We introduce the notion of a Poisson symmetric space and the associated infinitesimal object, a symmetric Lie bialgebra. They generalize corresponding notions for Lie groups due to V. G. Drinfel’d. We use them to give some geometric insight to certain Poisson brackets that have appeared before in the literature. 1 Motivation Let us recall briefly the best-known examples of Poisson manifolds. Th...

let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

In this paper, applying the concept of generalized A-valued norm on a right $H^*$-module and also the notion of ϕ-homomorphism of Finsler modules over $C^*$-algebras we first improve the definition of the Finsler module over $H^*$-algebra and then define ϕ-morphism of Finsler modules over $H^*$-algebras. Finally we present some results concerning these new ones.

20.1 Classical Hamiltonian Reduction of a Poisson Vertex Algebra Let V be a Poisson vertex algebra, and suppose we are given a triple (V0, I0, φ) where V0 is a Poisson vertex algebra, I0 ⊂ V0 is a Poisson vertex algebra ideal, and φ : V0 → V is a Poisson vertex algebra homomorphism. Then the Hamiltonian reduction associated to (V0, I0, φ) is the differential algebra W =W(V0, I0, φ) := (V/Vφ(I0)...

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule with the left and right module actions $pi_ell: Atimes Xrightarrow X$ and $pi_r: Xtimes Arightarrow X$, respectively. In this paper, we  study  the topological centers of the left module action $pi_{ell_n}: Atimes X^{(n)}rightarrow X^{(n)}$ and the right module action $pi_{r_n}:X^{(n)}times Arightarrow X^{(n)}$,  which inherit from th...

Many interesting C∗-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C∗-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, ...

we study topological von neumann regularity and principal von neumann regularity of banach algebras. our main objective is comparing these two types of banach algebras and some other known banach algebras with one another. in particular, we show that the class of topologically von neumann regular banach algebras contains all $c^*$-algebras, group algebras of compact abelian groups and cer...

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 ...

We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H , and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra as...