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Quantum dissipative systems. IV. Analog of Lie algebra and Lie group
Teoretičeskaâ i matematičeskaâ fizika, Tome 110 (1997) no. 2, pp. 214-227 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

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