1.6.8 Matrix Determinants and the Infinite Symmetry Group

In the last chap­ter, we de­fined the tan­gent space $\mathfrak{T}\left( \mathcal{u} \times\mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$ of the in­fi­nite sym­me­try group as

where we see that the el­e­ments of the Lie al­ge­bra could be points any­where on the com­plex unit circle. For the tan­gent space el­e­ments above, the top row is of the $\mathcal{X}$ form and the el­e­ments on the bottom row, the $\mathcal{Y}$ form. In order to group the Lie al­ge­bra to­geth­er, we change the vari­ables in the $\mathcal{X}$-type ma­tri­ces as fol­lows

where it be­comes ap­par­ent that each matrix entry can be any point on the com­plex unit circle. Adding the Lie al­ge­bra to­geth­er as one matrix, re­sults in a matrix de­not­ed as $\mathcal{M}$

which rep­re­sents the entire tan­gent space of the $\mathcal{X}$-type Lie al­ge­bra. The $\mathcal{Y}$-type el­e­ments are de­pen­dent on the $\mathcal{X}$-type el­e­ments and are the result of prod­ucts of the $\mathfrak{T}\left( \mathcal{X} \right)$ el­e­ments. The de­ter­mi­nant of the matrix $\mathcal{M}$ nicely demon­strates this fact, as

which sim­pli­fies to

and the dis­tinc­tion be­tween the $\mathcal{Y},\mathcal{X}$-types be­comes ap­par­ent, as the three ex­po­nen­tials on the top cycle though the $a,b,c$ el­e­ments in cyclic fash­ion, which is akin to the $\mathcal{Y}$-type el­e­ments of the $Z_{3}$ group, and the el­e­ments on the bottom are the sub-de­ter­mi­nants of the $\mathcal{X}$-type Lie al­ge­bras. To make this clear­er, we denote the de­pen­dent $\mathcal{Y}_{j}$ el­e­ments, as

and note how this co­in­cides with the three el­e­ments on the left side we found in $\det\left( \mathcal{M} \right)$.