JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (8): 28-37.doi: 10.6040/j.issn.1671-9352.0.2020.024

Previous Articles     Next Articles

Gröbner-Shirshov basis of irreducible modules over quantum group of type B2

HE Yu-xing, Abdukadir Obul*   

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, Xinjiang, China
  • Published:2020-07-14

PDF (PC)

593

Abstract: By using the known Gröbner-Shirshov basis of quantum group Uq(B2) and the double free module method, we give a Gröbner-Shirshov pair of finite dimensional irreducible Uq(B2)-module Vq(λ), first, then by specializing a suitable version of Uq(B2) at q=1, we get a Gröbner-Shirshov basis of U(B2) and a Gröbner-Shirshov pair of finite dimensional irreducible U(B2)-module V(λ).

Key words: Grö, bner-Shirshov pair, composition, quantum group, double free module

CLC Number: 

  • O153.3
[1] BUCHBERGER B. An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal[D]. Innsbruck: University of Innsbruck, 1965.
[2] SHIRSHOV A I. Some algorithmic problem for Lie algebras[J]. Sibirsk Mat Z, 1962, 3(2):292-296.
[3] BERGMAN G M. The diamond lemma for ring theory[J]. Adv in Math, 1978, 29:178-218.
[4] BOKUT L A, MALCOLMSON P. Gröbner-Shirshov bases for quantum enveloping algebras[J]. Israel Journal of Mathematics, 1996, 96:97-113.
[5] KANG S J, LEE K H. Gröbner-Shirshov bases for representation theory[J]. J Korean Math Soc, 2000, 37(1):55-72.
[6] KANG S J, LEE K H. Gröbner-Shirshov bases for irreducible sln+1-modules[J]. J Algebra, 2000, 232:1-20.
[7] KANG S J, LEE I S, LEE K H, et al. Hecke algebras, Specht modules and Gröbner-Shirshov bases[J]. J Algebra, 2002, 252:258-292.
[8] CHIBRIKOV E S. On free Lie conformal algebras[J]. Vestnik Novosibirsk State University, 2004, 4(1):65-83.
[9] CHEN Y Q, CHEN Y S, ZHONG C Y. Composition-diamond lemma for modules[J]. Czechoslovak Math J, 2010, 135(60), no.1:59-76.
[10] LEE Dong-IL. Gröbner-Shirshov bases for irreducible sp4-modules[J]. J Korean Math Soc, 2008, 45:711-725.
[11] KONG J S, OBUL A. Gröbner-Shirshov basis of quantum group of type B2[J]. Southeast Asian Bull Math, 2010, 34(4):693-704.
[12] JANTZEN J C. Lectures on quantum groups[M] // Graduate Studies in Mathematics, Volume 6, Amer Math Soc.[S.l.] : Providence, 1996.
[13] RINGEL C M. PBW-bases of quantum groups[J]. J Reine Angew Math, 1996, 470:51-88.
[14] BOKUT L A, KLEIN A A. Serre relations and Gröbner-Shirshov bases for simple Lie algebras, II[J]. Internat J of Algebra and Comput, 1996, 6:389-400.
[1] PANG Yongfeng, DU Yawei, YUE Huihui. Spectral decomposition of the Moore-Penrose inverse of operator with closed range [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2026, 61(2): 37-42.
[2] LIU Fuguo, LIU Yuanmeng, SHI Yufeng, TIAN Maozai. Multi-factor iron ore futures price prediction based on VMD-DBO-BiGRU [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(9): 121-132.
[3] LI Jing, YANG Yiping, ZHAO Peixin. Composite quantile regression estimation of linear mixed effects model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(3): 88-99.
[4] MA Nan, LI Yuan. Spectrum and norm of the combinations of two orthogonal projections [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(12): 102-108.
[5] Jin-yu FAN,Yang ZOU,Jian XIONG,Yongyi GU. Imagedata control chart based on nonnegative CP tensor decomposition [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(1): 27-34.
[6] CHENG Yang, CUI Yu-ru. Weak core inverse of core-EP decomposition [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(3): 19-24.
[7] WEI Fang, WANG Chang-peng. Low-rank matrix factorization with double Gaussian prior model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(3): 101-108.
[8] HU De-sheng, Abdukadir OBUL. Gröbner-Shirshov basis of modified Ringel-Hall algebra of type B2 [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(2): 33-43.
[9] LI Xin-yu, FAN Hui, LIU Jing-lei. Robust clustering based on adaptive graph regularization and low-rank matrix decomposition [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(8): 21-38.
[10] REN Xiao-qian, XU Yong-jun. PBW-deformations of type B2 quantum group and their symmetry property [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(8): 65-74.
[11] Gulshadam Yunus, Abdukadir Obul. Generalized inverse via Gröbner-Shirshov basis [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(4): 54-57.
[12] XU Xiu-juan, YAN Shuo, ZHU Ye-qing. Global regularity for very weak solutions to non-homogeneous A-harmonic equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(2): 48-56.
[13] Munayim Dilxat, Abdukadir Obul. Gröbner-Shirshov basis of degenerate affine Hecke algebras of type A [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(2): 106-110.
[14] YAN Yan, HAO Xiao-hong. Differential privacy partitioning algorithm based on adaptive density grids [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(9): 12-22.
[15] ZOU Shao-hui, ZHANG Tian. Interaction relationship between international carbon future price and domestic carbon price [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(5): 70-79.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd