  We consider the graded space $R$ of syzygies for the coordinate algebra $A$
of projective variety $X=G/P$ embedded into projective space as an orbit of the
highest weight vector of an irreducible representation of semisimple complex
Lie group $G$. We show that $R$ is isomorphic to the Lie algebra cohomology
$H=H^\bdot(\Lt,\CC)$, where $\Lt$ is graded Lie subalgebra of the graded Lie
s-algebra $L=L_1\oplus\Lt$ Koszul dual to $A$. We prove that the isomorphism
identifies the natural associative algebra structures on $R$ and $H$ coming
from their Koszul and Chevalley DGA resolutions respectively. For
subcanonically embedded $X$ a Frobenius algebra structure on the syzygies is
constructed. We illustrate the results by several examples including the
computation of syzygies for the Pl\"ucker embeddings of grassmannians
$\Gr(2,N)$.
