Superimposed D–branes have matrix–valued functions as their transverse coordinates, since the latter take values in the Lie algebra of the gauge group inside the stack of coincident branes. This leads to considering a classical dynamics where the multiplication law for coordinates and/or momenta, being given by matrix multiplication, is nonabelian. Quantisation further introduces noncommutativi...

2 Vertex algebras, associative filtered algebras and Poisson algebras 1 2.1 Vertex algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 The enveloping algebra U(V ) of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 The Poisson algebra Ps(V ) of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Poisson algebras and modules ...

We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as for example those given by Poisson or contact structures, but we consider degenerate structures as well. Introduction We shall admit different classes of smoothness, so by a manifold of ...

For any Poisson manifold P , the Poisson bracket on C∞(P ) extends to a Lie bracket on the space Ω(P ) of all differential one-forms, under which the space Z(P ) of closed one-forms and the space B(P ) of exact one-forms are Lie subalgebras. These Lie algebras are related by the exact sequence: 0 −→ R −→ C∞(P ) d −→ Z(P ) f −→ H(P,R) −→ 0, where H(P,R) is considered as a trivial Lie algebra, an...

It is well known that the Poisson Lie algebra is isomorphic to the Hamiltonian Lie algebra [1],[3],[13]. We show that the Poisson Lie algebra can be embedded properly in the special type Lie algebra [13]. We also generalize the Hamitonian Lie algebra using exponential functions, and we show that these Lie algebras are simple.

We extend to the sl(N) case the results that we previously obtained on the construction of Wq,p algebras from the elliptic algebra Aq,p(ŝl(2)c). The elliptic algebra Aq,p(ŝl(N)c) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(N −1)/2 integers, defining q-deformations of the WN algebra, are constructed. The op...

We give a full description of the Poisson structures on finitary incidence algebra $FI(P,R)$ an arbitrary poset $P$ over commutative unital ring $R$.

pC++ is an object-parallel extension to the C++ programming language. This paper describes the current language deenition and illustrates the programming style. Examples of parallel linear algebra operations are presented and a fast poisson solver is described in complete detail.

A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved and the Poisson K-rings are calculated for a number of examples. In particular, for the zero Poisson structure ...

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Riemannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Li...