1.2.1 Tangent Spaces and Differential Forms

The last sec­tion saw how the in­fi­nite in­te­gra­tion of a pro­ject­ed form of ori­ent­ed vec­tors could yield a sim­i­lar result to ex­po­nen­ti­a­tion. The abil­i­ty to in­te­grate these pro­ject­ed forms im­plies the ex­is­tence of dif­fer­en­tial forms. This chap­ter will dis­cuss tan­gent spaces and tan­gent vec­tors of Lie groups and their re­la­tion­ship to the Lie al­ge­bras of that group. Lie groups will typ­i­cal­ly be non-Eu­clid­ean, but re­mark­ably, it is pos­si­ble to reveal the Lie al­ge­bra un­der­ly­ing the group through its tan­gent space, which is a dis­tinct­ly Eu­clid­ean con­cept. For a smooth path through the Lie group, where smooth means that a de­riv­a­tive exists, the tan­gent space is cap­tured through the tan­gent vec­tors pass­ing through the iden­ti­ty.

Before get­ting into the de­tails sur­round­ing tan­gent spaces, we must con­sid­er matrix de­riv­a­tives. Our first step is to update our el­e­ment no­ta­tion to help avoid any con­fu­sion as we pro­ceed. For each of the $D_{3\mathbb{R}},\ D_{3\mathbb{C}},\ D_{3\mathbb{c}}$ groups, the el­e­ments $X_{j},Y_{j}$, and the sets for which $X_{j},Y_{j} \in A$, the fol­low­ing ter­mi­nol­o­gy is used

where we will refer to the $S_{3\mathbb{R}}/D_{3\mathbb{R}}$ group as $D_{3\mathbb{R}}$ in the chap­ter. The reader should note that $\mathbb{c}$ is the lower-case bold c, de­not­ing a com­plex ar­gu­ment of the form ${(a + ib) = re}^{ix_{j}}$, for ${r,x}_{j}\in \mathbb{R}$, and the groups $D_{3\mathbb{C}}$,${\ D}_{3\mathbb{c}}$ have two in­dices for the greater number of sub-el­e­ments in each. For the set $A$

the ex­po­nen­tial map­pings of the $X_{j}$ and $Y_{j}$ al­ge­bras $\in A$ define the matrix groups $G\left( {x_{j}X}_{j} \right),G\left( y_{j}Y_{j} \right)$

for vari­ables $x_{j},y_{j}\in \mathbb{R}.$The matrix de­riv­a­tives of $G\left( {x_{j}X}_{j} \right)$ and $G\left( y_{j}Y_{j} \right)$ are

which have de­riv­a­tive al­ge­bras de­fined as

and since the matrix log­a­rithm and ex­po­nen­tial func­tions are in­verse­ly re­lat­ed, we have the iden­ti­ties

In matrix cal­cu­lus, it should be noted that there are dif­fer­ent ap­proach­es to the dif­fer­en­ti­a­tion of a matrix group. This chap­ter will use the matrix-by-scalar de­riv­a­tive tech­nique to yield the tan­gent matrix. As the name im­plies, for an $m \times n$ matrix group $G$, its matrix-by-scalar de­riv­a­tive is the fol­low­ing

We return to the in­fi­nite in­te­gra­tion func­tions to con­sid­er their dif­fer­en­tial forms.

De­f­i­n­i­tions (1.2.8) – For the vector $\mathcal{c}x_{j}$, where $x_{j}$ is a unit basis vector of some $n$-di­men­sion­al vector space, and $\mathcal{c}$ is a scalar con­stant$,$ the de­riv­a­tive of the in­fi­nite in­te­gra­tion func­tions $f_{\mathbb{R}}^{- \infty}\left( \mathcal{c}x_{j} \right)\ and\ f_{\mathbb{C}}^{- \infty}\left( \mathcal{c}ix_{j} \right)$ start­ing at the zero point, for $\mathcal{c}x_{j}\in \mathbb{R},\ \mathcal{c}ix_{j}\in \mathbb{C}$, are de­fined as

The­o­rem (1.2.9) – Taking a unit basis vector $x_{j}$as a fixed form el­e­ment, the func­tions $D^{k}\left\lbrack f_{\mathbb{R}}^{- \infty}\left( \mathcal{c}x_{j} \right) \right\rbrack,D^{k}\left\lbrack f_{\mathbb{C}}^{- \infty}\left( \mathcal{c}ix_{j} \right) \right\rbrack$ are equal to the first de­riv­a­tives of the ex­po­nen­tial map­ping $exp:\mathfrak{g} \rightarrow G\in \mathbb{R},\mathbb{C}$ onto a con­tin­u­ous man­i­fold $\forall\ \mathfrak{g} =x_{j}$.

Proof -

as

and for $D^{1}\left\lbrack f_{\mathbb{C}}^{- \infty}\left( \mathcal{c}ix_{j} \right) \right\rbrack$

as

$\square$

We apply this de­riv­a­tive ap­proach for our ex­po­nen­ti­at­ed per­mu­ta­tions in the ex­am­ples above to find $D^{1}\left\lbrack \mathcal{c}e^{X_{3}} \right\rbrack$

and next, $D^{1}\left\lbrack \mathcal{c}e^{Y_{2}} \right\rbrack$

We can see that for $D^{1}\left\lbrack \mathcal{c}e^{Y_{2}} \right\rbrack$ , the de­riv­a­tive can be thought of as a cyclic shift of the three-way trigono­met­ric func­tions by way of the right shift­ing of its matrix en­tries. Al­though it is not shown here, the $D^{1}\left\lbrack \mathcal{c}e^{Y_{3}} \right\rbrack$ form is rep­re­sent­ed by a left-shift­ing of its func­tion en­tries. To gain the geo­met­ric in­tu­ition for de­riv­a­tives of the $U(3)$ matrix group gen­er­a­tors, we can take the $D^{1}\left\lbrack \mathcal{c}e^{iX_{3}} \right\rbrack = G^{'}\left( iX_{3} \right)$ de­riv­a­tive above, noting that this is the same as $f_{\mathbb{C}}'\left( iX_{3} \right)$, and rotate the iden­ti­ty poly­gon by $\frac{i\pi}{4}$ ra­di­ans, find­ing

We then com­pare our orig­i­nal gen­er­a­tor $G\left( X_{3}^{\mathbb{C}} \right)$, which is the anti-de­riv­a­tive of the above, using a sim­i­lar ro­ta­tion of $\frac{i\pi}{4}$ ra­di­ans

When com­par­ing the map­ping, $\phi\left( G\left( X_{3}^{\mathbb{C}}\left( \frac{\pi}{4} \right) \right):\epsilon \right)$, to the de­riv­a­tive map­ping, $\phi\left( G^{'}\left( X_{3}^{\mathbb{C}}\left( \frac{\pi}{4} \right) \right):\epsilon \right),$it can be seen that we are still ro­tat­ing the equi­lat­er­al tri­an­gle about the $X_{1}/X_{2}$-plane for both, but the de­riv­a­tive is ro­tat­ed coun­ter­clock­wise by $\frac{i\pi}{2}$ ra­di­ans in the $\mathbb{C}^{3}$ geo­met­ric space. This phase shift, seen in our de­rivate ex­am­ple, is anal­o­gous to the $\frac{i\pi}{2}$ ra­di­ans shift we see in the de­riv­a­tive of $e^{ix}$

co­in­cid­ing with the result, found by mul­ti­ply­ing the $\phi\left( G\left( X_{3}^{\mathbb{C}}\left( \frac{\pi}{4} \right) \right):\epsilon \right)$ vector map­ping by $i$

We plot both re­sults $\in \mathbb{C}^{3}$ space above in Fig­ures 1.2.1-2, to vi­su­al­ize the linear trans­for­ma­tions of the $U(3)$ gen­er­a­tor $G\left( X_{3}^{\mathbb{C}} \right)$ and its de­riv­a­tive $G^{'}\left( X_{3}^{\mathbb{C}} \right)$. The volume of the equi­lat­er­al tri­an­gle after the $\phi\left( G^{'}\left( X_{3}^{\mathbb{C}}\left( \frac{\pi}{4} \right) \right):\epsilon \right)$ map­ping ap­pears to be more promi­nent than the $\phi\left( G\left( X_{3}^{\mathbb{C}}\left( \frac{\pi}{4} \right) \right):\epsilon \right)$ map­ping, but being $\in \mathbb{C}^{3}$, the norms $\left\| \mathbf{z} \right\|$for both poly­gons in com­plex space yield the same mag­ni­tudes for re­sults, given that

is equal to