20.1.1 Poisson Process Recall the definition of a Poisson Process with rate γ, depicted in Figure 20.1. This is a stochastic process counting the number of arrivals up to time t starting from time 0, (Nt, t ≥ 0), where the interarrival times are exponentially distributed with mean 1/γ. we can refer to this as a Poisson (γ) process The number of points in an interval (a, b] is denoted N(a, b]. P...

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

In this paper, we compute the Gerstenhaber bracket on the Hochschild cohomology of C∞(M)⋊Γ. Using this computation, we classify all the noncommutative Poisson structures on C∞(M) ⋊ Γ when M is a symplectic manifold. We provide examples of deformation quantizations of these noncommutative Poisson structures.

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra R = C[X1, . . . , Xn] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a coro...

In this paper we introduce almost Poisson structures on Lie groups which generalize Poisson structures based on the use of the classical Yang-Baxter identity. Almost Poisson structures fail to be Poisson structures in the sense that they do not satisfy the Jacobi identity. In the case of cross products of Lie groups, we show that an almost Poisson structure can be used to derive a system which ...

In this paper, we compute the Gerstenhaber bracket on the Hochschild cohomology of C∞(M)⋊Γ. Using this computation, we classify all the noncommutative Poisson structures on C∞(M) ⋊ Γ when M is a symplectic manifold. We provide examples of deformation quantizations of these noncommutative Poisson structures.

Let G be a simple complex algebraic group equipped with a factorizable Poisson Lie structure. Let U~(g) be the corresponding quantum group. We study U~(g)-equivariant quantization C~[G] of the affine coordinate ring C[G] along the Semenov-Tian-Shansky Poisson Lie bracket. For a simply connected group G we prove an analog of the KostantRichardson theorem stating that C~[G] is a free module over ...

We prove that the standard Poisson structure on the Grassmannian Gr(k, n) is invariant under the action of the Coxeter element c = (12 . . . n). In particular, its symplectic foliation is invariant under c. As a corollary, we obtain a second, Poisson geometric proof of the result of Knutson, Lam, and Speyer that the Coxeter element interchanges the Lusztig strata of Gr(k, n). We also relate the...