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).
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