Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations

Abstract: We introduce a new elliptic quantum toroidal algebra U_{q,\kappa,p}(g_tor) associated with an arbitrary toroidal algebra g_tor. We show that U_{q,\kappa,p}(g_tor) contains two elliptic quantum algebras associated with a corresponding affine Lie algebra bg as subalgebras. They are analogue of the horizontal and the vertical subalgebras in the quantum toroidal algebra U_{q,\kappa}(g_tor). A Hopf algebroid structure is introduced as a co-algebra structure of U_{q,\kappa,p}(g_tor) using the Drinfeld comultiplication. We also investigate the Z-algebra structure of U_{q,\kappa,p}(g_tor) and show that the Z-algebra governs the irreducibility of the level (k(\not=0),l)-infinite dimensional U_{q,\kappa,p}(g_tor)-modules in the same way as in the elliptic quantum group U_{q,p}(g). As an example, we construct the level (1,l) irreducible representation of U_{q,\kappa,p}(g_tor) for the simply laced gtor. We also construct the level (0,1) representation of U_{q,\kappa,p}(g_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.