Call the forms $\gamma$ and $\beta$ compatible if

(2.1)

$\displaystyle 2\,\gamma(u,v,\gamma(u,v,\cdot)^{\dagger})=\beta(u,u)\beta(v,v)-\beta(u,v)^{2}$

for all $u,v\in V$ . An alternating trilinear form $\gamma:\textstyle\bigwedge^{3}V\to k$ is nondegenerate if there exists a compatible nondegenerate symmetric bilinear form on $V$ .

The circular measure $\varepsilon$ of a rooted bipartite graph $X$ is given by

$$d\varepsilon(q)=d\mu((q+q^{-1})^{2})$$

where $\mu$ is the real measure.

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

(1.1)

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

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

(1.3)

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

A $1$ - Hom-cochain is a map $f$ , where $f\in\mathcal{C}^{1}({\mathcal{G},\mathcal{G}})$ satisfying

(5.1)

$$f\circ\alpha=\alpha\circ f$$

We denote by $Hom\mathcal{C}^{1}({\mathcal{G},\mathcal{G}})$ the set of all $1$ - Hom-cochain of $\mathcal{G}$ .

Let $u$ be a harmonic function in $B$ . We denote by $\tilde{u}$ its conjugate harmonic function and we set

$$f=u+i\tilde{u}.$$

Let $M=\mathbf{k}$ and define a $SP$ -module structure by

$$p\cdot v=\pi(p)\alpha,$$

where $\pi:SP\rightarrow\mathbf{k}$ is the augmentation map.

(Supporting mountains) Fix $c:\mathop{\rm cl}(M\times\bar{M})\longrightarrow\mathbf{R}\cup\{+\infty\}$ . A mountain refers to any function $f$ on $\mathop{\rm cl}M$ of the form

$$f(\cdot)=-c(\cdot,\bar{x})+\lambda$$

with $(\lambda,\bar{x})\in\mathbf{R}\times\mathop{\rm cl}\bar{M}$ . The mountain is said to be focused at $\bar{x}$ . The mountain $f$ is said to support $u:\mathop{\rm cl}M\longrightarrow\mathbf{R}\cup\{+\infty\}$ at $x\in\mathop{\rm cl}M$ if $u(x)=f(x)<+\infty$ and

$$u(y)\geq f(y)$$

for all $y\in\mathop{\rm cl}M$ . The mountain is said to support $u$ to first order at $x\in M$ if $u(x)=f(x)<+\infty$ , and $M$ has a manifold structure near $x$ with

(A.1)

$\displaystyle u(y)\geq f(y)+o(|y-x|)\qquad{\rm as}\ y\to x.$

Given a valued field $(L,w)$ , define a $w$ -restricted exponential $\exp$ to be an isomorphism of groups between the valuation ideal of $L$ and the $1$ -units of $L$ (with respect to $w$ ) which is $w$ -compatible ; that is,

$$wa=w(1-\exp(a))\,.$$

Let $A$ be an algebra with automorphism $\sigma$ . Let $\delta$ be a $\sigma$ -derivation that satisfies

$$\delta(ab)=\sigma(a)\delta(b)+\delta(a)b$$

for all $a,b\in A$ . The Ore extension of $A$ associated with data $\{\sigma,\delta\}$ is a ring $B$ , containing $A$ as a subring, generated by elements in $A$ and a new variable $y$ and subject to the relation

(E1.1.1)

$$ya=\sigma(a)y+\delta(a)$$

for all $a\in A$ . The Ore extension ring $B$ is denoted by $A[y;\sigma,\delta]$ .

We say that functional ( $P_{C}$ ) is invariant under the one-parameter family of infinitesimal transformations

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

if and only if

$$\displaystyle\int_{t_{a}}^{t_{b}}\left[{\mathcal{H}}\left(t,q(t),u(t),p(t)\right)-p(t)\cdot\,_{a}^{C}D_{t}^{\alpha}q(t)\right]dt\\ \displaystyle=\int_{t_{a}}^{t_{b}}\left[{\mathcal{H}}\left(t,\bar{q}(t),\bar{u}(t),\bar{p}(t)\right)-\bar{p}(t)\cdot\,_{a}^{C}D_{t}^{\alpha}\bar{q}(t)\right]dt$$

(12)

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

Functional ( $P_{C}$ ) is said to be invariant under the one-parameter infinitesimal transformations

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

(19)

if and only if

$$\displaystyle\int_{t_{a}}^{t_{b}}\left[{\mathcal{H}}\left(t,q(t),u(t),p(t)\right)-p(t)\cdot{{}_{a}^{C}D_{t}^{\alpha}q(t)}\right]dt\\ \displaystyle=\int_{\bar{t}(t_{a})}^{\bar{t}(t_{b})}\left[{\mathcal{H}}\left(\bar{t},\bar{q}(\bar{t}),\bar{u}(\bar{t}),\bar{p}(\bar{t})\right)-\bar{p}(\bar{t})\cdot{{}_{a}^{C}D_{\bar{t}}^{\alpha}\bar{q}}(\bar{t})\right]d\bar{t}$$

(20)

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

Functional ( 6 ) is said to be invariant under the one-parameter family of infinitesimal transformations

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

(25)

if and only if

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

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

The extended nil Hecke ring ${\widetilde{\mathbb{A}}_{\rm Aff}}$ is the smashed product $Z\ltimes{\mathbb{A}_{\rm Aff}}$ of ${\mathbb{A}_{\rm Aff}}$ by $Z$ . As a $\mathbb{Z}$ -module, it is just the tensor product $Z\otimes{\mathbb{A}_{\rm Aff}}$ . The multiplication is defined by

$$(\tau\otimes a)(\sigma\otimes b)=\tau\sigma\otimes\sigma^{-1}(a)b.$$

An operation $f:D^{n}\rightarrow D$ , where $n\geq 2$ , is a weak near-unanimity function if $f$ is idempotent and

$$f(x,y,y,\ldots,y)=f(y,x,y,y,\ldots,y)=\ldots=f(y,\ldots,y,x)$$

for all $x,y\in D$ .

Let $X$ be a complex manifold with a holomorphic $2$ -vector $\beta$ . If the Schouten bracket vanishes, i.e., $[\beta,\beta]_{Sch}=0$ , we call $\beta$ a (holomorphic) Poisson structure on $X$ and the Poisson bracket is defined by

$$\{f,g\}=\beta(df\wedge dg).$$

:

[Monodromy around a boundary circle:] Let $(\pi:\tilde{\Sigma}\longrightarrow\Sigma,\{\tilde{p_{a}}\})$ be a $G$ -cover of $(\Sigma,\{p_{a}\}_{a\in A(\Sigma)})$ . Consider the $a$ th boundary circle, $S$ , where the base point on $S$ is $p_{a}$ and the point on the fiber above is $\tilde{p_{a}}$ . This $S$ has an orientation which comes from the orientation of the surface. Let $\alpha:[0,1]\rightarrow S$ be a parameterization of the boundary circle $S$ which also preserves the orientation of $S$ . We also assume that $\alpha(0)=\alpha(1)=p_{a}$ . Then there is a unique lifting of $\alpha$ to the $G$ -cover, say $\tilde{\alpha}:[0,1]\rightarrow\tilde{\Sigma}$ such that $\tilde{\alpha}(0)=\tilde{p_{a}}$ . We define the monodromy, $m\in G$ , of this $a$ th boundary circle to be that unique element of $G$ such that

(1)

$$m\tilde{\alpha}(0)=\tilde{\alpha}(1)$$