samples/texts/1749790/page_11.md

the algebra is not isomorphic to the one with which we started at $\varepsilon = 0$. The Gell-Mann formula itself, i.e., (5.2), now provides an analytic continuation of the representation of the $[\cdot, \cdot]_0$ structure that is given, each representation for $\varepsilon$ being a representation of the $[\cdot, \cdot]_e$ structure.

Let us now look for the interpretation of this in terms of cohomology. Let us change notations to conform with our earlier work. Suppose $G$ and $L$ are Lie algebras, with the bracket in $G$ given by $[X, Y]$, and suppose $\varphi$ is a homomorphism $G \to L$. Again, let $\varphi'$ be the homomorphism for $G$ into the linear transformations on $L$ given by:

$${\phi }^{\mathrm{\prime }}(X)(Z)=[\phi (X),Z]\text{for }X\in G,Z\in L\mathrm{.}\backslash varphi\text{'}(X)(Z)\; =\; [\backslash varphi(X),\; Z]\; \backslash quad\; \backslash text\{for\}\; \backslash \; X\; \backslash in\; G,\; Z\; \backslash in\; L.$$

Suppose a one-parameter family

$$([X,Y]{)}_{\lambda }([X,\; Y])\_\{\backslash lambda\}$$

of Lie algebra structures is given on $G$, reducing to the given one for $\lambda = 0$. Let $\gamma: G \to (\text{linear maps on } G)$ be the adjoint representation of the $\lambda = 0$ Lie algebra on $G$, i.e.,

$$\gamma (X)(Y)=[X,Y]\text{for }X,Y\in G\mathrm{.}\backslash gamma(X)(Y)\; =\; [X,\; Y]\; \backslash quad\; \backslash text\{for\; \}\; X,\; Y\; \backslash in\; G.$$

Then, we know that the formula:

$$\omega (X,Y)=\frac{d}{d\lambda }[X,Y{]}_{\lambda }{\mathrm{\mid }}_{\lambda =0}\backslash omega(X,\; Y)\; =\; \backslash frac\{d\}\{d\backslash lambda\}\; [X,\; Y]\_\{\backslash lambda\}|\_\{\backslash lambda=0\}$$

defines $\omega$ as a two-cocycle relative to $\gamma$, i.e., on element in $Z^2(\gamma)$, whose cohomology class in $H^2(\gamma)$ measures the "nonisomorphism" of the structure at $\gamma = 0$ and that for small, but nonzero $\gamma$.

Suppose further that, for each $\lambda$, $\varphi_\lambda$ is a linear mapping of $G \to L$ reducing to $\varphi$ for $\lambda = 0$, such that:

$${\phi }_{\lambda }([X,Y{]}_{\lambda })=[{\phi }_{\lambda }(X),{\phi }_{\lambda }(Y)]\text{for }X,Y\in G\mathrm{.}(5.4)\backslash varphi\_\{\backslash lambda\}([X,\; Y]\_\{\backslash lambda\})\; =\; [\backslash varphi\_\{\backslash lambda\}(X),\; \backslash varphi\_\{\backslash lambda\}(Y)]\; \backslash quad\; \backslash text\{for\; \}\; X,\; Y\; \backslash in\; G.\; \backslash quad\; (5.4)$$

Define $\varphi: G \to L$ by the formula:

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

$\theta$ is a one-cochain in $C^1(\varphi')$. However, it is not a cocycle. In fact, let us differentiate (5.4) and set $\lambda = 0$:

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

This gives the formula:

$$\phi (\omega )=d\theta (5.5)\backslash varphi(\backslash omega)\; =\; d\backslash theta\; \backslash qquad\; (5.5)$$

where $\varphi(\omega)$ is the two-chain in $C^2(\varphi')$ given by:

$$\phi (\omega )(X,Y)=\phi (\omega (X,Y))\mathrm{.}\backslash varphi(\backslash omega)(X,\; Y)\; =\; \backslash varphi(\backslash omega(X,\; Y)).$$

Thus, $\omega$ considered as a cocycle in $C^2(\gamma)$ is not necessarily a coboundary, but its image under $\varphi$, $\varphi(\omega)$, is a coboundary, and the element $\varphi$ in $C^1(\gamma)$ is the first term in the analytic continuation of $\varphi$.