Conics from the adjoint representation of \(SU(2)\). (English) Zbl 1538.11087
Summary: The aim of this paper is to introduce and study the class of conics provided by the symmetric matrices of the adjoint representation of the Lie group \(SU(2)=S^3\). This class depends on three real parameters as components of a point of sphere \(S^2\) and various relationships between these parameters give special subclasses of conics. A symmetric matrix inspired by one giving by Barning as Pythagorean triple preserving matrix and associated hyperbola are carefully analyzed. We extend this latter hyperbola to a class of hyperbolas with integral coefficients. A complex approach is also included.
MSC:
| 11D09 | Quadratic and bilinear Diophantine equations |
| 51N20 | Euclidean analytic geometry |
| 30C10 | Polynomials and rational functions of one complex variable |
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
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