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

Arindam Chakraborty

TL;DR

<3-5 sentence high-level summary> This paper develops Witt-type vector-field realizations of 2-D Cayley-Klein algebras with non-zero curvatures by constructing 2×2 matrix generators from bi-orthogonal systems and then expressing the corresponding Lie algebra actions as differential operators via Jacobi elliptic functions. It shows how the elliptic modulus, tied to bi-orthogonality, controls these realizations and demonstrates how modular transformations extend the constructions to arbitrary moduli, including cases with modulus k′ and general |λ|. The work presents multiple concrete realizations using elliptic, reciprocal, and modular-transformation-based methods, discusses an intrinsic Casimir operator to address representation-dependence of invariants, and analyzes limiting behavior where elliptic functions reduce to trigonometric or hyperbolic forms in some cases. Together, these results connect CK geometries with elliptic-function-based representations, offering a framework potentially extensible to higher dimensions.

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.

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

TL;DR

<3-5 sentence high-level summary> This paper develops Witt-type vector-field realizations of 2-D Cayley-Klein algebras with non-zero curvatures by constructing 2×2 matrix generators from bi-orthogonal systems and then expressing the corresponding Lie algebra actions as differential operators via Jacobi elliptic functions. It shows how the elliptic modulus, tied to bi-orthogonality, controls these realizations and demonstrates how modular transformations extend the constructions to arbitrary moduli, including cases with modulus k′ and general |λ|. The work presents multiple concrete realizations using elliptic, reciprocal, and modular-transformation-based methods, discusses an intrinsic Casimir operator to address representation-dependence of invariants, and analyzes limiting behavior where elliptic functions reduce to trigonometric or hyperbolic forms in some cases. Together, these results connect CK geometries with elliptic-function-based representations, offering a framework potentially extensible to higher dimensions.

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.

Paper Structure

This paper contains 12 sections, 1 theorem, 45 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Caley-Klein algebras and their normalized forms
  3. Bi-orthogonal system and Cayley-Klein Algebras
  4. Witt realizations of CK-algebras via elliptic functions
  5. Realizations by elliptic functions of modulus $k$
  6. Realizations by Reciprocals and Quotients of Jacobi Functions
  7. Realizations related to Modular Transformation
  8. Realizations by elliptic functions of modulus $k^{\prime}$
  9. Realizations by Jacobi functions of arbitrary absolute value of the modulus $\vert\lambda\vert\in (0, \infty)$
  10. Discussion on intrinsic Casimir operator
  11. Realizations at the limiting values of the moduli
  12. Comments

Key Result

Theorem 1

Given a pair of vectors $\{\vert v_j\rangle : j=1,2\}$ and an operator the set $\{\vert\phi_j\rangle= T \vert v_j\rangle: j=1, 2\}$ and $\{\vert\chi_j\rangle= (T^{-1})^\dagger\vert v_j\rangle: j=1, 2\}$ constitute bi-orthogonal system provided $\langle v_j\vert v_k\rangle=0$; $\{\sigma_m : m=1, 2, 3\}$ are Pauli matrices.

Theorems & Definitions (7)

  • Theorem 1
  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5