Let $U_{q}(\mathfrak{sl}_{2})$ denote the $\mathbb{K}$ -algebra with generators $k^{\pm 1},e,f$ and relations

	$\displaystyle\qquad kk^{-1}=k^{-1}k=1,$
	$\displaystyle ke=q^{2}ek,\quad\quad kf=q^{-2}fk,$
	$\displaystyle\qquad ef-fe=\frac{k-k^{-1}}{q-q^{-1}}.$

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

(2.2)

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

Define the map $\widetilde{h}:\mathbb{R}\to\mathbb{R}$ by:

$$\widetilde{h}(\varphi)=\psi\mbox{ where }\arg[h(re^{i\varphi})]=\psi$$

for any $r>0$ .

Given two morphisms $\mathfrak{f}$ and $\mathfrak{g}$ in $\mathfrak{Mor}[\mathbf{A},\mathbf{B}]$ and $\mathfrak{Mor}[\mathbf{B},\mathbf{C}]$ respectively, we define

$$\mathfrak{g}\circ\mathfrak{f}=\nu(\mathfrak{f}\mathop{\dblcolon}\mu(\mathfrak{g}))$$

A right brace is a set $G$ with two operations $+$ and $\cdot$ such that $(G,+)$ is an abelian group, $(G,\cdot)$ is a group and

$$(a+b)c+c=ac+bc,$$

(2)

for all $a,b,c\in G$ . We call $(G,+)$ the additive group and $(G,\cdot)$ the multiplicative group of the right brace.

(Invariance of ( 8 )) The integral functional ( 8 ) is said to be invariant under the one-parameter group of infinitesimal transformations

$$\begin{cases}\bar{t}=t+\varepsilon\tau(t,q)+o(\varepsilon)\,,\\ \bar{q}(t)=q(t)+\varepsilon\xi(t,q)+o(\varepsilon)\,,\\ \end{cases}$$

(16)

if

$$\int_{t_{a}}^{t_{b}}F\left(t,q(t),{{}_{a}D_{t}^{\alpha}q(t)},\lambda\right)dt=\int_{\bar{t}(t_{a})}^{\bar{t}(t_{b})}F\left(\bar{t},\bar{q}(\bar{t}),{{}_{a}D_{t}^{\alpha}\bar{q}(\bar{t})},\lambda\right)d\bar{t}$$

for any subinterval $[{t_{a}},{t_{b}}]\subseteq[a,b]$ .

(Variational invariance of ( 28 )) We say that the integral functional ( 28 ) is invariant under the one-parameter family of infinitesimal transformations

$$\begin{cases}\bar{t}=t+\varepsilon\tau(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \bar{q}(t)=q(t)+\varepsilon\xi(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \bar{u}(t)=u(t)+\varepsilon\varrho(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \bar{p}(t)=p(t)+\varepsilon\varsigma(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \end{cases}$$

(29)

if

$$\left[{\cal H}(\bar{t},\bar{q}(\bar{t}),\bar{u}(\bar{t}),\bar{p}(\bar{t}),\lambda)-\bar{p}(\bar{t})\cdot{{}_{\bar{a}}D_{\bar{t}}}^{\alpha}\bar{q}(\bar{t})\right]d\bar{t}=\left[{\cal H}(t,q(t),u(t),p(t),\lambda)-p(t)\cdot{{}_{a}D_{t}^{\alpha}}q(t)\right]dt\,.$$

(30)

A Courant algebroid on a smooth manifold $M$ consists of a vector bundle $E\to M$ , $\mathbb{R}$ -bilinear bracket $[\,,\,]:\Gamma(E)\otimes\Gamma(E)\to\Gamma(E)$ called the Dorfman bracket , non-degenerate bilinear form $\langle\,,\,\rangle$ and a bundle map $\rho:E\to TM$ called the anchor such that for all $a,b,c\in\Gamma(E)$ and $f\in\mathcal{C}^{\infty}(M)$ , we have

• (C1)

  $[a,[b,c]]=[[a,b],c]+[b,[a,c]]$ ,

• (C2)

  $\rho[a,b]=[\rho(a),\rho(b)]$ ,

• (C3)

  $[a,fb]=\rho(a)(f)b+f[a,b]$ ,

• (C4)

  $[a,b]+[b,a]=d\langle a,b\rangle$ ,

• (C5)

  $\rho(a)\langle b,c\rangle=\langle[a,b],c\rangle+\langle b,[a,c]\rangle$ ,

where $[\rho(a),\rho(b)]$ is the Lie bracket of vector fields and $d$ is the operator $d:\mathcal{C}^{\infty}(M)\to\Gamma(E)$ defined by $\langle df,a\rangle=\frac{1}{2}\rho(a)(f)$ .

A Courant algebroid $E\to M$ is exact if the sequence

$$\xymatrix{0\ar[r]&T^{*}M\ar[r]^{-}{\frac{1}{2}\rho^{*}}&E\ar[r]^{-}\rho&TM\ar[r]&0}$$

is exact. Here $\rho^{*}$ is the transpose $T^{*}M\to E^{*}$ of $\rho$ followed by the identification of $E$ and $E^{*}$ using the pairing.

Given two morphisms $\mathfrak{f}$ and $\mathfrak{g}$ in $\mathfrak{Mor}[\mathbf{A},\mathbf{B}]$ and $\mathfrak{Mor}[\mathbf{B},\mathbf{C}]$ respectively, we define

$$\mathfrak{g}\circ\mathfrak{f}=\nu(\mathfrak{f}\mathop{\dblcolon}\mu(\mathfrak{g}))$$

Let ${\cal A}$ be a vector space over the field ${\mathbb{C}}$ equipped with an inner operation $\cdot:{\cal A}\times{\cal A}\longrightarrow{\cal A}$ which associates $x\cdot y$ (or simply $xy$ ) to each pair $x,y\in{\cal A}$ . ${\cal A}$ is an algebra if this operation is distributive and associative, i.e., if for all $x,y,z\in{\cal A}$ and all $a,b\in{\mathbb{C}}$ holds:

• •

  $x\cdot(y\cdot z)=(x\cdot y)\cdot z$ ;

• •

  $x\cdot(y+z)=x\cdot y+x\cdot z$ ;

• •

  $(ab)(x\cdot y)=(ax)\cdot(by)$ .

A function $\bm{f}:\mathrm{I\kern-1.72ptR}^{p}\to\mathrm{I\kern-1.72ptR}^{q}$ is linear if for all $\bm{x},\bm{y}\in\mathrm{I\kern-1.72ptR}^{p}$ and $\alpha,\beta\in\mathrm{I\kern-1.72ptR}$ it satisfies the condition

$$\bm{f}(\alpha\bm{x}+\beta\bm{y})=\alpha\bm{f}(\bm{x})+\beta\bm{f}(\bm{y}).$$

(2.6)

We will describe the mapping of Proposition 2 with the following bijection $f:[0,3]\rightarrow[0,3]$ between indices of the set of Bell states:

$$f(0)=2,\;f(1)=3,\;f(2)=0,\;f(3)=1.$$

A weak BCC-algebra $G$ is called branchwise implicative , if

$$x\cdot yx=x$$

holds for all $x,y$ belonging to the same branch of $G$ .

A weak BCC-algebra $G$ is called $\varphi$ -implicative , if it satisfies the identity

(21)

$$xy=xy\cdot y(0\cdot 0y),$$

i.e.,

$$xy=xy\cdot y\varphi^{2}(y).$$

If ( 21 ) is satisfied only by elements belonging to the same branch, then we say that this weak BCC-algebra is branchwise $\varphi$ -implicative .

A weak BCC-algebra $G$ is called weakly positive implicative , if it satisfies the identity

(22)

$$xy\cdot z=(xz\cdot z)\cdot yz.$$

If ( 22 ) is satisfied only by elements belonging to the same branch, then we say that this weak BCC-algebra is branchwise weakly positive implicative .

A right quasigroup is a set $X$ with a binary operation $\circ$ such that for each $a,b\in X$ there exists a unique $x\in X$ such that $a\,\circ\,x\,=\,b$ . We write the solution of this equation $x\,=\,a\,\bullet\,b$ .

An idempotent right quasigroup (irq) is a right quasigroup $(X,\circ)$ such that for any $x\in X$ $x\,\circ\,x\,=\,x$ . Equivalently, it can be seen as a set $X$ endowed with two operations $\circ$ and $\bullet$ , which satisfy the following axioms: for any $x,y\in X$

1. (R1)

   $\displaystyle x\,\circ\,x\,=\,x\,\bullet\,x\,=\,x$

2. (R2)

   $\displaystyle x\,\circ\,\left(x\,\bullet\,y\right)\,=\,x\,\bullet\,\left(x\,\circ\,y\right)\,=\,y$

A Killing inner metric is an inner metric $h$ such that

$$h([\xi,\gamma],\eta)+h(\gamma,[\xi,\eta])=0$$

for any $\gamma,\eta,\xi\in{{\mathbf{\mathsf{L}}}}$ .

A locally constant inner metric is an inner metric $h$ such that the local components $h_{ab}$ of $h$ are constant functions in any local trivializations of ${{\mathbf{\mathsf{A}}}}$ .

Let $x,z\in\alpha\cap\beta$ . A broken $(\alpha,\beta)$ -heart from $x$ to $z$ is a triple

$$h=(u,y,v)$$

such that $y\in\alpha\cap\beta$ , $u$ is a smooth $(\alpha,\beta)$ -lune from $x$ to $y$ , and $v$ is a smooth $(\alpha,\beta)$ -lune from $y$ to $z$ . The point $y$ is called the midpoint of the heart. By Theorem 6.8 the broken $(\alpha,\beta)$ -heart $h$ is uniquely determined by the septuple

$$\Lambda_{h}:=(x,y,z,u({\mathbb{D}}\cap{\mathbb{R}}),v({\mathbb{D}}\cap{\mathbb{R}}),u({\mathbb{D}}\cap S^{1}),v({\mathbb{D}}\cap S^{1})).$$

Two broken $(\alpha,\beta)$ -hearts $h=(u,y,z)$ and $h^{\prime}=(u^{\prime},y^{\prime},z^{\prime})$ from $x$ to $z$ are called equivalent if $y^{\prime}=y$ , $u^{\prime}$ is equivalent to $u$ , and $v^{\prime}$ is equivalent to $v$ . The equivalence class of $h$ is denoted by $[h]=([u],y,[v])$ . The set of equivalence classes of broken $(\alpha,\beta)$ -hearts from $x$ to $z$ will be denoted by ${\mathcal{H}}(x,z)$ .