$\mathfrak{C}$, and inverse Cartan matrix $\mathfrak{C}^{-1}$ of $A_n$ are as follows: $$\begin{gathered}
\parbox{.6\linewidth}
{\setlength{\unitlength}{1pt}
\def\kr{\circle{10}}
\thicklines
\begin{picture}(20,30)
\put(10,14){\kr}
\put(6,0){$\alpha_1$}
\put(15,14){\line(1,0){10}}
\put(30,14){\kr}
\put(26,0){$\alpha_2$}
\put(35,14){\line(1,0){10}}
\put(50,14){\kr}
\put(46,0){$\alpha_3$}
\put(55,14){\line(1,0){10}}
\put(70,13.5){$\ \,\ldots$}
\put(95,14){\line(1,0){10}}
\put(110,14){\kr}
\put(104,0){$\alpha_{n\!-\!1}$}
\put(115,14){\line(1,0){10}}
\put(130,14){\kr}
\put(126,0){$\alpha_n$}
\end{picture}
}
\hspace{-110 pt}
\mathfrak{C}{=}\left(\begin{smallmatrix}
\phantom{-}2&{-}1&\phantom{-}0&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0\\
{-}1&\phantom{-}2&{-}1&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0\\
\phantom{-}0&{-}1&\phantom{-}2&{-}1&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0\\[-1ex]
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\[0.5ex]
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&{-}1&\phantom{-}2&{-}1\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&{-}1&\phantom{-}2
\end{smallmatrix}\right),
\\[2ex]
\mathfrak{C}^{-1}=\frac{1}{n+1}\left(\begin{smallmatrix}
1\cdot n\;    &1\cdot(n-1)\;&1\cdot(n-2)\;&1\cdot(n-3)\;&\dots &1\cdot 3\;   &1\cdot 2\;   &1\cdot 1\\
1\cdot (n-1)\;&2\cdot(n-1)\;&2\cdot(n-2)\;&2\cdot(n-3)\;&\dots &2\cdot 3\;   &2\cdot 2\;   &2\cdot 1\\
1\cdot (n-2)\;&2\cdot(n-2)\;&3\cdot(n-2)\;&3\cdot(n-3)\;&\dots &3\cdot 3\;   &3\cdot 2\;   &3\cdot 1\\
1\cdot (n-3)\;&2\cdot(n-3)\;&3\cdot(n-3)\;&4\cdot(n-3)\;&\dots &4\cdot 3\;   &4\cdot 2\;   &4\cdot 1\\[-1 ex]
\vdots        &\vdots       &\vdots       &\vdots       &\ddots&\vdots       &\vdots       &\vdots\\[0.5ex]
1\cdot 3\;    &2\cdot3\;    &3\cdot3\;    &4\cdot3\;    &\dots &(n-2)\cdot3\;&(n-2)\cdot2\;&(n-2)\cdot 1\\
1\cdot 2\;    &2\cdot2\;    &3\cdot2\;    &4\cdot2\;    &\dots &(n-2)\cdot2\;&(n-1)\cdot2\;&(n-1)\cdot 1\\
1\cdot 1\;    &2\cdot1\;    &3\cdot1\;    &4\cdot1\;    &\dots &(n-2)\cdot1\;&(n-1)\cdot1\;&n\cdot 1\
                     \end{smallmatrix}\right).\end{gathered}$$

The simple roots $\alpha_i$, $1\le i\le n$ of $A_n$ form a basis ($\a$-basis) of a real Euclidean space $\R^n$. We choose them in ${\mathcal H}$: $$\begin{gathered}
\alpha_i = e_i - e_{i+1}, \quad i=1,\dots,n.\end{gathered}$$ This choice fixes the lengths and relative angles of the simple roots. Their length is equal to $\sqrt 2$ with relative angles between $\alpha_k$ and $\alpha_{k+1}$ $(1\leq k\leq n-1)$ equal to $\frac{2\pi}{3}$, and $\tfrac\pi2$ for any other pair.

In addition to $e$- and $\a$-bases, we introduce the $\w$-basis as the $\Z$-dual basis to 