  Given a simple, simply laced, complex Lie algebra $\bfg$ corresponding to the
Lie group $G$, let $\bfnp$ be the subalgebra generated by the positive roots.
In this paper we construct a BV-algebra $\fA[\bfg]$ whose underlying graded
commutative algebra is given by the cohomology, with respect to $\bfnp$, of the
algebra of regular functions on $G$ with values in $\mywedge
(\bfnp\backslash\bfg)$. We conjecture that $\fA[\bfg]$ describes the algebra of
{\it all} physical (i.e., BRST invariant) operators of the noncritical
$\cW[\bfg]$ string. The conjecture is verified in the two explicitly known
cases, $\bfg=\sltw$ (the Virasoro string) and $\bfg=\slth$ (the $\cW_3$
string).
