Quasi-Hopf superalgebras and elliptic quantum supergroups

Abstract

We introduce the quasi-Hopf superalgebras which are Z2-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 Bq,λ(G) and the vertex type Aq,p[sl(n|∧n)] (and Aq,p[gl(n|∧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 Uq[sl(n|∧n) in a nonstandard system of simple roots, all of which are fermionic.