Half-metallic ferromagnetism and spin-polarization in CrO

Definition 2.3 (Distributive Lattice)

Distributive lattices are lattices $\langle L,\sqsubseteq,\sqcap,\sqcup,\bot,\top\rangle$ that satisfy the distributive law

a\sqcap(b\sqcup c)=(a\sqcap b)\sqcup(a\sqcap c)

for any elements $a$ , $b$ , and $c$ of the lattice.

Definition 2.4 (homomorphism)

A function between distributive lattices $f:L\rightarrow K$ , is a homomorphism if

	$\displaystyle f(a\sqcap b)=f(a)\sqcap f(b)$
	$\displaystyle f(a\sqcup b)=f(a)\sqcup f(b)$
	$\displaystyle f(\bot)=\bot$
	$\displaystyle f(\top)=\top$

Definition 4.4 .

Given two twisted commutative algebras $A$ and $B$ , together with a bilinear pairing $(\cdot\,,\cdot):\Gamma\otimes\Gamma\rightarrow\mathbb{Z}$ ( $\Gamma=\Gamma_{A}\oplus\Gamma_{B}$ ) extending ²⁹ ²⁹ 29 In the case that is of interest to us, this extension is not the trivial one. those on $\Gamma_{A}$ and $\Gamma_{B}$ , we can form the twisted tensor product $A\widetilde{\otimes}B$ , again a twisted commutative algebra, by letting

a\otimes b\cdot a^{\prime}\otimes b^{\prime}=(-1)^{p(\lambda)p(\chi)+(\lambda,% \chi)}a\cdot a^{\prime}\otimes b\cdot b^{\prime}

for $b\in B^{\lambda}$ and $a^{\prime}\in A^{\chi}$ .

Definition 2.4 (Hörmander)

Let $\Lambda$ be a $C^{\infty}$ conic Lagrangian submanifold in $T^{\ast}(X)$ . A Lagrangian distribution $u(x)$ of order $\mu$ defined by $\Lambda$ is a locally finite (in $X$ ) sum of integrals of the form

u(x)=\int e^{i\phi(x,\theta)}a(x,\theta)\,d\theta,

where $\theta\in\mathbb{R}^{N}$ , $\phi$ satisfies ( 2.6 ), $\Lambda_{\phi}$ is a piece of $\Lambda$ and

a(x,\theta)\in S^{\mu-\frac{N}{2}+\frac{\dim X}{4}}(X\times\mathbb{R}^{N}).

Definition 3.3

For our convenience, we introduce the map

(\,\cdot\,,\cdot\,)^{\dagger}:\;(p,q)\mapsto(p,q)^{\dagger}=(1/p,1/q).

(3.5)

Definition 3.2 .

A function

e:X_{n}\longrightarrow M

is called an $M$ -valued Manin symbol over $\Gamma_{0}(p^{n})$ if $e$ satisfies the following “Manin relations” for all ${\bf x}=(x,y)\in X_{n}$ and $\lambda\in\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{\times}$ :

(1)

$e(\lambda{\bf x})=\chi(\lambda)\cdot e({\bf x})$ ;
(2)

$e(x,y)+e(y,-x)=0$ ; and
(3)

$e(x,y)+e(y,-x-y)+e(-x-y,x)=0$ .

We denote by

{{\hbox{\rm Manin}}_{\Gamma_{0}(p^{n})}}(M).

the group of all $M$ -valued Manin symbols over ${\Gamma_{0}(p^{n})}$ .

Definition 1.1 .

A Hom-associative algebra is a triple $(V,\mu,\alpha)$ consisting of a linear space $V$ , a bilinear map $\mu:V\times V\rightarrow V$ and a homomorphism $\alpha:V\rightarrow V$ satisfying

\mu(\alpha(x),\mu(y,z))=\mu(\mu(x,y),\alpha(z))

Definition 1.2 .

A Hom-Leibniz algebra is a triple $(V,[\cdot,\cdot],\alpha)$ consisting of a linear space $V$ , bilinear map $[\cdot,\cdot]:V\times V\rightarrow V$ and a homomorphism $\alpha:V\rightarrow V$ satisfying

[[x,y],\alpha(z)]=[[x,z],\alpha(y)]+[\alpha(x),[y,z]]

(1.1)

Definition 1.3 (Hartwig, Larsson and Silvestrov [ 4 ] ) .

A Hom-Lie algebra is a triple $(V,[\cdot,\cdot],\alpha)$ consisting of a linear space $V$ , bilinear map $[\cdot,\cdot]:V\times V\rightarrow V$ and a linear space homomorphism $\alpha:V\rightarrow V$ satisfying

	$\displaystyle[x,y]=-[y,x]$		(skew-symmetry)		(1.4)
	$\displaystyle{}\circlearrowleft_{x,y,z}{[\alpha(x),[y,z]]}=0$		(Hom-Jacobi identity)		(1.5)

for all $x,y,z$ from $V$ , where $\circlearrowleft_{x,y,z}$ denotes summation over the cyclic permutation on $x,y,z$ .

Definition 2.1 .

Let $\mathcal{A}$ be a Hom-algebra structure on $V$ defined by the multiplication $\mu$ and a homomorphism $\alpha$ . Then $\mathcal{A}$ is said to be Hom-Lie admissible algebra over $V$ if the bracket defined for all $x,y\in V$ by

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

(2.1)

satisfies the Hom-Jacobi identity $\circlearrowleft_{x,y,z}{[\alpha(x),[y,z]]}=0$ for all $x,y,z\in V$ .

Definition 3.5 .

A Hom-Vinberg algebra is a triple $(V,\mu,\alpha)$ consisting of a linear space $V$ , a bilinear map $\mu:V\times V\rightarrow V$ and a homomorphism $\alpha$ satisfying

\mu(\alpha(x),\mu(y,z))-\mu(\alpha(y),\mu(x,z))=\mu(\mu(x,y),\alpha(z))-\mu(% \mu(y,x),\alpha(z))

(3.5)

Definition 3.6 .

A Hom-pre-Lie algebra is a triple $(V,\mu,\alpha)$ consisting of a linear space $V$ , a bilinear map $\mu:V\times V\rightarrow V$ and a homomorphism $\alpha$ satisfying

\mu(\alpha(x),\mu(y,z))-\mu(\alpha(x),\mu(z,y))=\mu(\mu(x,y),\alpha(z))-\mu(% \mu(x,z),\alpha(y))

(3.6)

Definition 4.1 .

A Hom-algebra $\mathcal{A}=(V,\mu,\alpha)$ is called flexible if for any $x,y$ in $V$

\mu(\mu(x,y),\alpha(x))=\mu(\alpha(x),\mu(y,x)))

(4.1)

Definition 2

Let $(X,O)$ be a germ of normal surface singularity, $\pi:\tilde{X}\longrightarrow X$ be the minimal resolution of singularities and $E_{1},\ldots,E_{n}$ be the components of the exceptional divisor. Let $K_{\tilde{X}}$ the canonical divisor on ${\tilde{X}}$ . We say that $(X,O)$ satisfies numerical Nash condition for $(i,j)$ if the following condition is fulfilled

(N N_{(i, j)}) \exists E = \sum_{k = 1}^{n} n_{k} E_{k}, n_{k} \in {𝐼𝑁}^{*} with n_{i} < n_{j} and - E \cdot E_{k} \geq 2 K_{\tilde{X}} \cdot E_{k}, \forall k = 1, \dots, n

We also say that $(X,O)$ satisfies numerical Nash condition, (NN), if $(NN_{(i,j)})$ is true for all couples $(i,j)$ , with $i\not=j$ .

Definition II.1

birk2nd A lattice is an algebra ${\rm L}=\langle\mathcal{L}_{\rm O},\cap,\cup\rangle$ such that the following conditions are satisfied for any $a,b,c\in\mathcal{L}_{\rm O}$ :

\displaystyle\begin{array}[]{ccc}a\cup b=b\cup a\hfill&\hbox to 56.905512pt{% \hfill}&a\cap b=b\cap a\hfill\\ (a\cup b)\cup c=a\cup(b\cup c)&&(a\cap b)\cap c=a\cap(b\cap c)\\ a\cap(a\cup b)=a\hfill&&a\cup(a\cap b)=a\hfill\end{array}

Definition II.3

birk3rd An ortholattice (OL) is an algebra $\langle\mathcal{L}_{\rm O},^{\prime},\cap,\cup,0,1\rangle$ such that $\langle\mathcal{L}_{\rm O},\cap,\cup\rangle$ is a lattice with unary operation ${}^{\prime}$ called orthocomlementation which satisfies the following conditions for $a,b\in\mathcal{L}_{\rm O}$ ( $a^{\prime}$ is called the orthocomplement of $a$ ):

	$\displaystyle a\cup a^{\prime}=1,\qquad\qquad a\cap a^{\prime}=0$		(2)
	$\displaystyle a\leq b\quad\Rightarrow\quad b^{\prime}\leq a^{\prime}$		(3)
	$\displaystyle a^{\prime\prime}=a$		(4)

Definition II.5

zeman We say that $a$ and $b$ commute in OML , and write $aCb$ , when either of the following equivalent equations hold:

	$\displaystyle a=((a\cap b)\cup(a\cap b^{\prime}))$		(6)
	$\displaystyle a\cap(a^{\prime}\cup b)\leq b$		(7)

Definition 4.1

(a) Let $A$ be a $\mathbb{H}$ -vector space as well as a $\mathbb{R}$ -algebra (with respect to the induced $\mathbb{R}$ -vector space structure on $A$ ). Then $A$ is called a $\mathbb{H}$ -algebra if the following conditions are satisfied for any $a,b\in A$ and $\gamma\in\mathbb{H}$ :

(\gamma a)b\ =\ \gamma(ab),\qquad(a\gamma)b\ =\ a(\gamma b)\qquad{\rm and}% \qquad(ab)\gamma=a(b\gamma).

Moreover, a $\mathbb{R}$ -involution ${}^{*}$ on $A$ is called a $\mathbb{H}$ -involution if for any $\alpha\in\mathbb{H}$ and $b\in A$ ,

(\alpha b)^{*}\ =\ b^{*}\alpha^{*}\qquad{\rm and}\qquad(b\alpha)^{*}\ =\ % \alpha^{*}b^{*}.

In this case, $A$ is called a $\mathbb{H}$ -involutive algebra .

(b) Suppose that $A$ is a $\mathbb{H}$ -involutive algebra with a $\mathbb{H}$ -norm $\|\cdot\|$ . Then $A$ is called a $\mathbb{H}$ - $B^{*}$ -algebra if it is complete under $\|\cdot\|$ and $\|\cdot\|$ satisfies the following properties:

\|ab\|\ \leq\ \|a\|\ \|b\|\qquad{\rm and}\qquad\|a^{*}a\|=\|a\|^{2}\qquad% \qquad(a,b\in A).

Definition 4.3 .

Let $G$ be an amenable group acting on a probability space $X$ . The spectrum of this action is the associated $G$ -representation on $L_{2}(X)$ given by

(\rho(g)f)(x)=f(g^{-1}x).

The spectrum is called Lebesgue with multiplicity $N$ (which can be infinity) if $L_{2}(X)$ decomposes into direct sum of $N$ copies of the regular representation of $G$ .