The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. This notion is however not so straightforward even in Euclidian signature, and the generalization to Minkowskian signature brings in further problems. The Cauchy-Riemann-Fuerter conditions make however sense also in Minkowskian quaternionic situation and the problem is whether they allow the physically expected solutions. One should also show that the possible generalization is consistent with (co)-associativity.
In this article these problems are considered. Also a comparison with Igor Frenkel's ideas about hierarchy of Lie algebras, loop, algebras and double look algebras and their quantum variants is made: it seems that TGD as a generalization of string models replacing string world sheets with space-time surfaces gives rise to the analogs of double loop algebras and they quantum variants and Yangians. The straightforward generalization of double loop algebras seems to make sense only at the light-like boundaries of causal diamonds and at light-like orbits of partonic 2-surfaces but that in the interior of space-time surface the simple form of the conformal generators is not preserved. The twistor lift of TGD in turn corresponds nicely to the heuristic proposal of Frenkel for the realization of double loop algebras.
See the article Are Preferred Extremals Quaternion-Analytic in Some Sense? or the chapter Unified Number Theoretical Vision
of "TGD as Generalized Number Theory".
For a summary of earlier postings see Latest progress in TGD.