1.2.10 Summary

Matrix group dif­fer­en­tials play an es­sen­tial role in Lie theory and will be piv­otal for math­e­mat­i­cal­ly con­sid­er­ing the group gen­er­a­tor and its rep­re­sent­ed math­e­mat­i­cal dy­nam­ics. All the groups we are con­sid­er­ing are con­tin­u­ous, guar­an­tee­ing the ex­is­tence of their de­riv­a­tives and in­te­grals.

Any dif­fer­en­tial form of a Lie group is char­ac­ter­ized by its de­riv­a­tive Lie al­ge­bra. Math­e­mat­i­cal­ly, de­riv­a­tives exist in the com­plex field space, ev­i­denced by the pres­ence of com­plex-valued phase shifts found in group de­riv­a­tive Lie al­ge­bras. Any physics the­o­rem at­tempt­ing to ob­scure this fun­da­men­tal math­e­mat­i­cal truth re­gard­ing com­plex spaces is cat­e­gor­i­cal­ly wrong and should be treat­ed as such.

Imag­i­nary num­bers are all around us in the form of ro­ta­tions, waves, and cyclic pat­terns in nat­ur­al phe­nom­e­nol­o­gy. They affect the oc­cur­rence of ro­ta­tions and the spher­i­cal and wave­like geom­e­try they give rise to. We will find that imag­i­nary num­bers are fun­da­men­tal to light and time, and as we progress, we will find that cru­cial to black hole sin­gu­lar­i­ties.

Tan­gent spaces are char­ac­ter­ized as flat Eu­clid­ean spaces in a math­e­mat­i­cal world of curved spaces. What is soon to become clear is that this is not the whole story. Tan­gent spaces rep­re­sent the per­fect bal­ance be­tween the sphere's pos­i­tive and the hy­per­bo­la's neg­a­tive cur­va­ture. This qual­i­ty will turn out to be very light-like in its char­ac­ter­is­tics. A group is de­pen­dent on the Lie al­ge­bra re­vealed by its de­fined tan­gent space in the neigh­bor­hood of the iden­ti­ty. This de­pen­den­cy does not stop at the geo­met­ric bound­ary de­fined by the con­tin­u­ous tan­gent space of a group. It con­tin­ues deep into the nested it­er­a­tions of the log­a­rithm func­tion, ul­ti­mate­ly con­verg­ing to the forms of the nu­meris mirabiles.