Witt type Realizations of 2-D Cayley-Klein Algebras with non-zero curvatures

TL;DR

This paper develops Witt type vector field realizations of 2-D Cayley-Klein algebras with non-zero curvature, utilizing Jacobi elliptic functions and modular transformations to handle various moduli.

Contribution

It introduces new Witt type realizations involving elliptic functions for 2-D Cayley-Klein algebras with non-zero curvature, including arbitrary moduli via modular transformations.

Findings

Realizations involve Jacobi elliptic functions with moduli in (0,1).

Modular transformations extend realizations to arbitrary moduli.

Parameter of biorthogonality influences the vector field expressions.

Abstract

The article presents various Witt type vector field realizations of 2-D Cayley-Klein algebras with non-vanishing curvatures. The expressions of the vector fields involve Jacobi elliptic functions whose moduli are directly related to the parameters that appear in the corresponding matrix representation obtained from a bi-orthogonal set of vectors. First, the realizations are obtained with the values of the moduli lying in the unit interval (0, 1). The parameter of biorthogonality plays a crucial role in this context. Later, with the help of modular transformation, realizations involving arbitrary moduli have been obtained.