Abstract
Let U be a quantized enveloping algebra and \(\dot U\) its modified form. Lusztig gives some symmetries on U and \(\dot U\). In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig’s symmetries on U and their important properties, for instance, the braid relations. In this paper, we define a modified form 
of the Ringel-Hall algebra and realize the Lusztig’s symmetries on \(\dot U\) by applying the BGP-reflection functors to 
.
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References
Bernstein, I. N., Gelfand, I. M., Ponomarev, A.: Coxeter functors and Gabriel’s theorem. Uspekhi Mat. Nauk, 28, 19–33 (1973)
Dlab, V., Ringel, C. M.: Indecomposable Representation of Graphs and Algebras. Mem. Amer. Math. Soc., 173, American Mathematical Society, Providence, 1976
Deng, B., Xiao, J.: On double Ringel-Hall algebras. J. Algebra, 251, 110–149 (2002)
Deng, B., Xiao, J.: Ringel-Hall algebras and Lusztig’s symmetries. J. Algebra, 255, 357–372 (2002)
Green, J. A.: Hall algebras, hereditary algebras and quantum groups. Invent. Math., 120, 361–377 (1995)
Kac, V. G.: Infinite Dimensional Lie Algebras, Third edition, Cambridge Univ. Press, Cambridge, 1990
Lusztig, G.: Introduction to Quantum Groups, Birkhäuser, Boston, 1993
Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math., 70, 237–249 (1988)
Lusztig, G.: Quantum groups at roots of 1. Geom. Dedicata, 35, 89–113 (1990)
Ringel, C. M.: Hall algebras and quantum groups. Invent. Math., 101, 583–591 (1990)
Ringel, C. M.: PBW-bases of quantum groups. J. Reine Angew. Math., 470, 51–88 (1996)
Sevenhant, B., Van den Bergh, M.: A relation between a conjecture of Kac and the structure of the Hall algebra. J. Pure Appl. Algebra, 160, 319–332 (2001)
Sevenhant, B., Van den Bergh, M.: On the double of the Hall algebra of a quiver. J. Algebra, 221, 135–160 (1999)
Xiao, J.: Drinfeld double and Ringel-Green theory of Hall algebras. J. Algebra, 190, 100–144 (1997)
Xiao, J., Yang, S.: BGP-reflection functors and Lusztig’s symmetries: a Ringel-Hall algebra approach to quantum groups. J. Algebra, 241, 204–246 (2001)
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Supported by National Natural Science Foundation of China (Grant No. 11131001) and the Fundamental Research Funds for the Central Universities (Grant No. BLX2013014)
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Xiao, J., Zhao, M.H. BGP-reflection functors and Lusztig’s symmetries of modified quantized enveloping algebras. Acta. Math. Sin.-English Ser. 29, 1833–1856 (2013). https://doi.org/10.1007/s10114-013-2295-9
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DOI: https://doi.org/10.1007/s10114-013-2295-9