Let $\eta\colon\mathfrak{m}\to\mathfrak{g}$ and $\mu\colon\mathfrak{n}\to\mathfrak{g}$ be two Leibniz crossed modules in the previous setting. We define the non-abelian exterior product $\mathfrak{m}\curlywedge\mathfrak{n}$ of $\mathfrak{m}$ and $\mathfrak{n}$ by

$$\mathfrak{m}\curlywedge\mathfrak{n}=\dfrac{\mathfrak{m}\star\mathfrak{n}}{\mathfrak{m}\square\mathfrak{n}}.$$

The cosets of $m*n$ and $n*m$ will be denoted by $m\curlywedge n$ and $n\curlywedge m$ , respectively.

Let $\mathbb{G}$ be a groupoid. The Inertia groupoid $I(\mathbb{G})$ of $\mathbb{G}$ is defined as follows.

An object $a$ is an arrow in $\mathbb{G}$ such that its source and target are equal. A morphism $v$ joining two objects $a$ and $b$ is an arrow $v$ in $\mathbb{G}$ such that

$$v\circ a=b\circ v.$$

In other words, $b$ is the conjugate of $a$ by $v$ , $b=v\circ a\circ v^{-1}$ .

The torsion Inertia groupoid $I^{tors}(\mathbb{G})$ of $\mathbb{G}$ is a full subgroupoid of of $I(\mathbb{G})$ with only objects of finite order.

An $n$ -dimensional standard static spacetime is a Lorentzian manifold $(X^{\circ},\tilde{g})$ of the form $X^{\circ}={\mathbb{R}}\times\Sigma^{\circ}$ , with $\Sigma^{\circ}$ a manifold of dimension $n-1$ , and such that the metric $\tilde{g}$ is of the form

$$\tilde{g}=\beta dt^{2}-\pi^{*}\tilde{h},$$

where the static time coordinate $t:X^{\circ}\to{\mathbb{R}}$ is the canonical projection onto the first factor, $\pi:X^{\circ}\to\Sigma^{\circ}$ is the canonical projection onto the second factor, $\tilde{h}$ is a Riemmanian metric on $\Sigma^{\circ}$ , and $\beta\in\pazocal{C}^{\infty}(\Sigma^{\circ})$ satisfies $\beta>0$ .

For $\gamma\in\overline{C}(\mu)(c_{1},c_{2})$ and $\delta\in\overline{C}(\mu)(c_{2},c_{3})$ , define $\delta\circ\gamma\in\overline{C}(\mu)(c_{1},c_{3})$ by

$$\delta\circ\gamma=r(\gamma*\delta).$$

A commutative automorphic Lie triple system $(T,[a,b,c])$ is a Lie triple system $(T,[a,b,c])$ that satisfies

$$[[a,b,c],a^{\prime},b^{\prime}]=[[a,a^{\prime},b^{\prime}],b,c]+[a,[b,a^{\prime},b^{\prime}],c]+[a,b,[c,a^{\prime},b^{\prime}]]$$

for any $a,b,c,a^{\prime},b^{\prime}\in T$ .

A Manin triple $(\mathfrak{d},\mathfrak{g},\mathfrak{h})$ is a metrized Lie algebra $\mathfrak{d}$ , together with two Lagrangian Lie subalgebra $\mathfrak{g},\ \mathfrak{h}$ of $\mathfrak{d}$ such that

$$\mathfrak{d}=\mathfrak{g}\oplus\mathfrak{h}$$

as a vector space. Let $G$ be a Lie group integrating $\mathfrak{g}$ . Given an extension of the adjoint action $g\mapsto\operatorname{Ad}_{g}$ of $G$ on $\mathfrak{g}$ to an action by Lie algebra automorphisms of $\mathfrak{d}$ preserving the metric, with infinitesimal action the given adjoint action $\operatorname{ad}$ of $\mathfrak{g}\subseteq\mathfrak{d}$ , then $(\mathfrak{d},\mathfrak{g},\mathfrak{h})$ is called a $G$ -equivariant Manin triple .

$$\begin{cases}&\displaystyle{n\brack k}=\displaystyle{n-1\brack k}+{n\choose k}{n-1\brack k-1}\\ \\ &\displaystyle{n\brack 0}=1,\ {n\brack n}=1\end{cases}$$

We define an action of $K_{F/E}$ on $H^{*}(E)\backslash H^{*}(\mathbf{A}_{E,\mathrm{f}})/U^{*}$ as follows. Given $a\in F^{\times+}$ representing a class in $K_{F/E}$ , there exists $\gamma\in H(E)$ such that $\operatorname{nrd}(\gamma)=a$ , and $u\in U$ such that $a\operatorname{nrd}(u)\in\mathbf{A}_{E,\mathrm{f}}^{\times}\subset\mathbf{A}_{F,\mathrm{f}}^{\times}$ . Then we define the action by

$$a\cdot[x]=[\gamma xu],$$

which clearly is independent of the choice of $\gamma$ and $u$ , and preserves the fibres of $\psi$ .

A control system $\Sigma=(X,U,W,\mathcal{U},\mathcal{W},f)$ consists of a state space $X$ , an input space $U$ and a disturbance space $W$ where $X\subseteq\real{n}$ , $U\subseteq\real{m}$ and $W\subseteq\real{p}$ are compact subsets of normed vector spaces (of appropriate dimensions) containing the origins, sets $\mathcal{U}$ and $\mathcal{W}$ , of input and disturbance signals consisting of measurable essentially bounded functions ${\mu}:{{\real{}_{\geq 0}}}\rightarrow{U}$ and ${\nu}:{{\real{}_{\geq 0}}}\rightarrow{W}$ , respectively, and a continuous state transition function ${f}:{X\times U\times W}\rightarrow{X}$ satisfying the following Lipschitz assumption: there exists a constant $L>0$ such that $\|\!~{}f(x,u,w)-f(y,u,w)~{}\!\|\leq L\|\!~{}x-y~{}\!\|$ for all $x,y\in X$ , $u\in U$ , and $w\in W$ , where $\|\!~{}\cdot~{}\!\|$ represents a norm.

A trajectory ${\xi}:{(a,b)}\rightarrow{\mathbb{R}^{n}}$ associated with the control system $\Sigma{}$ and signals $\mu\in\mathcal{U}$ and $\nu\in\mathcal{W}$ is an absolutely continuous curve satisfying:

$$\dot{\xi}(t)=f(\xi(t),\mu(t),\nu(t))$$

(1)

for almost all $t\in(a,b)$ . Although we define trajectories over open intervals, we talk about trajectories $\xi:[0,\tau]\rightarrow X$ for $\tau\in\real{}_{>0}$ , with the understanding that $\xi$ is the restriction to $[0,\tau]$ of some trajectory defined on an open interval containing $[0,\tau]$ . We write $\xi_{x\mu\nu}(t)$ for the state reached by the trajectory $\xi$ starting from the initial condition $x$ and with input and disturbance signals $\mu$ and $\nu$ , respectively.

A Smale space $(X,\varphi)$ consists of a compact metric space $(X,d)$ and a homeomorphism $\varphi:X\to X$ such that there exist constants $\epsilon_{X}>0,0<\lambda<1$ and a continuous partially defined map:

$$\{(x,y)\in X\times X\mid d(x,y)\leq\epsilon_{X}\}\mapsto[x,y]\in X$$

satisfying the following axioms:

• B1

  $\left[x,x\right]=x$ ,

• B2

  $\left[x,[y,z]\right]=[x,z]$ ,

• B3

  $\left[[x,y],z\right]=[x,z]$ , and

• B4

  $\varphi[x,y]=[\varphi(x),\varphi(y)]$ ;

where in these axioms, $x$ , $y$ , and $z$ are in $X$ and in each axiom both sides are assumed to be well-defined. In addition, $(X,\varphi)$ is required to satisfy

• C1

  For $x,y\in X$ such that $[x,y]=y$ , we have $d(\varphi(x),\varphi(y))\leq\lambda d(x,y)$ and

• C2

  For $x,y\in X$ such that $[x,y]=x$ , we have $d(\varphi^{-1}(x),\varphi^{-1}(y))\leq\lambda d(x,y)$ .

Let $\mathcal{C}$ be a cooperad and $\mathcal{P}$ be an operad, then we define a operad structure on the space of linear maps $Hom(\mathcal{C},\mathcal{P})$ as follows.

The arity $n$ part of the convolution operad is defined as the space $Hom_{\mathbb{K}}(\mathcal{C}(n),\mathcal{P}(n))$ . The symmetric group action on
$Hom_{\mathbb{K}}(\mathcal{C}(n),\mathcal{P}(n))$ is defined by

$$f^{\sigma}(x)=\sigma(f(x^{\sigma^{-1}})).$$

(DC functions [21]) Let $\mathcal{C}$ be a convex subset of $\mathcal{R}^{l}$ . A real-valued function $f:\mathcal{C}\mapsto\mathcal{R}$ is called DC on $\mathcal{C}$ , if there exist two convex functions $g,h:\mathcal{C}\mapsto\mathcal{R}$ such that $f$ can be expressed in the form

$$f(x)=g(x)-h(x).$$

(44)

If $\mathcal{C}=\mathcal{R}^{l}$ , then $f$ is simply called a DC function. Each representation of the form (44) is said to be a DC decomposition of $f$ .

An involution on a (complex) algebra $\mathcal{A}$ is a map $*:\mathcal{A}\rightarrow\mathcal{A}$ such that, for any $a,b\in\mathcal{A}$ and $\mu,\nu\in\mathds{C}$ :

• i.

  $(\lambda a+\mu b)^{*}=\bar{\lambda}a^{*}+\bar{\mu}b^{*}$

• ii.

  $(ab)^{*}=b^{*}a^{*}$

• iii.

  $(a^{*})^{*}=a$

Notice that this is, basically, an abstraction of the adjoint operation on $\mathcal{B}(\mathcal{H})$ .

(Mesh Layers in $v$ - and $v_{e}$ -tetrahedra) Let $T_{(0)}=\triangle^{4}x_{0}x_{1}x_{2}x_{3}\in\mathcal{T}_{0}$ be either a $v$ - or a $v_{e}$ -tetrahedron with $x_{0}\in\mathcal{V}$ or $x_{0}\in e\in\mathcal{E}$ . We use a local Cartesian coordinate system, such that $x_{0}$ is the origin. For $1\leq i\leq n$ , the $i$ th refinement on $T_{(0)}$ produces a small tetrahedron with $x_{0}$ as a vertex and with one face, denoted by $P_{v,i}$ , parallel to the face $\triangle^{3}x_{1}x_{2}x_{3}$ of $T_{(0)}$ . See Figure 1 for example.

Then, after $n$ refinements, we define the $i$ th mesh layer $L_{v,i}$ of $T_{(0)}$ , $1\leq i<n$ , as the region in $T_{(0)}$ between $P_{v,i}$ and $P_{v,i+1}$ . We denote by $L_{v,0}$ the region in $T_{(0)}$ between $\triangle^{3}x_{1}x_{2}x_{3}$ and $P_{v,1}$ ; and let $L_{v,n}$ be the small tetrahedron with $x_{0}$ as a vertex that is bounded by $P_{v,n}$ and three faces of $T_{(0)}$ . Since it is clear that $x_{0}$ is the only point for the special refinement, we drop the sub-index $\ell$ in the grading parameter ( 14 ). Namely, for such $T_{(0)}$ , we use

$$\kappa=2^{-m/a}$$

to denote the grading parameter near $x_{0}$ ( $\kappa=\kappa_{ev}$ if $x_{0}\in\mathcal{V}$ and $\kappa=\kappa_{e}$ if $x_{0}\in e\in\mathcal{E}$ ). Define the dilation matrix

(16)

$\displaystyle\mathbf{B}_{v,i}:=\begin{pmatrix}\kappa^{-i}&0&0\\ 0&\kappa^{-i}&0\\ 0&0&\kappa^{-i}\end{pmatrix}.$

Then, by Algorithm 3.2 , $\mathbf{B}_{v,i}$ maps $L_{v,i}$ to $L_{v,0}$ for $0\leq i<n$ , and maps $L_{v,n}$ to $T_{(0)}$ . We define the initial triangulation of $L_{v,i}$ , $0\leq i<n$ , to be the first decomposition of $L_{v,i}$ into tetrahedra. Thus, the initial triangulation of $L_{v,i}$ consists of those tetrahedra in $\mathcal{T}_{i+1}$ that are contained in the layer $L_{v,i}$ .

For any configuration $c=\langle{{z},\;{\sigma\!\mid\!i\!\mid\!j},\;{t}}\rangle$ , we define the set of reachable gold brackets (with respect to gold tree $t_{\mathrm{G}}$ ) as

$$\mathit{reach}(c)=\mathit{left}(c)\cup\mathit{right}(c)$$

where the left- and right-reachable brackets are

	$\displaystyle\mathit{left}(c)$	$\displaystyle\!=\!\{\!\ _{{p}}{{X}}_{{q}}\in t_{\mathrm{G}}\mid(i,j)\prec(p,q),\,p\in\sigma\!\mid\!i\}$
	$\displaystyle\mathit{right}(c)$	$\displaystyle\!=\!\{\!\ _{{p}}{{X}}_{{q}}\in t_{\mathrm{G}}\mid p\geq j\}$

for even $z$ , with the $\prec$ replaced by $\preceq$ for odd $z$ .

Special case (initial): $\mathit{reach}(\langle{{0},\;{[0]},\;{\emptyset}}\rangle)=t_{\mathrm{G}}$ .

Let $\mathfrak{sl}_{2}$ denote the Lie algebra over $\mathbb{K}$ that has a basis $h,\,e,\,f$ and Lie bracket

$\displaystyle[h,e]=2e,\qquad[h,f]=-2f,\qquad[e,f]=h.$