An automorphism $\sigma$ on $X_{C}\otimes X_{C}$ is a Yang-Baxter operator on $X_{C}$ , iff $\sigma$ satisfies

$$(\mathrm{id}\bar{\otimes}\sigma)a(\sigma\bar{\otimes}\mathrm{id})a^{-1}(\mathrm{id}\bar{\otimes}\sigma)a=a(\sigma\bar{\otimes}\mathrm{id})a^{-1}(\mathrm{id}\bar{\otimes}\sigma)a(\sigma\bar{\otimes}\mathrm{id}).$$

Here, $a=a_{X_{C},X_{C},X_{C}}$ and $\mathrm{id}=\mathrm{id}_{X_{C}}$ .

A deformation retract datum of complexes of $R$ -modules consists of a diagram

$$\xymatrix@C+2pc{(L,b)\ar@<-0.8ex>[r]_{\iota}&(M,b),\ar@<-0.8ex>[l]_{p}}\quad h$$

where $(L,b)$ and $(M,b)$ are complexes, $p$ and $\iota$ are morphisms of complexes, and $h$ is a degree one $R$ -linear map $M\longrightarrow M$ , which together satisfy the following two conditions:

• (i)

  $p\iota=1$ ,

• (ii)

  $\iota p=1+bh+hb$ .

Notice that in particular $p$ is a homotopy equivalence with inverse $\iota$ .

A symplectic Lie algebra is a Lie algebra $\mathfrak{s}$ endowed with a symplectic structure $\omega$ such that $\forall x$ , $y$ , $z\in\mathfrak{s}$ ,

$$\omega([x,y],z)+\omega([y,z],x)+\omega([z,x],y)=0.$$

(B.83)

For any relative prime integers $q$ and $a$ , define

$$c(q;a)=-1+\#\{x{\text{\rm\ (mod~{}$q$)}}\colon x^{2}\equiv a{\text{\rm\ (mod~{}$q$)}}\}.$$

Note that $c(q;a)$ takes only the values $-1$ and $\rho(q)-1$ . Now, with $X_{\gamma}$ as defined in Definition 2.2 , define the random variable

$$X_{q;a,b}=c(q,b)-c(q,a)+2\sum_{\chi{\text{\rm\ (mod~{}$q$)}}}|\chi(b)-\chi(a)|\sum_{\begin{subarray}{c}\gamma>0\\ L(1/2+i\gamma,\chi)=0\end{subarray}}\frac{X_{\gamma}}{\sqrt{\frac{1}{4}+\gamma^{2}}}.$$

Note that the expectation of the random variable $X_{q;a,b}$ is either $\pm\rho(q)$ or 0, depending on the values of $c(q,a)$ and $c(q,b)$ . $\diamondsuit$

. The bijection $\lambda$ is defined by $\lambda(x)$ satisfying the properties that $\lambda(x)=x$ for $x\in X$ and that

$$\lambda(l)=[\lambda(m),\lambda(n)]$$

(5.7)

for $l\in Ly(X)=X$ , where $\sigma(l)=(m,n)$ is the standard factorization of $l$ .

[ 25 ] , see also [ 3 , Definition 2, p.543] . The group $\tilde{G}_{2}$ is defined as the subgroup $\{g\in GL({\mathbb{R}}^{7})|\,g^{*}(\tilde{\phi})=\tilde{\phi}\}$ where

$$\tilde{\phi}={\omega}^{123}-{\omega}^{145}-{\omega}^{167}-{\omega}^{246}+{\omega}^{257}+{\omega}^{347}+{\omega}^{356}.$$

[ 25 ] , see also [ 13 , IV.1.A, p.114] , and [ 3 , Definition 1, p.539] . The group $G_{2}$ is defined as the subgroup $\{g\in GL({\mathbb{R}}^{7})|\,g^{*}(\tilde{\phi})=\tilde{\phi}\}$ where

$$\phi={\omega}^{123}+{\omega}^{145}+{\omega}^{167}+{\omega}^{246}-{\omega}^{257}-{\omega}^{347}-{\omega}^{356}.$$

The algebra $\mathsf{bv}$ is an associative algebra with two generators $x,y$ and two relations

	$$\displaystyle y^{2}=0,$$
	$$\displaystyle x^{2}y+xyx+yx^{2}=0.$$

The anti-associative operad $\widetilde{\operatorname{As}}$ is the nonsymmetric operad with one generator $f(\textrm{-},\textrm{-})\in\widetilde{\operatorname{As}}(2)$ and one relation

(5)

$$f(f(\textrm{-},\textrm{-}),\textrm{-})+f(\textrm{-},f(\textrm{-},\textrm{-}))=0.$$