samples/texts/1749790/page_12.md

Now, this does not quite reflect the situation in the case developed above; $\omega$ defined as the first derivative is zero, since the parameter $\lambda$ occurs to different order in the continuation of the representation and the Lie algebra structure. Suppose then that

$$\frac{d}{d\lambda }[X,Y]{\mathrm{\mid }}_{\lambda =0}=0\text{for }X,Y\in G\mathrm{.}\backslash frac\{d\}\{d\backslash lambda\}\; [X,\; Y]|\_\{\backslash lambda=0\}\; =\; 0\; \backslash quad\; \backslash text\{for\; \}\; X,\; Y\; \backslash in\; G.$$

Define now

$${\omega }_2(X,Y)=\frac{d^2}{d{\lambda }^2}[X,Y]{\mathrm{\mid }}_{\lambda =0}\text{for }X,Y\in G\mathrm{.}\backslash omega\_2(X,\; Y)\; =\; \backslash frac\{d^2\}\{d\backslash lambda^2\}\; [X,\; Y]|\_\{\backslash lambda=0\}\; \backslash quad\; \backslash text\{for\; \}\; X,\; Y\; \backslash in\; G.$$

Since the first derivations are zero, it is readily seen that $\omega_2$ so defined also satisfies the cocycle condition. Then,

$$d\theta =0,d\backslash theta\; =\; 0,$$

i.e., $\theta$ itself is a cocycle. Let

$${\theta }_2(X)=\frac{d^2}{d{\lambda }^2}{\phi }_{\lambda }(X){\mathrm{\mid }}_{\lambda =0}\mathrm{.}\backslash theta\_2(X)\; =\; \backslash frac\{d^2\}\{d\backslash lambda^2\}\; \backslash varphi\_\backslash lambda(X)|\_\{\backslash lambda=0\}.$$

Differentiating (5.4) twice gives now:

This can be rewritten as

$$-d{\theta }_2(X,Y)+\phi \omega (X,Y)=2[{\theta }_1(X),{\theta }_1(Y)]\mathrm{.}-d\backslash theta\_2(X,\; Y)\; +\; \backslash varphi\backslash omega(X,\; Y)\; =\; 2[\backslash theta\_1(X),\; \backslash theta\_1(Y)].$$

Now, the right-hand side obviously is a two-cocycle in $C^2(\varphi')$ since the left-hand side is such a cocycle. Let us denote this cocycle by

$$[{\theta }_1,{\theta }_1]\mathrm{.}[\backslash theta\_1,\; \backslash theta\_1].$$

(This operation is discussed in the review article by NIJENHUIS and RICHARDSON [6]. It turns out to depend only on the cohomology class determined by $\theta_1$ in $H^1(\varphi')$). Then, we can write the relation as:

$$\phi \omega =d{\theta }_2+2[{\theta }_1,{\theta }_1]\backslash varphi\backslash omega\; =\; d\backslash theta\_2\; +\; 2[\backslash theta\_1,\; \backslash theta\_1]$$

i.e., the cohomology class determined by $\varphi\omega$ in $H^1(\varphi')$ can be written as a "square" of an element of $H^1(\varphi')$.

In summary, we have shown that there are interesting relations between the deformation theory and the analytic continuation problems that are of importance for the application of group-theoretical ideas to elementary particle physics. Before proceeding further with the general theory (in a later paper) it is appropriate to work out a further example that is of the greatest importance for physics.

6. Contraction of the Poincaré group into the Galilean group

Let T be a vector space over the real numbers, considered as an Abelian Lie algebra. (One might think of T as the Lie algebra of the group of space-time translations.) Denote elements of T by such letters