I have an optimization problem with linear objective function. The constraints are in two different groups. The first set of constraints are linear while the second set is nonlinear. The nonlinear constraints are in the form: $ab-cd \geq 0$.

-The optimization problem should be an instance of a convex optimization problem. right? - Is it polynomial solvable? - Is it possible to transformed it to a semidefinite optimization problem (SDP)?

No, constraints of the form $ab-cd\geq 0$ are not convex.
We can prove this by showing that the set $$\mathcal{C}\triangleq \{(a,b,c,d)\,|\,ab-cd\geq 0\}$$ fails the standard midpoint test for convex sets. That is, given any two points $(a_1,b_1,c_1,d_1)$ and $(a_2,b_2,c_2,d_2)$ in $\mathcal{C}$, the midpoint $$(a_3,b_3,c_3,d_3)=(a_1+a_2,b_1+b_2,c_1+c_2,d_1+d_2)/2$$ must be in $\mathcal{C}$ as well. Let us choose $$(a_1,b_1,c_1,d_1)=(2,2,1,1) \quad \text{and} \quad (a_2,b_2,c_2,d_2)=(-2,-2,1,1).$$ Since $a_ib_i-c_id_i=4-1=3\geq 0$ in both cases, both points are in $\mathcal{C}$. But the midpoint $$(a_3,b_3,c_3,d_3)=(0,0,1,1)$$ does not, since $a_3b_3-c_3d_3=0-1=-1\not\geq 0$. So the set is not convex.