It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra Ĝk can be associated with each parabolic subalgebra P = (G0+G+) of the Lie algebra G, where in the standard case G0 is the Cartan and P is the Borel subalgebra. In this letter we obtain an explicit formula for the Wakimoto realization in the general case. Using Hamiltoni...

has recently been studied by Cartan' with the aid of ingenious geometric methods peculiar to his way of looking at an affinely connected manifold.2 From the analytic and invariant-theoretic standpoint, however, Cartan's geometric methods are by no means the last word on the subject. Furthermore, the proof of many of his theorems bring into play the variables xl, x2, .. ., x". It is the purpose ...

Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a ...

In this paper, we study the tensor commutativity degree of pair of finite groups. Erdos introduced the relative commutativity degree and studied its properties. Then, Mr. Niroomand introduced the tensor relative commutativity degree, calculated tensor relative degree for some groups, and studied its properties. Also, he explained its relation with relative commutativity degree. In this paper, w...

The Riemannian geometry of elastica in one and two dimensions is considered. Two examples from engineering science are given. The first is the deflexion or Frenet curvature of the elastic filament rod where the Riemannian curvature vanishes, since the curve is one dimensional. However the Frenet curvature scalar appears on the LeviCivita-Christoffel symbol of the Riemannian geometry. A second e...

We consider a Weyssenhoff fluid assuming that the spacetime is homogeneous and isotropic, therefore being relevant for cosmological considerations of gravity theories with torsion. In this paper, it is explicitely shown that the Weyssenhoff fluids obeying the Frenkel condition or the Papapetrou-Corinaldesi condition are incompatible with the cosmological principle, which restricts the torsion t...

In this paper, using the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a quantum-deformed exterior calculus on the phase-space of an arbitrary hamiltonian system. Introducing additional bosonic and fermionic coordinates we construct a supermanifold which is closely related to the tangent and cotangent bundle over phase-space. Scalar functions on the super-manifold become equival...

A non-Riemannian geometrical approach to the investigation of an acoustic black hole in irrotational mean flows, based on the Lighthill vortex sound theory is given. This additional example of analog gravity based on classical fluids is used to investigate the acoustic Lorentz violation. An example is given where the contortion vector is distributed along a ring inside the fluid which can be gr...

A new notion of Cartan pairs as a substitute of notion of vector fields in the noncommutative geometry is proposed. The correspondence between Cartan pairs and differential calculi is established.