Here it will be convenient to work in local coordinates $(x,y)=(\log|\xi|,\log|\eta|)$ on $Y$.

Let $(x_{0},y_{0})\in{\bf R}^{2}$ be a point, $\alpha_{1},\alpha_{2}\in({\bf Z}^{2})^{\ast}$ be $1$-covectors such that $\alpha_{1}\wedge\alpha_{2}>0$. Denote by $V=V_{(x_{0},y_{0}),\alpha_{1},\alpha_{2}}$ the closed angle

$$\{(x,y)\in{\bf R}^{2}|\langle\alpha_{i},(x,y)-(x_{0},y_{0})\rangle\geq 0,i=1,2\,\}\,\,.$$

Let ${\cal O}(V)$ be a $K$-algebra consisting of series $f=\sum_{n,m\in{\bf Z}}c_{n,m}\xi^{n}\eta^{m}$, such that $c_{n,m}\in K$ and for all $(x,y)\in V$ we have:

1. 1.

   if $c_{n,m}\neq 0$ then $\langle(n,m),(x,y)-(x_{0},y_{0})\rangle\leq 0$, where we identified $(n,m)\in{\bf Z}^{2}$ with a covector in $(T_{p}^{\ast}Y)^{{\bf Z}}$;

2. 2.

   $\log|c_{n,m}|+nx+my\to-\infty$ as long as $|n|+|m|\to+\infty$.

Algebra ${\cal O}(V)$ is a Poisson algebra with respect to the bracket $\{\xi,\eta\}=\xi\eta$.

For an integer covector $\mu=adx+bdy\in({\bf Z}^{2})^{*}$ we denote by $R_{\mu}$ the monomial $\xi^{a}\eta^{b}$.

Let us consider a pro-nilpotent Lie algebra ${\bf g}:={\bf g}_{\alpha_{1},\alpha_{2},V}\subset{\cal O}(V)$ consisting of series

$$f=\sum_{n_{1},n_{2}\geq 0,n_{1}+n_{2}>0}c_{n_{1},n_{2}}R_{\alpha_{1}}^{-n_{1}}R_{\alpha_{2}}^{-n_{2}}$$

satisfying the condition

$$\log|c_{n,m}|-n_{1}\langle\alpha_{1},(x,y)\rangle-n_{2}\langle\alpha_{2},(x,y)\rangle\leq 0\,\,\,\,\forall\,(x,y)\in V\,\,.$$

The latter condition is equivalent to $\log|c_{n,m}|-\langle n_{1}\alpha_{1}+n_{2}\alpha_{2},(x_{0},y_{0})\rangle\leq 0$.

Lie algebra ${\bf g}$ admits a filtration by Lie subalgebras ${\bf g}^{\geq k},k\in{\bf Z},k\geq 1$, ${\bf g}={\bf g}^{\geq 1}$, such that ${\bf g}^{\geq k}$ consists of the above series which satisfy the condition $n_{1}+n_{2}\geq k$. Clearly $[{\bf g}^{\geq k_{1}},{\bf g}^{\geq k_{2}}]\subset{\bf g}^{\geq k_{1}+k_{2}}$, and ${\bf g}=\varprojlim_{k\to+\infty}{\bf g}/{\bf g}^{\geq k}$.

Thus, ${\bf g}$ is a topological complete pro-nilpotent Lie algebra over $K$. We denote by $G$ the corresponding pro-nilpotent Lie group $\exp({\bf g})$. It inherits the filtration by normal subgroups $G^{\geq k}$ obtained from the corresponding Lie algebras.