1.5.10 Summary

In this chap­ter we in­tro­duced the Baker-Camp­bell-Haus­dorff the­o­rem and the rev­e­la­tion that the matrix log­a­rithm of Lie group prod­ucts with non-com­mut­ing al­ge­bras is com­posed of Lie brack­et­ed terms. Anal­o­gous to the Fouri­er trans­form, a Lie group's es­sen­tial in­for­ma­tion is con­tained in its Lie al­ge­bra, which can be ac­cessed by the matrix log­a­rithm and by eval­u­at­ing the tan­gent space in the neigh­bor­hood of the group iden­ti­ty. By ex­tend­ing min­i­mal el­e­ments to the in­fin­i­tes­i­mal scale, we were able to define the in­fin­i­ty sym­me­try group $S_{3\mathbb{C}}^{\infty}$, a group de­fined by its smooth and con­tin­u­ous tan­gent space, $\mathfrak{g}_{3\mathbb{C}}^{\infty}$. This con­tin­u­ous tan­gent space nat­u­ral­ly led to the de­f­i­n­i­tion of its sub­tan­gent space com­prised of angle func­tions, whose ex­po­nen­tial map­pings de­ter­mine the forms of the tan­gent space. We have ex­tend­ed the de­f­i­n­i­tion of an iden­ti­ty el­e­ment $\epsilon$, to that of the iden­ti­ty group $S_{3}^{\epsilon}$, an in­tu­itive step an­chored in the math­e­mat­i­cal forms of $S_{3\mathbb{C}}^{\infty}$ and its sub­tan­gent space $\mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$. The ex­ten­sion of Lie theory to in­clude a de­fined con­tin­u­ous tan­gent space and sub­tan­gent space, gives rise to the pos­si­bil­i­ty of con­sid­er­ing not only cir­cu­lar motion and me­chan­ics but that of non-zero ec­cen­tric­i­ty groups.

Our take­away is that group dy­nam­ics may be iso­lat­ed to sub­group geome­tries in the in­fin­i­ty sym­me­try group $S_{3\mathbb{C}}^{\infty}$, in­clud­ing dif­fer­ing man­ners of move­ment, like ro­ta­tions, boosts, and ex­ten­sion. Move­ment in the func­tion angle space is math­e­mat­i­cal­ly pri­mor­dial and not bound­ed to any sub­group's geom­e­try. This pri­mor­dial move­ment in the func­tion angle space is not lim­it­ed by space or time and is un­bound from a phe­nom­e­nal form. The angle space rep­re­sents a fit­ting start­ing point to ex­plore the domain of the mind, the sub­ject matter of our future work.