## Quantum Toroidal Superalgebras

thesis

posted on 30.04.2020 by Luan Pereira Bezerra#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

We introduce the quantum toroidal superalgebra E

_{m|n }associated with the Lie superalgebra gl_{m|n}and initiate its study. For each choice of parity "s" of gl_{m|n}, a corresponding quantum toroidal superalgebra E_{s}is defined.To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed.

The superalgebra E

_{s}contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra U_{q}sl̂_{m|n}with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E_{s}, which exchanges the vertical and horizontal subalgebras.If

*m*and*n*are different and "s" is standard, we give a construction of level 1 E_{m|n}-modules through vertex operators. We also construct an evaluation map from E_{m|n}(q_{1},q_{2},q_{3}) to the quantum affine algebra U_{q}gl̂_{m|n}at level c=q_{3}^{(m-n)/2}.