Let
$V$
be a hypersurface with an isolated singularity at the origin defined by the holomorphic function
$f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$
. The Yau algebra
$L(V)$
is defined to be the Lie algebra of derivations of the moduli algebra
$A(V):={\mathcal{O}}_{n}/(f,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{n})$
, that is,
$L(V)=\text{Der}(A(V),A(V))$
. It is known that
$L(V)$
is finite dimensional and its dimension
$\unicode[STIX]{x1D706}(V)$
is called the Yau number. We introduce a new series of Lie algebras, that is,
$k$
th Yau algebras
$L^{k}(V)$
,
$k\geq 0$
, which are a generalization of the Yau algebra. The algebra
$L^{k}(V)$
is defined to be the Lie algebra of derivations of the
$k$
th moduli algebra
$A^{k}(V):={\mathcal{O}}_{n}/(f,m^{k}J(f)),k\geq 0$
, that is,
$L^{k}(V)=\text{Der}(A^{k}(V),A^{k}(V))$
, where
$m$
is the maximal ideal of
${\mathcal{O}}_{n}$
. The
$k$
th Yau number is the dimension of
$L^{k}(V)$
, which we denote by
$\unicode[STIX]{x1D706}^{k}(V)$
. In particular,
$L^{0}(V)$
is exactly the Yau algebra, that is,
$L^{0}(V)=L(V),\unicode[STIX]{x1D706}^{0}(V)=\unicode[STIX]{x1D706}(V)$
. These numbers
$\unicode[STIX]{x1D706}^{k}(V)$
are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that
$\unicode[STIX]{x1D706}^{(k+1)}(V)>\unicode[STIX]{x1D706}^{k}(V),k\geq 0.$
We prove this conjecture for a large class of singularities.
Abelian extensions and crossed modules of Hom-Lie algebras
