${\mathbb C}[v_1,\ldots,v_N]$-module, the space ${{\widehat{\mathcal{A}}}}_W$ has dimension $N$ with a basis given by the classes $[\phi_1],\ldots,[\phi_N]\in{{\widehat{\mathcal{A}}}}_W$ of the polynomials $\phi_1,\ldots,\phi_N$. Identifying the ${\mathbb C}[v_1,\ldots,v_N]$-modules ${\mathcal{T}}_{{\mathbb C}^N}^{\mathrm{alg}}({\mathbb C}^N)$ and ${{\widehat{\mathcal{A}}}}_W$ via the isomorphism $\Psi_W$ defined by $$\begin{gathered}
\Psi_W\left(\frac{{{\partial}}}{{{\partial}}v_k}\right) := \left[\phi_k\right], \quad 1 \le k \le N,\end{gathered}$$ by Remark \[remark:algebraic construction of multiplication\], we endow the tangent spaces $T_p {\mathbb C}^N$ with a multiplication. The structure constants of it are polynomials in the coordinates $v_1,\ldots,v_N$.

A flat metric can be introduced as follows. It is easy to see that there exist unique positive rational numbers $q_x,q_y,q_1,\ldots,q_N$ such that $$q_x x \frac{{{\partial}}\Lambda_W}{{{\partial}}x} + q_y y \frac{{{\partial}}\Lambda_W}{{{\partial}}y} + \sum_{k=1}^N q_k v_k \frac{{{\partial}}\Lambda_W}{\d v_k}=\Lambda_W.$$ There is a unique index $1\le l\le N$ such that the number $q_l$ is minimal. For the singularities $A_N$ and $E_N$ we have $l=N$ and in the $D_N$-case we have $l=N-1$. Denote by $(c_v)_{i,j}^k$ the structure constants of multiplication on ${\mathbb C}^N$ in the coordinates $v_1,\ldots,v_N$. Define a bilinear form $\eta_W$ on ${\mathbb C}^N$ by $$\eta_W\left(\frac{\d}{\d v_i},\frac{\d}{\d v_j}\right):=(c_v)^l_{i,j}.$$ This bilinear form together with the multiplication, introduced above, define a Frobenius manifold structure on ${\mathbb C}^N$, often called the Saito Frobenius manifold. It is semisimple and conformal with an Euler vector field 