Quasi-Hopf Superalgebras and Elliptic Quantum Supergroups

Details Quasi-Hopf Superalgebras and Elliptic Quantum Supergroups Quasi-Hopf Superalgebras and Elliptic Quantum Supergroups

1998-09-28

arxiv.org/abs/math/9809156v1

J.Math.Phys. 40 (1999) 5264-5282

Mathematics

Quantum Algebra

22 pages, Latex file

Scientific paper

10.1063/1.533029

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 $B_{q,\lambda}(G)$ and the vertex type $A_{q,p}[\hat{sl(n|n)}]$ (and $A_{q,p}[\hat{gl(n|n)}]$), where $G$ is any Kac-Moody superalgebra with symmetrizable generalized Cartan matrix. It appears that the vertex type twistor can be constructed only for $U_q[\hat{sl(n|n)}]$ in a non-standard system of simple roots, all of which are fermionic.

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