Duflo-Moore Operator for The Square-Integrable Representation of 2-Dimensional Affine Lie Group

In this paper, we study the quasi-regular and the irreducible unitary representation of affine Lie group  of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier transform of quasi-regular representation of . The second, in such the representation of affine Lie group   is square-integrable then we compute its Duflo-Moore operator instead of using Fourier transform as in F hr’s work

Geometry of the Abel Equation of the first kind

We study the first kind Abel differential equation $$\dfrac{dy}{dx}=c_0(x)+3c_1(x)y+3c_2y^2+c_3(x)y^3,$$ where the functions $c_i$ are real analytic. The first step of our analysis is through the Cartan equivalence method, then we use techniques from representation theory; this latter mean allows us to exhibit an affine connection and hence a covariant derivative on the space of differential invariants of the Abel equation. In the context of Abel equation, this affine connection is similar to the connection (also called Frobenius-Stickelberger connexion) given by the quasi-modular Eisenstein series $E_2$, which permits to define a covariant derivative in the space of modular forms and is a solution of a Chazy type equation.Comment: 12 page

Twisted Parafermions

A new type of nonlocal currents (quasi-particles), which we call twisted parafermions, and its corresponding twisted $Z$-algebra are found. The system consists of one spin-1 bosonic field and six nonlocal fields of fractional spins. Jacobi-type identities for the twisted parafermions are derived, and a new conformal field theory is constructed from these currents. As an application, a parafermionic representation of the twisted affine current algebra $A^{(2)}_2$ is given.Comment: RevTex 5 pages; Cosmetic changes, to appear in Phys.Lett.