(see [ 5 , Definition 15] ) A crossed module of Lie algebras consists of a Lie algebra homomorphism $\mu:\mathfrak{m}\to\mathfrak{n}$ together with an action of $\mathfrak{n}$ on $\mathfrak{m}$ by derivations, that is a Lie algebra homomorphism

$$\nu:\mathfrak{n}\to\mathsf{Der}(\mathfrak{m})$$

from $\mathfrak{n}$ to the Lie algebra $\mathsf{Der}(\mathfrak{m})$ of derivations of $\mathfrak{m}$ so that for all $m,m^{\prime}\in\mathfrak{m}$ , $n\in\mathfrak{n}$

• (i)

  $\mu(\nu(n)m)=[n,\mu(m)]$ and

• (ii)

  $\nu(\mu(m))m^{\prime}=[m,m^{\prime}]$ .

Given the RQS $(\mathcal{G},\mu)$ on $\mathcal{X}$ , we define $\mathcal{F}=\mathcal{F}(\mathcal{G},\mu)$ as the set of functions $f:\mathcal{X}\mapsto[0,\infty)$ such that $\mu[f]=1$ , and

$\displaystyle\mu[f\log(\rho/\mu)]=\mu[\log(\rho/\mu)],$

(2.14)

for all stationary $\rho\in\mathcal{P}_{+}(\mathcal{X})$ . Moreover, we say that the RQS satisfies the entropy production estimate with constant $\delta>0$ if for all $f\in\mathcal{F}$ one has

$\displaystyle D(f,f)\;\geqslant\;\delta\,{\rm Ent}(f),$

(2.15)

where ${\rm Ent}(f)=\mu[f\log f]-\mu[f]\log\mu[f]$ .

(Pavičić and Megill, Pavičić and Megill ( 1999 , 2008 ) .) A weakly distributive ortholattice (WDL) is an OL in which condition ( 36 ) (called commensurability ) holds

$\displaystyle(a\cap b)\cup(a\cap b^{\prime})\cup(a^{\prime}\cap b)\cup(a^{\prime}\cap b^{\prime})=1.$

(36)

Let $g$ be a function. We define:

• •

  $g^{0}(b)=b$ ,

• •

  $g^{\alpha}(b)=g^{\alpha[b]}(g(b))$ .

Let $c_{c}(G)$ denote the linear space consisting of functions $G\rightarrow{\mathbb{C}}$ with finite support. We turn $c_{c}(G)$ to a right $\varepsilon(E)$ -module by setting

$$(\xi f)(g)=\xi(g)f(gg^{*})$$

for all $\xi\in c_{c}(G),f\in c_{0}(E)$ and $g\in G$ . This module is endowed with an $\varepsilon(E)$ -valued inner product given by

$$\langle\xi,\eta\rangle(e)=\sum_{g\in G,gg^{*}=e}\overline{\xi(g)}\eta(g).$$

The space $c_{c}(G)$ will be equipped with the $G$ -action

$$(h\xi)(g)=\xi(h^{*}g)\,[hh^{*}\geq gg^{*}]$$

for all $\xi\in c_{c}(G)$ and $h,g\in G$ .

The closure of $c_{c}(G)$ under the norm induced by the inner product is a $G$ -Hilbert $\varepsilon(E)$ -module denoted by $\ell^{2}(G)$ .

Let $G$ be an abelian group. A map $\varepsilon:G\times G\rightarrow{\bf\mathbb{K}^{*}}$ is called a skew-symmetric bicharacter on $G$ if the following identities hold,

1. (i)

   $\varepsilon(a,b)\varepsilon(b,a)=1$ ,

2. (ii)

   $\varepsilon(a,b+c)=\varepsilon(a,b)\varepsilon(a,c)$ ,

3. (iii)

   $\varepsilon(a+b,c)=\varepsilon(a,c)\varepsilon(b,c)$ ,

$a,b,c\in G$ ,

Let $(A,\cdot,\varepsilon,\alpha)$ and $(A^{\prime},\cdot^{\prime},\varepsilon,\alpha^{\prime})$ be two color Hom-algebras. An even linear map $f:(A,\cdot,\varepsilon,\alpha)\rightarrow(A^{\prime},\cdot^{\prime},\varepsilon,\alpha^{\prime})$ is called a weak morphism if, for any $x,y\in A$ ,

$$f(x\cdot y)=f(x)\cdot^{\prime}f(y).$$

If in addition $f\circ\alpha=\alpha^{\prime}\circ f$ , $f$ is said to be a morphism of color Hom-algebras.

[ 34 ] A Hom-associative color algebra is a color Hom-algebra $(A,\cdot,\varepsilon,\alpha)$ such that

$\displaystyle\alpha(x)\cdot(y\cdot z)=(x\cdot y)\cdot\alpha(z),$

(2.1)

for any $x,y,z\in\mathcal{H}(A)$ .

[ 9 ] A color Hom-algebra $(A,\mu,\varepsilon,\alpha)$ is said to be a Hom-Lie admissible color algebra if, for any hogeneous elements $x,y\in A$ , the bracket $[\cdot,\cdot]:A\times A\rightarrow A$ defined by

$$[x,y]=\mu(x,y)-\varepsilon(x,y)\mu(y,x)$$

satisfies the $\varepsilon$ -Hom-Jacobi identity.

Given a reflection group $G$ acting on $\mathbb{C}^{p}$ , a $G$ -reflection map $f\colon X\to\mathbb{C}^{p}$ is a map of the form

$$f=\omega\circ h,$$

where $h\colon X\hookrightarrow\mathbb{C}^{p}$ is an embedding and $\omega\colon\mathbb{C}^{p}\to\mathbb{C}^{p}$ is the orbit map of $G$ .

Some results do not require $h$ to be an embedding. If we only ask $h\colon X\to\mathbb{C}^{p}$ to be a finite map, then $f=\omega\circ h$ is a generalized reflection map .

Two projections $p{},q{}\in\mathcal{B}(\mathcal{H})$ are said to be in generic position if

$$p{}\wedge q{}=p{}\wedge q{}^{\perp}=p{}^{\perp}\wedge q{}=p{}^{\perp}\wedge q{}^{\perp}=0.$$

For $f\in E_{r}$ , let $\mathcal{N}^{\perp}(f)$ be a dynamical system of the type

$$\dfrac{dz}{dt}=\dfrac{-if(z)}{f^{{}^{\prime}}(z)}.$$

For $A$ a semisimple algebra, a trace function on $A$ is a $\mathbb{C}$ -linear function $\tau:A\rightarrow\mathbb{C}$ such that for all $a,b\in A$ ,

$$\tau(ab)=\tau(ba).$$

Let $f,g\in R$ with $\text{class}(g)=x_{j}$ . We first write $f,g$ as univariate polynomials in $x_{j}$ . The pseudoremainder of $f$ by $g$ is the remainder obtained from considering the coefficients of $f,g$ as coming from the field $k(x_{1},x_{2},\ldots,x_{j-1},x_{j+1},\ldots,x_{n})$ , performing univariate polynomial division of $f$ by $g$ with coefficients in this field, and then clearing denominators to obtain some equation of the form:

$\displaystyle\operatorname{lc}{(g)}^{\alpha}f=qg+r$

with $\deg_{x_{j}}(r)<\deg_{x_{j}}(g),\alpha\in\mathbb{N}$ . We require that $\alpha\leq\deg_{x_{j}}(f)-\deg_{x_{j}}(g)+1$ . (For uniqueness of $q,r$ , one takes $\alpha$ minimal.) We sometimes denote $r$ by $\text{prem}(f,g)$ .

Let a linear subspace $D(q)\subset\mathcal{H}$ be fixed. A mapping $q:D(q)\times D(q)\longrightarrow\mathbb{C}$ which satisfies

$$q[\alpha u+\beta v,w]=\alpha q[u,w]+\beta q[v,w]$$

and

$$q[u,\alpha v+\beta w]=\overline{\alpha}q[u,v]+\overline{\beta}q[u,w]$$

for all $u,v,w\in D(q)$ is called a sesquilinear form. If $D(q)$ is dense in $\mathcal{H}$ , then $q$ is densely defined.

A quasifree state $\omega_{s}\in E_{\mathfrak{A}}$ induced by the two-point operator $s\in{\mathcal{L}}({{\mathfrak{h}}})$ is called translation invariant if

$\displaystyle[s,u]=0,$

(33)

where $u\in{\mathcal{L}}({{\mathfrak{h}}})$ is the right translation from ( 11 ).