Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations
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arXiv
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| Format: | Preprint |
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2024
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| author | Konno, Hitoshi Oshima, Kazuyuki |
| author_facet | Konno, Hitoshi Oshima, Kazuyuki |
| contents | We introduce a new elliptic quantum toroidal algebra U_{q,κ,p}(g_tor) associated with an arbitrary toroidal algebra g_tor. We show that U_{q,κ,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,κ}(g_tor). A Hopf algebroid structure is introduced as a co-algebra structure of U_{q,κ,p}(g_tor) using the Drinfeld comultiplication. We also investigate the Z-algebra structure of U_{q,κ,p}(g_tor) and show that the Z-algebra governs the irreducibility of the level (k(\not=0),l)-infinite dimensional U_{q,κ,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,κ,p}(g_tor) for the simply laced gtor. We also construct the level (0,1) representation of U_{q,κ,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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11177 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations Konno, Hitoshi Oshima, Kazuyuki Representation Theory High Energy Physics - Theory Algebraic Geometry Quantum Algebra 17B37 We introduce a new elliptic quantum toroidal algebra U_{q,κ,p}(g_tor) associated with an arbitrary toroidal algebra g_tor. We show that U_{q,κ,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,κ}(g_tor). A Hopf algebroid structure is introduced as a co-algebra structure of U_{q,κ,p}(g_tor) using the Drinfeld comultiplication. We also investigate the Z-algebra structure of U_{q,κ,p}(g_tor) and show that the Z-algebra governs the irreducibility of the level (k(\not=0),l)-infinite dimensional U_{q,κ,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,κ,p}(g_tor) for the simply laced gtor. We also construct the level (0,1) representation of U_{q,κ,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. |
| title | Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations |
| topic | Representation Theory High Energy Physics - Theory Algebraic Geometry Quantum Algebra 17B37 |
| url | https://arxiv.org/abs/2405.11177 |