On projective representations of nilpotent groups

Let k be a field containing a primitive n-root of unity ζ. Let A = (a,b) _n be a symbol algebra over k with generators x,y satisfying xⁿ = a, yⁿ = b and xy = ζyx. It is easy to see that the group 〈x,y〉 k^×/k^× is isomorphic to Z_n×Z_n and, in fact, A is isomorphic to a twisted group algebra k^α(Z_n×Z_n) for suitable α ∈ H²(Z_n×Z_n,k^×). We refer to Z_n×Z_n as a projective basis of A. There is a wide class of groups G which are projective bases of central simple algebras, like the group Z_n×Z_n above, (these are the so called groups of central type). Few of these groups are projective bases of central division algebras. We present a complete classification of such groups and use it to obtain a bound on the Schur index of projective representation of nilpotent groups.