A new effective-field technique for the ferromagnetic spin-1 Blume-Capel model in a transverse crystal field

Definition 2.1 .

Let $\mathcal{A}$ be a Banach algebra and let $\sigma\in Hom(\mathcal{A})$ . Then $\mathcal{A}$ is $\sigma$ -biflat if there exists a bounded linear map $\rho:(\mathcal{A}\widehat{\otimes}\mathcal{A})^{*}\longrightarrow\mathcal{A}^{*}$ satisfying

\rho(\sigma(a)\cdot\lambda)=a\cdot\rho(\lambda)\ \ \ \text{and}\ \ \ \rho(% \lambda\cdot\sigma(a))=\rho(\lambda)\cdot a\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

for $a\in\mathcal{A},\lambda\in\mathcal{A}^{\ast}$ such that $\rho\circ\pi^{*}=\sigma^{*}$ .

Definition 2.15 (Orientation) .

Let $x,y,z$ be three points in $X$ . Define

\mathrm{sign}(x,y,z)=\left\{\begin{aligned} \displaystyle 0&\displaystyle\text% { if any two points coincide},\\ \displaystyle 1&\displaystyle\text{ if }\{x,y,z\}\text{ is counterclockwise},% \\ \displaystyle-1&\displaystyle\text{ otherwise}.\end{aligned}\right.

Definition 1.1

A left (respectively right) pre-Lie algebra is a linear space $A$ endowed with a linear map $\cdot:A\otimes A\rightarrow A$ satisfying (for all $x,y,z\in A$ )

\displaystyle x\cdot(y\cdot z)-(x\cdot y)\cdot z=y\cdot(x\cdot z)-(y\cdot x)% \cdot z,

(1.1)

respectively

\displaystyle x\cdot(y\cdot z)-(x\cdot y)\cdot z=x\cdot(z\cdot y)-(x\cdot z)% \cdot y.

(1.2)

Definition 1.2

( [ 13 ] ) A left (respectively right) Leibniz algebra is a linear space $L$ endowed with a bilinear map $[\cdot,\cdot]:L\times L\rightarrow L$ satisfying (for all $x,y,z\in L$ ):

\displaystyle[x,[y,z]]=[[x,y],z]+[y,[x,z]],

(1.3)

respectively

\displaystyle[[x,y],z]=[[x,z],y]+[x,[y,z]].

(1.4)

A morphism of (left or right) Leibniz algebras $L$ and $L^{\prime}$ is a linear map $f:L\rightarrow L^{\prime}$ such that $f([x,y])=[f(x),f(y)]$ , for all $x,y\in L$ .

Definition 1.4

A BiHom-associative algebra over $\Bbbk$ is a 4-tuple $\left(A,\mu,\alpha,\beta\right)$ , where $A$ is a $\Bbbk$ -linear space, $\alpha:A\rightarrow A$ , $\beta:A\rightarrow A$ and $\mu:A\otimes A\rightarrow A$ are linear maps, with notation $\mu(x\otimes y)=xy$ , for all $x,y\in A$ , satisfying the following conditions, for all $x,y,z\in A$ :

	$\displaystyle\alpha\circ\beta=\beta\circ\alpha,$		(1.6)
	$\displaystyle\alpha(xy)=\alpha(x)\alpha(y)\text{ and }\beta(xy)=\beta(x)\beta(% y),\quad\text{(multiplicativity)}$		(1.7)
	$\displaystyle\alpha(x)(yz)=(xy)\beta(z).\quad\text{(BiHom-associativity)}$		(1.8)

We call $\alpha$ and $\beta$ (in this order) the structure maps of $A$ .

Definition 1.5

A BiHom-Lie algebra over $\Bbbk$ is a 4-tuple $\left(L,\left[\cdot,\cdot\right],\alpha,\beta\right)$ , where $L$ is a $\Bbbk$ -linear space, $\alpha,\beta:L\rightarrow L$ are linear maps and $\left[\cdot,\cdot\right]:L\times L\rightarrow L$ is a bilinear map, satisfying the following conditions, for all $x,y,z\in L:$

	$\displaystyle\alpha\circ\beta=\beta\circ\alpha,$
	$\displaystyle\alpha(\left[x,y\right])=\left[\alpha\left(x\right),\alpha(y)% \right]\;\;\text{ and }\;\;\beta(\left[x,y\right])=\left[\beta\left(x\right),% \beta\left(y\right)\right],$
	$\displaystyle\left[\beta\left(x\right),\alpha\left(y\right)\right]=-\left[% \beta\left(y\right),\alpha\left(x\right)\right],\;\;\;\;\text{ (BiHom-skew-% symmetry)}$
	$\displaystyle\left[\beta^{2}\left(x\right),\left[\beta\left(y\right),\alpha% \left(z\right)\right]\right]+\left[\beta^{2}\left(y\right),\left[\beta\left(z% \right),\alpha\left(x\right)\right]\right]+\left[\beta^{2}\left(z\right),\left% [\beta\left(x\right),\alpha\left(y\right)\right]\right]=0.$
	(BiHom-Jacobi condition)

We call $\alpha$ and $\beta$ (in this order) the structure maps of $L$ .

A morphism $f:\left(L,\left[\cdot,\cdot\right],\alpha,\beta\right)\rightarrow\left(L^{% \prime},\left[\cdot,\cdot\right]^{\prime},\alpha^{\prime},\beta^{\prime}\right)$ of BiHom-Lie algebras is a linear map $f:L\rightarrow L^{\prime}$ such that $\alpha^{\prime}\circ f=f\circ\alpha$ , $\beta^{\prime}\circ f=f\circ\beta$ and $f(\left[x,y\right])=[f(x),f(y)]^{\prime}$ , for all $x,y\in L$ .

Definition 2.1

A left (respectively right) BiHom-pre-Lie algebra is a 4-tuple $(A,\cdot,\alpha,\beta)$ , where $A$ is a linear space and $\cdot:A\otimes A\rightarrow A$ and $\alpha,\beta:A\rightarrow A$ are linear maps satifying $\alpha\circ\beta=\beta\circ\alpha$ , $\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y)$ , $\beta(x\cdot y)=\beta(x)\cdot\beta(y)$ and

\displaystyle\alpha\beta(x)\cdot(\alpha(y)\cdot z)-(\beta(x)\cdot\alpha(y))% \cdot\beta(z)=\alpha\beta(y)\cdot(\alpha(x)\cdot z)-(\beta(y)\cdot\alpha(x))% \cdot\beta(z),

(2.1)

respectively

\displaystyle\alpha(x)\cdot(\beta(y)\cdot\alpha(z))-(x\cdot\beta(y))\cdot% \alpha\beta(z)=\alpha(x)\cdot(\beta(z)\cdot\alpha(y))-(x\cdot\beta(z))\cdot% \alpha\beta(y),

(2.2)

for all $x,y,z\in A$ . We call $\alpha$ and $\beta$ (in this order) the structure maps of $A$ .

A morphism $f:(A,\cdot,\alpha,\beta)\rightarrow(A^{\prime},\cdot^{\prime},\alpha^{\prime},% \beta^{\prime})$ of left or right BiHom-pre-Lie algebras is a linear map $f:A\rightarrow A^{\prime}$ satisfying $f(x\cdot y)=f(x)\cdot^{\prime}f(y)$ , for all $x,y\in A$ , as well as $f\circ\alpha=\alpha^{\prime}\circ f$ and $f\circ\beta=\beta^{\prime}\circ f$ .

Definition 2.5

( [ 12 ] A BiHom-dendriform algebra is a 5-tuple $(A,\prec,\succ,\alpha,\beta)$ consisting of a linear space $A$ and linear maps $\prec,\succ:A\otimes A\rightarrow A$ and $\alpha,\beta:A\rightarrow A$ satisfying the following conditions (for all $x,y,z\in A$ ):

	$\displaystyle\alpha\circ\beta=\beta\circ\alpha,$		(2.3)
	$\displaystyle\alpha(x\prec y)=\alpha(x)\prec\alpha(y),~{}\alpha(x\succ y)=% \alpha(x)\succ\alpha(y),$		(2.4)
	$\displaystyle\beta(x\prec y)=\beta(x)\prec\beta(y),~{}\beta(x\succ y)=\beta(x)% \succ\beta(y),$		(2.5)
	$\displaystyle(x\prec y)\prec\beta(z)=\alpha(x)\prec(y\prec z+y\succ z),$		(2.6)
	$\displaystyle(x\succ y)\prec\beta(z)=\alpha(x)\succ(y\prec z),$		(2.7)
	$\displaystyle\alpha(x)\succ(y\succ z)=(x\prec y+x\succ y)\succ\beta(z).$		(2.8)

We call $\alpha$ and $\beta$ (in this order) the structure maps of $A$ .

Definition 3.1

A left (respectively right) BiHom-Leibniz algebra is a 4-tuple $(L,[\cdot,\cdot],\alpha,\beta)$ , where $L$ is a linear space, $[\cdot,\cdot]:L\times L\rightarrow L$ is a bilinear map and $\alpha,\beta:L\rightarrow L$ are linear maps satisfying $\alpha\circ\beta=\beta\circ\alpha$ , $\alpha([x,y])=[\alpha(x),\alpha(y)]$ , $\beta([x,y])=[\beta(x),\beta(y)]$ and

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

(3.1)

respectively

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

(3.2)

for all $x,y,z\in L$ . We call $\alpha$ and $\beta$ (in this order) the structure maps of $L$ .

A morphism $f:(L,[\cdot,\cdot],\alpha,\beta)\rightarrow(L^{\prime},[\cdot,\cdot]^{\prime},% \alpha^{\prime},\beta^{\prime})$ of BiHom-Leibniz algebras is a linear map $f:L\rightarrow L^{\prime}$ such that $\alpha^{\prime}\circ f=f\circ\alpha$ , $\beta^{\prime}\circ f=f\circ\beta$ and $f([x,y])=[f(x),f(y)]^{\prime}$ , for all $x,y\in L$ .

Definition 2.5 .

We say that the fragmentation process (or the characteristics $\big{(}(c_{i})_{i\in[K]},(\nu_{i})_{i\in[K]}\big{)})$ is Malthusian if it is irreducible and there exists a number $p^{*}\in(0,1]$ called the Malthusian exponent such that

\lambda(p^{*})=0.

Definition 2 .

A magic element of type $(n,0)$ is inductively defined by the following formula:

\raisebox{-4.0pt}{\includegraphics[]{onezero.pdf}}=\raisebox{-3.0pt}{% \includegraphics[]{one.pdf}}

\raisebox{-4.0pt}{\includegraphics[]{nzero.pdf}}=\raisebox{-13.0pt}{% \includegraphics[]{none.pdf}}-\frac{[n-1]}{[n]}\quad\raisebox{-14.0pt}{% \includegraphics[]{ntwo.pdf}}

Definition 7 .

A compatible almost complex structure $J$ on $(M,\omega)$ is called special if there exists $\lambda\in\mathbf{R}$ such that the Chern-Ricci form of $J$ satisfies

\rho=\lambda\omega.

(12)

6.1 Definition .

For $0<a<1$ , let

c(a)=c^{+}(a)\cup c^{-}(a),

where $c^{\pm}(a)$ are the circles $(\partial\mathbf{B})\cap\{z=\pm a\}$ . We orient $c^{-}(a)$ by $d\theta$ and $c^{+}(a)$ by $-d\theta$ . Let

\mathcal{M}(a)

be the set of rotationally invariant area-minimizing surfaces bounded by $c(a)$ .

Definition 2.4 .

Fix $\kappa\in\mathbb{R}$ . Let $(\Sigma,h)$ a complete three-dimensional Riemannian manifold with constant sectional curvature $\kappa$ and $I\subset\mathbb{R}$ an open interval. Let $t$ be a coordinate for $I$ . Define $M=I\times\Sigma$ . Let $\eta$ and $\pi$ be the projections of $M$ onto the first and second factors, respectively. Define the Lorentzian metric $g=-\eta^{*}dt^{2}+(a\circ\pi)^{2}\pi^{*}h$ where $a:I\to(0,\infty)$ is a $C^{k}$ function. Then we say $(M,g)$ is a $C^{k}$ FLRW spacetime . $a(t)$ is called the scale factor . We will abuse notation and simply write

g=-dt^{2}+a^{2}h

(2.2)

for the metric. The comoving observers are the integral curves of $U=\partial/\partial t$ . If $a(t)\to 0$ as $t\searrow t_{i}:=\inf I$ , then we say $(M,g)$ admits a big bang .

Definition 2 .

A para-Hermitian connection on an almost para-Hermitian manifold
$(\mathcal{P},\eta,K)$ is a connection $\nabla$ which preserves both $\eta$ and $\omega$ , i.e.

\nabla\eta=\nabla\omega=0.

(3.6)

Alternatively, a para-Hermitian connection preserves $\eta$ and $K$ . The Levi-Civita connection $\mathring{\nabla}$ is the unique connection which preserves $\eta$ and is torsionless.

Definition 3.2 .

Let $\Gamma=(V,E)$ be a finite graph with $n$ vertices and adjacency matrix $\varepsilon\in M_{n}(\{0,1\})$ . The quantum automorphism group $\mathrm{QAut_{Ban}}(\Gamma)$ is the compact matrix quantum group $(C(\mathrm{QAut_{Ban}}(\Gamma)),u)$ , where $C(\mathrm{QAut_{Ban}}(\Gamma))$ is the universal $C^{*}$ -algebra with generators $u_{ij},1\leq i,j\leq n$ and Relations ( 3.1 ), ( 3.2 ) together with

(3.7)

\displaystyle u\varepsilon=\varepsilon u,

which is nothing but $\sum_{k}u_{ik}\varepsilon_{kj}=\sum_{k}\varepsilon_{ik}u_{kj}$ .

Definition B.1 .

Define the algebra $\mathcal{U}$ to be the unital associative algebra over $\mathbb{Q}(q,t)$ generated by the currents $x^{\pm}(z)=\sum_{n\in\mathbb{Z}}x^{\pm}_{n}z^{-n}$ , $\psi^{\pm}(z)=\sum_{\pm n\in\mathbb{Z}_{\geq 0}}\psi^{\pm}_{n}z^{-n}$ and the central element $\gamma^{\pm 1/2}$ satisfying the defining relations

	$\displaystyle\psi^{\pm}(z)\psi^{\pm}(w)=\psi^{\pm}(w)\psi^{\pm}(z),\quad\psi^{% +}(z)\psi^{-}(w)=\dfrac{g(\gamma^{+1}w/z)}{g(\gamma^{-1}w/z)}\psi^{-}(w)\psi^{% +}(z),$		(B.2)
	$\displaystyle\psi^{+}(z)x^{\pm}(w)=g(\gamma^{\mp 1/2}w/z)^{\mp 1}x^{\pm}(w)% \psi^{+}(z),$		(B.3)
	$\displaystyle\psi^{-}(z)x^{\pm}(w)=g(\gamma^{\mp 1/2}z/w)^{\pm 1}x^{\pm}(w)% \psi^{-}(z),$		(B.4)
	$\displaystyle[x^{+}(z),x^{-}(w)]=\dfrac{(1-q)(1-1/t)}{1-q/t}\big{(}\delta(% \gamma^{-1}z/w)\psi^{+}(\gamma^{1/2}w)-\delta(\gamma z/w)\psi^{-}(\gamma^{-1/2% }w)\big{)},$		(B.5)
	$\displaystyle G^{\mp}(z/w)x^{\pm}(z)x^{\pm}(w)=G^{\pm}(z/w)x^{\pm}(w)x^{\pm}(z).$		(B.6)

Definition 1.2 ( [ Mac05 , definition 3.3.1] ) :

Let $M$ be a manifold. A Lie algebroid $\mathcal{A}$ is a triple $(A\to M,\mathsf{a},[\cdot,\cdot])$ , where $A\to M$ is a vector bundle on $M$ , $\mathsf{a}$ is an anchor on $A\to M$ , and $[\cdot,\cdot]$ is a $\mathbb{R}$ -bilinear operation on the $\mathcal{C}^{\infty}(M)$ -module $\Gamma(A)$ of sections of $A\to M$ called the bracket , such that $(\Gamma(A),[\cdot,\cdot])$ is a Lie algebra and a right Leibniz rule is satisfied:

[u,fv]=f[u,v]+(\mathsf{a}(u)\cdot f)v,

for all $u,v\in\Gamma(A)$ and $f\in\mathcal{C}^{\infty}(M)$ .

Definition 1 (Agent Invariance) .

If for any permutation (bijection) $p:\{1,...,d\}\to\{1,...,d\}$ ,

\pi(p(\bar{h}))=p(\pi(\bar{h}))

we say that $\pi$ is agent invariant .

Definition 2.1 .

Given a graded $k$ -algebra $T=\bigoplus_{i\geq 0}[T]_{i}$ , a differential on $T$ is a linear map $\delta:T\smash{\mathop{\hbox to 20.0pt{\rightarrowfill}}\limits}T$ such that $\delta([T]_{i})\subseteq[T]_{i+1}$ , for all $i$ , and such that for any homogeneous elements $a,b\in T$ , the following Leibniz rule holds:

\delta(ab)=\delta(a)b+(-1)^{\deg(a)}a\delta(b).

Definition 2.1 .

Let $Q$ be a graph. The Brauer monoid ${\rm BrM}(Q)$ is the monoid generated by the symbols $R_{i}$ and $E_{i}$ , for each node $i$ of $Q$ and $\delta$ , $\delta^{-1}$ subject to the following relation, where $\sim$ denotes adjacency between nodes of $Q$ .

\delta\delta^{-1}=1

(2.1)

R_{i}^{2}=1

(2.2)

R_{i}E_{i}=E_{i}R_{i}=E_{i}

(2.3)

E_{i}^{2}=\delta E_{i}

(2.4)

R_{i}R_{j}=R_{j}R_{i},\,\,\mbox{for}\,\it{i\nsim j}

(2.5)

E_{i}R_{j}=R_{j}E_{i},\,\,\mbox{for}\,\it{i\nsim j}

(2.6)

E_{i}E_{j}=E_{j}E_{i},\,\,\mbox{for}\,\it{i\nsim j}

(2.7)

R_{i}R_{j}R_{i}=R_{j}R_{i}R_{j},\,\,\mbox{for}\,\it{i\sim j}

(2.8)

R_{j}R_{i}E_{j}=E_{i}E_{j},\,\,\mbox{for}\,\it{i\sim j}

(2.9)

R_{i}E_{j}R_{i}=R_{j}E_{i}R_{j},\,\,\mbox{for}\,\it{i\sim j}

(2.10)

The Brauer algebra ${\rm Br}(Q)$ is the the free $\mathbb{Z}$ -algebra for Brauer monoid ${\rm BrM}(Q)$ .

Definition 2

The discrete dynamical system $(\mathcal{X},f)$ is topologically semi-conjugate to the system $(\mathcal{Y},g)$ if it exists a function $\varphi:\mathcal{X}\longrightarrow\mathcal{Y}$ , both continuous and onto, such that:

\varphi\circ f=g\circ\varphi,

that is, which makes commutative the following diagram [ 7 ] .

\begin{CD}\mathcal{X}@>{f}>{}>\mathcal{X}\\ @V{\varphi}V{}V@V{}V{\varphi}V\\ \mathcal{Y}@>{}>{g}>\mathcal{Y}\end{CD}

In this case, the system $(\mathcal{Y},g)$ is called a factor of the system $(\mathcal{X},f)$ . ${}_{\Box}$

Definition 2 .

A skew-symmetric Hochschild $2$ -cocycle $p$ that satisfies the Jacobi identity

p(a\otimes p(b\otimes c))+p(b\otimes p(c\otimes a))+p(c\otimes p(a\otimes b))=0

is called an (algebraic) Poisson structure (or a Poisson bracket ). A commutative algebra together with a Poisson bracket is called a Poisson algebra . Its spectrum is called an affine Poisson variety .

Definition 3 .

A one-parameter formal deformation of an associative algebra $A$ is an associative algebra $(A[[\hslash]],*)$ , such that

a*b=ab(\leavevmode\nobreak\ \text{mod}\leavevmode\nobreak\ \hslash),

for each $a,b\in A$ . We require that $*$ is associative, $k[[\hslash]]$ -bilinear and continuous, which means that

(\sum_{m\geq 0}b_{m}\hslash^{m})*(\sum_{n\geq 0}c_{n}\hslash^{n})=\sum_{m,n% \geq 0}(b_{m}*c_{n})\hslash^{m+n}.

Definition 1 (Generalized curvature) .

The derivative of $f(x):\mathbb{R}\rightarrow\mathbb{R}$ , can be written as

f^{\prime}(x)=h(x)(x-x^{*})

(3)

for some $h(x)\in\mathbb{R}$ , where $x^{*}$ is the global minimum of $f(x)$ . We call $h(x)$ the generalized curvature .

Definition 2.1 .

Given a division ring $K$ , an injective homomorphism ${\sigma:K\to K}$ , and a $\sigma$ -derivation $\delta$ (an additive homomorphism $\delta:K\to K$ such that $\delta(ab)=\sigma(a)\delta(b)+\delta(a)b$ for all $a,b\in K$ ), let $R$ be the set of polynomials of the form $f(x)=\sum_{i=0}^{n}a_{i}x^{i}$ , where $n\in\mathbb{N}$ and $a_{i}\in K$ for $i\in\{0,\dots,n\}$ . The skew polynomial ring denoted $R=K[x;\sigma,\delta]$ is the set of polynomials with standard addition, but with multiplication determined by the rule

xa=\sigma(a)x+\delta(a)

for all $a\in K$ . We will call the elements of $R$ skew polynomials.

Definition 3.4 .

Let $N$ be an homogeneous norm on $\mathbb{H}^{n}$ . We say that $N$ is horizontally strictly convex if for all $p,p^{\prime}\neq e$ it holds

	$\displaystyle N(p*p^{\prime})=N(p)+N(p^{\prime})\Rightarrow p,p^{\prime}\text{% lie on a horizontal line through the origin, i.e.,}$
	$\displaystyle\exists z\in\mathbb{R}^{2n}\backslash\{0\},s,s^{\prime}\in\mathbb% {R},\ \text{such that}\ p=(sz,0)\ \text{and}\ p^{\prime}=(s^{\prime}z,0).$

Definition 2.2.4 .

Let $X$ be a strict $C^{1}$ submanifold of $K^{n}$ . A distribution $u$ on $X$ is by definition a ${\mathbb{C}}$ -linear map from ${\mathcal{S}}(X)$ to ${\mathbb{C}}$ . Write ${\mathcal{S}}^{\prime}(X)$ to denote the collection of distributions on $X$ . For any Schwartz-Bruhat function $\varphi$ in ${\mathcal{S}}(X)$ , depending on the context we will denote the evaluation of $u$ on $\varphi$ by $u(\varphi)$ or $\langle u,\varphi\rangle$ . The vector space ${\mathcal{S}}^{\prime}(X)$ has a structure of $\mathcal{C}^{\infty}(X)$ -module by the following operation: For any $\mathcal{C}^{\infty}$ function $\phi$ on $X$ and any distribution $u$ in ${\mathcal{S}}^{\prime}(X)$ , the distribution $\phi u$ is defined by

\langle\phi u,\varphi\rangle=\langle u,\phi\varphi\rangle

for any Schwartz-Bruhat function $\varphi$ in ${\mathcal{S}}(X)$ .

Definition 2.1 .

A rack is set $R$ with binary operations $(*,/)$ , such that every $x$ , $y$ and $z$ :

(1)

Self-Distributivity: $(x*y)*z=(x*z)*(y*z)$ and $(x/y)/z=(x/z)/(y/z)$ .
(2)

Existence of Right Inverse: $(x*y)/y=x$ , and $(x/y)*y=x$ .

Definition 2.5 (Legendrian rack) .

Let $n\in\mathbb{N}$ . An $n$ -Legendrian rack $(LR_{n},*)$ is a rack such that:

\forall x.\ x^{2n+2}=x

Definition 2.23 .

The renormalisation tangle below defines the nonassociative algebra structure $!$ on $Q_{4}$ :

x!y=\mbox{$\begin{array}[c]{l}\psfig{width=108.405pt}\end{array}$}

Definition 3.6 .

An HNN extension of a group $G$ is given by a subgroup $H$ which is embedded in $G$ in two different ways. Let $\varphi,\psi:H\to G$ be two such group monomorphisms. Then $N(H,G,\varphi,\psi)$ is the quotient of the free product $G\ast\left<t\right>$ of $G$ with the free group on one generator $t$ modulo the relation

t\varphi(h)=\psi(h)t

for every $h\in H$ .