## Tokyo Journal of Mathematics

### Inductive Construction of Nilpotent Modules of Quantum Groups at Roots of Unity

Yuuki ABE

#### Abstract

The purpose of this paper is to prove that we can inductively construct all finite-dimensional irreducible nilpotent modules of type 1 by using the Schnizer homomorphisms for quantum algebras at roots of unity of type $A_n$, $B_n$, $C_n$, $D_n$, or $G_2$.

#### Article information

Source
Tokyo J. Math., Volume 30, Number 2 (2007), 351-371.

Dates
First available in Project Euclid: 4 February 2008

https://projecteuclid.org/euclid.tjm/1202136681

Digital Object Identifier
doi:10.3836/tjm/1202136681

Mathematical Reviews number (MathSciNet)
MR2376514

Zentralblatt MATH identifier
1149.17006

#### Citation

ABE, Yuuki. Inductive Construction of Nilpotent Modules of Quantum Groups at Roots of Unity. Tokyo J. Math. 30 (2007), no. 2, 351--371. doi:10.3836/tjm/1202136681. https://projecteuclid.org/euclid.tjm/1202136681

#### References

• S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, JAMS, 17 (2004), 595–678.
• Y. Abe and T. Nakashima, Nilpotent representations of classical quantum groups at roots of unity, J. Math. Phys., 46 (2005), no. 12, 113505 1–19.
• V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994.
• E. Date, M. Jimbo, K. Miki and T. Miwa, Cyclic Representations of $U_q({\mathfrak s}{\mathfrak l}(n+1,\bbC))$ at $q^N=1$, Publ. RIMS, Kyoto Univ., 27 (1991), 366–437.
• C. De Concini and V. G. Kac, Representations of Quantum Groups at Roots of $1$, Actes du Colloque en l'honneur de Jacques Diximier, edited by A. Connes, M. Duflo, A. Joseph and R. Rentschler (Prog. Math. Birkhauser), 92 (1990), 471–506.
• J. C. Jantzen, Lectures on Quantum Groups, GSM. vol. 6, 1996.
• G. Lusztig, Modular representations and quantum groups, Contemp. Math., 82 (1989), 59–77.
• G. Lusztig, Finite-dimensional Hopf algebras arising from quantized unuversal enveloping algebra, J. Amer. Math. Soc., 3 (1990), no. 1, 257–296.
• T. Nakashima, Irreducible modules of finite dimensional quantum algebras of type A at roots of unity, J. Math. Phys., 43, (2002) no. 4, 2000–2014.
• W. A. Schnizer, Roots of unity: Representations for symplectic and orthogonal quantum groups, J. Math. Phys., 34 (1993), 4340–4363.
• W. A. Schnizer, Roots of unity: Representations of Quantum Group, Commun. Math. Phys., 163 (1994), 293–306.
• T. Tanisaki, Character formulas of Kazhdan-Lusztig type, Fields Institute Communications, 40 (2004), 147–176.