Ω_0 from the Apparent Evolution of Gas Fractions in Rich Clusters of Galaxies

Definition 6.1 .

We’ll say that a (weighted) graph morphism $\pi:G_{1}\rightarrow G_{2}$ is a covering, iff it is a surjective local isomorphism, i. e. for each vertex $v\in G_{2}$ there is a vertex $u\in G_{1}$ such that

v=\pi(u),

(6.1)

and each vertex $u\in G_{1}$ satisfying ( 6.1 ) has the same count ( and weights) of both incoming and outcoming edges starting or ending in $u$ as the edges starting and ending in $v$ .

Definition 1

We say that $M$ is a braided $T({\bf g}_{q})$ –module (or, simply braided module), if $M$ is equiped with a structure of $U_{q}(sl(n))$ –module and of $T({\bf g}_{q})$ –module, and these structures are related as

u\cdot(gm)=\left(u^{(1)}\cdot g\right)\left(u^{(2)}\cdot m\right)

(5.1)

for any $u\in U_{q}(sl(n))$ , $g\in T({\bf g}_{q})$ and $m\in M$ .

Definition 6.1

Let $M$ be a differentiable manifold. A Poisson bracket (PB) on ${\cal F}(M)$ is a bilinear mapping $\{\cdot,\cdot\}:{\cal F}(M)\times{\cal F}(M)\rightarrow{\cal F}(M)$ that satisfies ( $f,g,h\in{\cal F}(M)$ )

a)

Skew-symmetry

\{f,g\}=-\{g,f\}\quad,

(6.1)

b)

Leibniz’s rule,

\{f,gh\}=g\{f,h\}+\{f,g\}h\quad,

(6.2)

c)

Jacobi identity

\hbox{Alt}\{f,\{g,h\}\}=\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0\quad.

(6.3)

A PB on $M$ defines a PS.