Finite dimensional modules over quantum toroidal algebras

Abstract

For all generic q\mathrm{\in }{C}^{*}, when g is not of type A₁; we prove that the quantum toroidal algebra U_q(g_tor) has no nontrivial finite dimensional simple module.

Keywords

Quantum toroidal algebra / finite dimensional module

References

[1] Beck J. Convex bases of PBW type for quantum affine algebras. Comm Math Phys, 1994, 165: 193–199

[2] Block R E. The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra. Adv Math, 1981, 39: 69–110

[3] Chari V, Pressley A. Quantum affine algebras. Comm Math Phys, 1991, 142: 261–283

[4] Chari V, Pressley A. Minimal affinizations of representations of quantum groups: The nonsimply-laced case. Lett Math Phys, 1995, 35: 99–114

[5] Chari V, Pressley A. Minimal affinizations of representations of quantum groups: The simply-laced case. J Algebra, 1996, 184: 1–30

[6] Drinfeld V. A new realization of Yangians and quantized affine algebras. Soviet Math Dokl, 1988, 36: 212–216

[7] Frenkel I, Jing N, Wang W. Quantum vertex representations via finite groups and the McKay correspondence . Comm Math Phys,2000, 211: 365–393

[8] Ginzburg V, Kapranov M, Vasserot E. Langlands reciprocity for algebraic surfaces. Math Res Lett, 1995, 2: 147–160

[9] Hernandez D. Quantum toroidal algebras and their representations. Selecta Math (N S), 2009, 14: 701–725

[10] Jantzen J C. Lectures on Quantum Groups. Grad Stud Math, Vol 6. Providence: Amer Math Soc, 1996

[11] Miki K. Toroidal and level 0 U0 q(b sln+1) actions on Uq(b gln+1)-modules. J Math Phys, 1999, 40: 3191–3210

[12] Miki K. Representations of quantum toroidal algebra Uq(sln+1,tor)(n>2). J Math Phys, 2000, 41: 7079–7098

[13] Miki K. Quantum toroidal algebra Uq(sl2,tor) and R matrices. J Math Phys, 2001, 42: 2293–2308

[14] Saito Y. Quantum toroidal algebras and their vertex representations. Publ RIMS Kyoto Univ, 1998, 34: 155–177

[15] Varagnolo M, Vasserot E. Schur duality in the toroidal setting. Comm Math Phys, 1996, 182: 469–484

RIGHTS & PERMISSIONS

Higher Education Press