  We introduce a quotient of Fomin-Kirillov algebra $FK(n)$ denoted
$\overline{FK}_{C_n}(n)$, over the ideal generated by the edges of a complete
graph on n vertexes that are missing in the $n$-cycle graph $C_n$. For this
quotient algebra $\overline{FK}_{C_n}(n)$, we show that the basis is in
one-to-one correspondence with the set of matchings in an $n$-cycle graph. We
also prove that the dimension of $\overline{FK}_{C_n}(n)$ equals the Lucas
Number $L_n$ and its Hilbert series is $q$-Lucas polynomial. We find the
character map of this quotient algebra over Dihedral group $D_n$.
