Quasi-Hopf superalgebras and elliptic quantum supergroups

We introduce the quasi-Hopf superalgebras which are Z(2)-graded versions of Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is, the face type B-q,B-lambda(G) and the vertex type A(q,p)[sl(((n\) over cap\n)] (and A(q,p)[gl ((n\) over cap n)], where G is any Kac-Moody superalgebra with symmetrizable generalized Cartan matrix. It appears that the vertex type twistor can be constructed only for U-q[sl ((n\) over cap n) in a nonstandard system of simple roots, all of which are fermionic. (C) 1999 American Institute of Physics. [S0022-2488(99)00210-8]