarasov},
     title = {Quantum dissipative systems. {IV.~Analog} of {Lie} algebra and {Lie} group},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {214--227},
     year = {1997},
     volume = {110},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1997_110_2_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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