Let
$V$ be a
hypersurface with an isolated singularity at the origin defined by the holomorphic function
$f:\left({\u2102}^{n},0\right)\to \left(\u2102,0\right)$. The Yau
algebra
$L\left(V\right)$
is defined to be the Lie algebra of derivations of the moduli algebra
$A\left(V\right):={\mathcal{\mathcal{O}}}_{n}\u2215\left(f,\frac{\partial f}{\partial {x}_{1}},\cdots \phantom{\rule{0.3em}{0ex}},\frac{\partial f}{\partial {x}_{n}}\right)$, i.e.,
$L\left(V\right)=Der\left(A\left(V\right),A\left(V\right)\right)$. It is known that
$L\left(V\right)$ is finite dimensional
and its dimension
$\lambda \left(V\right)$
is called the Yau number. We introduced a new Lie algebra
${L}^{\ast}\left(V\right)$ which
was defined to be the Lie algebra of derivations of
$${A}^{\ast}\left(V\right)={\mathcal{\mathcal{O}}}_{n}/\left(f,\frac{\partial f}{\partial {x}_{1}},\dots ,\frac{\partial f}{\partial {x}_{n}},Det{\left(\frac{{\partial}^{2}f}{\partial {x}_{i}\partial {x}_{j}}\right)}_{\phantom{\rule{0.17em}{0ex}}i,j=1,\dots ,n}\right),$$
i.e.,
${L}^{\ast}\left(V\right)=Der\left({A}^{\ast}\left(V\right),{A}^{\ast}\left(V\right)\right)$.
${L}^{\ast}\left(V\right)$ is finite
dimensional and
${\lambda}^{\ast}\left(V\right)$ is
the dimension of
${L}^{\ast}\left(V\right)$.
In this paper we compute the generalized Cartan matrix
$C\left(V\right)$
and other various invariants arising from the new Lie algebra
${L}^{\ast}\left(V\right)$
for simple elliptic singularities and simple hypersurface singularities. We
use the generalized Cartan matrix to characterize the ADE singularities.
