Anisotropic X-ray emission in AGN accretion discs.

Definition .

Let $\varphi$ be as above. We say that $\varphi$ is a limiting Carleman weight (for the Laplacian) if

\{a,b\}=0\quad\text{when }a=b=0.

(24)

Definition 2

Two crossed modules $\mu:{\mathfrak{m}}\to{\mathfrak{n}}$ (with action $\eta$ ) and $\mu^{\prime}:{\mathfrak{m}}^{\prime}\to{\mathfrak{n}}^{\prime}$ (with action $\eta^{\prime}$ ) such that $ker(\mu)\,=\,ker(\mu^{\prime})=:V$ and $coker(\mu)\,=\,coker(\mu^{\prime})=:{\mathfrak{g}}$ are called elementary equivalent if there are morphisms of Lie algebras $\phi:{\mathfrak{m}}\to{\mathfrak{m}}^{\prime}$ and $\psi:{\mathfrak{n}}\to{\mathfrak{n}}^{\prime}$ such that they are compatible with the actions, meaning:

\phi(\eta(n)\cdot m)\,=\,\eta^{\prime}(\psi(n))\cdot\phi(m)\,\,\,\,\,\,\,% \forall n\in{\mathfrak{n}}\,\,\,\forall m\in{\mathfrak{m}},

and such that the following diagram is commutative:

Definition 24 .

Let $W$ be any integer and let $(t,d)\in(\mathbb{Z}/W\mathbb{Z})^{2}$ be any pair of integers modulo $W$ with $d\in(\mathbb{Z}/W\mathbb{Z})^{*}$ . Suppose that $G\subseteq GL_{2}(\mathbb{Z}/W\mathbb{Z})$ is any subgroup. We say that $G$ represents the pair $(t,d)$ if there is a matrix $g\in G$ satisfying

\textrm{tr}\,(g)=t,\quad\det(g)=d.

Definition 4.2

A generalized function $u\in\mathcal{G}(\mathbb{R}^{d})$ (resp. $\mathcal{G}(\mathbb{R}^{d}\setminus\{0\})$ ) is called homogeneous of degree $\alpha$ , if for each $\lambda\in\mathbb{R}^{+}$ ,

u(\lambda x)=\lambda^{\alpha}u(x)

(4.4)

holds in $\mathcal{G}(\mathbb{R}^{d})$ $($ resp. $\mathcal{G}(\mathbb{R}^{d}\setminus\{0\}))$ .

Definition 3.1 .

A vertex algebroid is a septuple $\mathcal{A}=(A,T,\Omega,\partial,\gamma,\langle,\rangle,c)$ where $A$ is a commutative algebra, $T$ is a Lie algebra acting by derivations on $A$ and equipped with the structure of an $A$ -module, $\Omega$ is an $A$ -module equipped with the structure of a module over the Lie algebra $T$ , $\partial:A\to\Omega$ is an $A$ -derivation and a morphism of $T$ -modules, $\langle,\rangle:(T\oplus\Omega)\times(T\oplus\Omega)\to A$ is a symmetric bilinear pairing which is zero on $\Omega\times\Omega$ , $c:T\times T\to\Omega$ is a skew symmetric bilinear pairing and $\gamma:A\times T\to\Omega$ is a bilinear map such that the following properties are satisfied for any $a,b\in A$ , $\tau,\tau_{i},\nu\in T$ and $\omega\in\Omega$ :

[\tau,a\nu]=a[\tau,\nu]+\tau(a)\nu

(a\tau)(b)=a\tau(b)

\langle\tau,\partial a\rangle=\tau(a)

\tau(a\omega)=\tau(a)\omega+a\tau(w)

(a\tau)(\omega)=a\tau(\omega)+\langle\tau,\omega\rangle\partial a

\tau(\langle\nu,\omega\rangle)=\langle[\tau,\nu],\omega\rangle+\langle\nu,\tau% (\omega)\rangle

\gamma(a,b\tau)=\gamma(ab,\tau)-a\gamma(b,\tau)-\tau(a)\partial b-\tau(b)\partial a

\langle a\tau_{1},\tau_{2}\rangle=a\langle\tau_{1},\tau_{2}\rangle+\langle% \gamma(a,\tau_{1}),\tau_{2}\rangle-\tau_{1}\tau_{2}(a)

c(a\tau_{1},\tau_{2})=ac(\tau_{1},\tau_{2})+\gamma(a,[\tau_{1},\tau_{2}])% \gamma(\tau_{2}(a),\tau_{1})+\tau_{2}(\gamma(a,\tau_{1}))

-\frac{1}{2}\langle\tau_{1},\tau_{2}\rangle\partial a+\frac{1}{2}\partial\tau_% {1}\tau_{2}(a)-\frac{1}{2}\partial\langle\tau_{2},\gamma(a,\tau_{1})\rangle

\langle[\tau_{1},\tau_{2}],\tau_{3}\rangle+\langle\tau_{2},[\tau_{1},\tau_{3}]% \rangle=\tau_{1}(\langle\tau_{2},\tau_{3}\rangle)-\frac{1}{2}\tau_{2}(\langle% \tau_{1},\tau_{3}\rangle)-\frac{1}{2}\tau_{3}(\langle\tau_{1},\tau_{2}\rangle)

+\langle\tau_{2},c(\tau_{1},\tau_{3})\rangle+\langle\tau_{3},c(\tau_{1},\tau_{% 2})\rangle

d_{\textrm{Lie}}c(\tau_{1},\tau_{2},\tau_{3})=-\frac{1}{2}\partial\{\langle[% \tau_{1},\tau_{2}],\tau_{3}\rangle+\langle[\tau_{1},\tau_{3}],\tau_{2}\rangle-% \langle[\tau_{2},\tau_{3}],\tau_{1}\rangle-\tau_{1}(\langle\tau_{2},\tau_{3}\rangle)

+\tau_{2}(\langle\tau_{1},\tau_{3}\rangle)-2\langle\tau_{3},c(\tau_{1},\tau_{2% })\rangle\}

where $d_{\textrm{Lie}}c(\tau_{1},\tau_{2},\tau_{3})=\tau_{1}c(\tau_{2},\tau_{3})-% \tau_{2}c(\tau_{1},\tau_{3})+\tau_{3}c(\tau_{1},\tau_{2})-c([\tau_{1},\tau_{2}% ],\tau_{3})+c([\tau_{1},\tau_{3}],\tau_{2})-c([\tau_{2},\tau_{3}],\tau_{1})$ . (cf. [GMS1] 1.1, 1.4)

Definition 4 .

The double Livernet–Loday operad ${\mathrsfs{L}\!\mathrsfs{L}_{\hbar}}$ is a quadratic operad over $\mathbbold{k}[[\hbar]]$ generated by a commutative operation $a,b\mapsto a\cdot b$ and a skew-commutative operation $a,b\mapsto[a,b]$ satisfying the identities

(i)

$[a\cdot b,c]=a\cdot[b,c]+[a,c]\cdot b,$
(ii)

$[a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0,$
(iii)

$(a\cdot b)\cdot c-a\cdot(b\cdot c)=\hbar^{2}[b,[a,c]]$ .

Definition 3.5 .

Let $T\subset V$ be an open cone, $x$ be in $\partial T$ , and $h$ be a function from $T$ to $\mathbb{R}$ satisfying

h((1-\lambda)x+\lambda y)=\log\lambda+h(y)

(8)

whenever $y$ and $(1-\lambda)x+\lambda y$ are in $T$ and $\lambda>0$ . We define the extension of $h$ to $\tau(T,x)$ by

h|^{\tau(T,x)}(y):=-\log\lambda_{y}+h((1-\lambda_{y})x+\lambda_{y}y),\qquad% \text{for all $y\in\tau(T,x),$}

where $\lambda_{y}>0$ is chosen so that $(1-\lambda_{y})x+\lambda_{y}y$ is in $T$ .

Définition 2 .

Un fibré $E$ est dit $c$ -réel s’il existe une involution antilinéaire de $E$ dans $E$ relevant $c$ . Dans ce cas, on désigne par $\kappa:\Gamma(E)\to\Gamma(E)$ l’involution sur l’espace des sections de $E$ induite par $\hat{c}$ :

\kappa(s)=\hat{c}^{-1}\circ s\circ c.

Une section $s$ de $E$ est dite symétrique si $\kappa(s)=s$ .

Definition 7.1

5.1 By ordering $\sigma$ we mean an enriched permutation it the following sense. Given $d$ positive real numbers $n_{1},\cdots,n_{d}$ , $\sigma$ acts by a permutation on the indices so that $n_{\sigma(1)}\leq\cdots\leq n_{\sigma(d)}$ . The permutation is enriched in the following way. It keep track whether after the permutation two consecutive numbers are equal. To set the notation, given $5$ numbers, if

\sigma=(5(31)24)

then

n_{5}<n_{3}=n_{1}<n_{2}<n_{4}.

Also, $\sigma(1)=5$ , $\sigma(2)=3$ , $\sigma(3)=1$ , $\sigma(4)=2$ , $\sigma(5)=2$ . And

(5(31)24)=(5(13)24).

Definition 2.1 .

A germ of a curve singularity in characteristic $\neq 2$ is called an $A_{n}$ -singularity if it is formally isomorphic to

y^{2}-x^{n+1}=0,

(see Artin [ 4 ] , and Greuel and Kröning [ 8 ] .)

Definition 12

For any integer $h>1$ we define the map $\phi:\mathbb{Z}_{2^{h}}\to\mathbb{Z}_{2}^{2^{h-1}}$ as follows. If $0\leq a\leq 2^{h-1}$ , the image $\phi(a)$ is the binary word $(000\cdots 111)$ with Hamming weight equal to $a$ . If $2^{h-1}<a<2^{h}$ , the image $\phi(a)$ is the binary word $(111\cdots 000)$ with Hamming weight equal to $2^{h}-a$ . We also define the maps $\beta,\gamma:\mathbb{Z}_{2^{h}}\to\mathbb{Z}_{2}^{2^{h-2}}$ such that for each $a\in\mathbb{Z}_{2^{h}}$

\phi(a)=(\beta(a),\gamma(a)).

Finally the maps $\phi$ , $\beta$ , and $\gamma$ are extended in the obvious way to act on words in $\mathbb{Z}_{2^{h}}^{n}$ .

Definition 2.1 .

An algebra $L$ over a field $F$ is called a Leibniz algebra if the Leibniz identity

[x,[y,z]]=[[x,y],z]-[[x,z],y]

holds for any $x,y,z\in L,$ where $[\ ,\ ]$ is the multiplication in $L.$

Definition 1 .

A group $G$ acting faithfully on the tree $X^{*}$ is said to be self-similar if for every $g\in G$ and every $x\in X$ there exist $h\in G$ and $y\in X$ such that

g(xw)=yh(w)

for all $w\in X^{*}$ .

Definition .

Let $\mathsf{A}$ be a $C^{*}$ -algebra with representations $(\pi,\mathsf{h})$ and $(\pi^{\prime},\mathsf{h}^{\prime})$ . A map $\delta:\mathsf{A}\to B(\mathsf{h};\mathsf{h}^{\prime})$ is called a $(\pi^{\prime},\pi)$ - derivation if it satisfies

\delta(ab)=\delta(a)\pi(b)+\pi^{\prime}(a)\delta(b);

it is inner if it is implemented by an operator $T\in B(\mathsf{h};\mathsf{h}^{\prime})$ in the sense that

\delta:a\to\pi^{\prime}(a)T-T\pi(a).

Definition 3.7 .

Let $B$ be a $G$ -algebra and $\mathbb{M}$ a functor of $\mathcal{B}$ -modules. We say that $\mathbb{M}$ is a ${\mathcal{B}}G$ -module if it has a $G$ -module structure which is compatible with the $\mathcal{B}$ -module structure, that is,

g(b\cdot m)=g(b)\cdot g(m)

for every $g\in G$ , $b\in{\mathcal{B}}$ and $m\in\mathbb{M}$ .