In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra such that commutators of this algebra generate Lie subalgebra. It was shown that classical connection between analytic group (Lie group) and Lie algebra, proved by Lie theorems, exists between analytic loop, commutant of which is associative subloop (group), and commutant Lie algebra (an ...

It is well known that a measured groupoid G defines a von Neumann algebra W ∗(G), and that a Lie groupoid G canonically defines both a C∗-algebra C∗(G) and a Poisson manifold A∗(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C∗-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of obj...

Let A+(k) denote the ring l~[ t] / t k+t and let fr be a reductive complex Lie algebra with exponents m 1 . . . . . m,. This paper concerns the Lie algebra cohomology of ~| considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we call weight, is inherited from the obvious grading of f f | § (k)). We conjecture that this Lie algebra cohomology is a...

We shall define a Lie algebra associated with this system to be the Lie subalgebra of g`(n, C) generated by all the matrices A(x), x ∈ R, and we shall denote it by LA. One of the main objectives of this paper is to demonstrate that the classical structure theory of this Lie algebra has important consequences for stability theory and chaotic motion. In fact, we shall see that the well-known chao...

This paper consists of two parts. In the first part we show that any Poisson algebraic group over a field of characteristic zero and any Poisson Lie group admits a local quantization. This answers positively a question of Drinfeld and generalizes the results of [BFGP] and [BP]. In the second part we apply our techniques of quan-tization to obtain some nontrivial examples of quantization of Pois...

We integrate the Lifting cocycles Ψ2n+1,Ψ2n+3,Ψ2n+5, . . . ([Sh1], [Sh2]) on the Lie algebra Difn of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle λ on an n-dimensional complex manifold M in the sense of Gelfand–Fuks cohomology [GF] (more precisely, we integrate th...