Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations

We introduce a new elliptic quantum toroidal algebra \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\) associated with an arbitrary toroidal algebra \({\mathfrak {g}}_{tor}\). We show that \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\) contains two elliptic quantum algebras associated with a corresponding affine Lie algebra \(\widehat{\mathfrak {g}}\) as subalgebras. They are analogue of the horizontal and the vertical subalgebras in the quantum toroidal algebra \(U_{q,\kappa }({\mathfrak {g}}_{tor})\). A Hopf algebroid structure is introduced as a co-algebra structure of \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\) using the Drinfeld comultiplication. We also investigate the Z-algebra structure of \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\) and show that the Z-algebra governs the irreducibility of the level \((k (\ne 0),l)\)-infinite dimensional \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\)-modules in the same way as in the elliptic quantum group \(U_{q,p}(\widehat{\mathfrak {g}})\). As an example, we construct the level (1, l) irreducible representation of \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\) for the simply laced \({\mathfrak {g}}_{tor}\). We also construct the level (0, 1) representation of \(U_{q,\kappa ,p}({\mathfrak {gl}}_{N,tor})\) and discuss a conjecture on its geometric interpretation as an action on the torus equivariant elliptic cohomology of the affine \(A_{N-1}\) quiver variety.