SIGMA 10 (2014), 047, 20 pages      arXiv:1310.5321      https://doi.org/10.3842/SIGMA.2014.047
Contribution to the Special Issue on New Directions in Lie Theory

Graded Limits of Minimal Affinizations in Type D

Katsuyuki Naoi
Institute of Engineering, Tokyo University of Agriculture and Technology, 3-8-1 Harumi-cho, Fuchu-shi, Tokyo, Japan

Received October 30, 2013, in final form April 14, 2014; Published online April 20, 2014

Abstract
We study the graded limits of minimal affinizations over a quantum loop algebra of type D in the regular case. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and also give their defining relations. As a corollary we obtain a character formula for the minimal affinizations in terms of Demazure operators, and a multiplicity formula for a special class of the minimal affinizations.

Key words: minimal affinizations; quantum affine algebras; current algebras.

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References

1. Chari V., Minimal affinizations of representations of quantum groups: the rank 2 case, Publ. Res. Inst. Math. Sci. 31 (1995), 873-911, hep-th/9410022.
2. Chari V., On the fermionic formula and the Kirillov-Reshetikhin conjecture, Int. Math. Res. Not. 2001 (2001), 629-654, math.QA/0006090.
3. Chari V., Braid group actions and tensor products, Int. Math. Res. Not. 2002 (2002), 357-382, math.QA/0106241.
4. Chari V., Greenstein J., Minimal affinizations as projective objects, J. Geom. Phys. 61 (2011), 594-609, arXiv:1009.4494.
5. Chari V., Moura A., The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras, Comm. Math. Phys. 266 (2006), 431-454, math.RT/0507584.
6. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
7. Chari V., Pressley A., Minimal affinizations of representations of quantum groups: the nonsimply-laced case, Lett. Math. Phys. 35 (1995), 99-114.
8. Chari V., Pressley A., Quantum affine algebras and their representations, in Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., Vol. 16, Amer. Math. Soc., Providence, RI, 1995, 59-78, hep-th/9411145.
9. Chari V., Pressley A., Minimal affinizations of representations of quantum groups: the irregular case, Lett. Math. Phys. 36 (1996), 247-266.
10. Chari V., Pressley A., Minimal affinizations of representations of quantum groups: the simply laced case, J. Algebra 184 (1996), 1-30, hep-th/9410036.
11. Fourier G., Littelmann P., Tensor product structure of affine Demazure modules and limit constructions, Nagoya Math. J. 182 (2006), 171-198, math.RT/0412432.
12. Hernandez D., On minimal affinizations of representations of quantum groups, Comm. Math. Phys. 276 (2007), 221-259, math.QA/0607527.
13. Hernandez D., Leclerc B., Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), 265-341, arXiv:0903.1452.
14. Joseph A., On the Demazure character formula, Ann. Sci. École Norm. Sup. (4) 18 (1985), 389-419.
15. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
16. Lakshmibai V., Littelmann P., Magyar P., Standard monomial theory for Bott-Samelson varieties, Compositio Math. 130 (2002), 293-318, alg-geom/9703020.
17. Lusztig G., Introduction to quantum groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.
18. Moura A., Restricted limits of minimal affinizations, Pacific J. Math. 244 (2010), 359-397, arXiv:0812.2238.
19. Mukhin E., Young C.A.S., Affinization of category O for quantum groups, Trans. Amer. Math. Soc., to appear, arXiv:1204.2769.
20. Naoi K., Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math. 229 (2012), 875-934, arXiv:1012.5480.
21. Naoi K., Demazure modules and graded limits of minimal affinizations, Represent. Theory 17 (2013), 524-556, arXiv:1210.0175.
22. Sam S.V., Jacobi-Trudi determinants and characters of minimal affinizations, Pacific J. Math., to appear, arXiv:1307.6630.