An Invitation to Split Quaternionic Analysis
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Six years after William Rowan Hamilton’s discovery of quaternions, in 1849 James Cockle introduced the algebra of split quaternions. (He called them “coquaternions.”) In this paper we define regular functions on split quaternions and prove two different analogues of the Cauchy-Fueter formula for these functions.
In the paper “Split quaternionic analysis and the separation of the series for SL(2, ℝ) and SL(2, ℂ)/SL(2, ℝ)” joint with Igor Frenkel we naturally apply the methods and formulas of quaternionic analysis to solve the problems of harmonic analysis on SL(2,ℝ) and the imaginary Lobachevski space SL(2, ℂ)/SL(2, ℝ).
KeywordsCauchy-Fueter formula split quaternions Minkowski space harmonic analysis on SL(2, ℝ) and SL(2, ℝ)/SL(2, ℝ) imaginary Lobachevski space Cayley transform
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