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Quasi-Hopf superalgebras and elliptic quantum supergroups

Sep, 1998
22 pages
Published in:
  • J.Math.Phys. 40 (1999) 5264-5282
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Abstract:
We introduce the quasi-Hopf superalgebras which are Z2Z_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 Bq,λ(G)B_{q,\lambda}(G) and the vertex type Aq,p[sl(n∣n)^]A_{q,p}[\hat{sl(n|n)}] (and Aq,p[gl(n∣n)^]A_{q,p}[\hat{gl(n|n)}]), where GG 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)^]U_q[\hat{sl(n|n)}] in a non-standard system of simple roots, all of which are fermionic.