Decomposition of Abelian Group Schemes

This is actually our second decomposition theorem for abelian groups in (6.8) we decomposed finite abelian group schemes into connected and etale factors. Moreover, that result is of the same type, since by (8.5) we see it is equivalent to a decomposition of the dual into unipotent and multiplicative parts. As this suggests, the theorem in fact holds for all abelian affine group schemes. To introduce the version of duality needed for this extension, we first prove separately a result of some interest in itself. 

If A and A differ, the coupling mode g is non-totally symmetric (coined g ). It can thus not coincide with (first-order) Franck-Condon active modes which are characterized by the decomposition of A <8> A or A <8 A and are totally symmetric (in Abelian point groups). The latter are also called tuning modes in the LVC scheme which forms the body of our early work in the field. The conical intersection is usually termed symmetry-allowed in such a case since it normally occurs (for a single coupling mode) in the subspace g = 0 where A 7 A and a free crossing is possible. 

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