Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution for arbitrary holomorphic Poisson manifold. More precisely, any manifold $(X, \pi)$, prove that there exists symplectic structure in neighborhood $Y$ of the zero section $T^*X$ such projection map given manifold, and moreover Lagrangian submanifold. We describe...

Let A be a C∗-algebra. Let E and F be Hilbert A-modules with E being full. Suppose that θ : E → F is a linear map preserving orthogonality, i.e., 〈θ(x), θ(y)〉 = 0 whenever 〈x, y〉 = 0. We show in this article that if, in addition, A has real rank zero, and θ is an A-module map (not assumed to be bounded), then there exists a central positive multiplier u ∈M(A) such that 〈θ(x), θ(y)〉 = u〈x, y〉 (x...

We present an example of a zero-dimensional F-space that is not strongly zero-dimensional.

A mapping f : M → N between Hilbert C∗-modules approximately preserves the inner product if ‖〈f(x), f(y)〉 − 〈x, y〉‖ ≤ φ(x, y), for an appropriate control function φ(x, y) and all x, y ∈ M. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert C∗modules on more general restricted domains. In particular, we investigate some asympt...

The free loop space of a Lie group is homeomorphic to the product itself and its based space. We show that coproduct on homology was introduced by Goresky Hingston splits into diagonal map so-called This result implies trivial for even-dimensional groups. Using results Bott Samelson, we as well large family simply connected

In this appendix we give another algebraic proof of the Weyl character formula, using methods that have many other applications in Lie theory. We begin by setting up the machinery of Lie algebra cohomology (without assuming any previous background in homological algebra). We define the cohomology spaces for a Lie algebra representation in terms of a cochain complex and differential, with the co...