Pure and Applied Mathematics Journal

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An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint

Received: Apr. 13, 2023    Accepted: May 02, 2023    Published: Aug. 11, 2023
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Abstract

In the study of quantum groups, quantized matrix algebras have been widely investigated from the viewpoints of representation theory and noncommutative geometry. This paper addresses a computational approach to the investigation of quantized matrix algebra , namely, by employing the Shirshov algorithmic method, it is shown that the defining relations of constitute a Gröbner-Shirshov basis; by constructing an appropriate monomial ordering on , it is shown that is a solvable polynomial algebra. Consequently, it is shown that several further structural properties of , such as being a Noetherian domain, having Hilbert series , having GK dimension n2, having global homological dimension n2, and being a classical quadratic Koszul algebra, may be derived in a constructive-computational way. Moreover, applying the foregoing structural properties in turn to investigate several structural properties of modules over , such as constructing finite free resolutions of finitely generated modules, establishing the stability of finitely generated projective modules, establishing the K0-groups of , computing minimal graded generating sets of finitely generated graded modules, and establishing the elimination property of one-sided ideals (finitely generated modules), it is shown that all of those properties may be obtained and realized in a computational way.

DOI 10.11648/j.pamj.20231203.11
Published in Pure and Applied Mathematics Journal ( Volume 12, Issue 3, June 2023 )
Page(s) 40-48
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Quantized Matrix Algebra, Gröbner-Shirshov Basis, PBW Basis, Solvable Polynomial Algebra

References
[1] M. Artin, W. F. Schelter, Graded algebras of global dimension 3. Adv. Math., 66 (2) (1987), 171-216.
[2] L. Bokut et al., Gröbner-Shirshov Bases: Normal Forms, Combinatorial and Decision Problems in Algebra. World Scientific Publishing, 2020. https://doi.org/10.1142/9287
[3] R. Dipper and S. Donkin, Quantum GLn. Proc, London Math. Soc., 63 (1991), 156-211.
[4] L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Lie algebras. Algebraic Analysis, Academic Press (1988), 129-140.
[5] H. P. Jakobsen and H. Zhang , A class of quadratic matrix algebras arising from the quantized enveloping algebra Uq(A2n−1). J. Math. Phys., (41) (2000), 2310-2336.
[6] A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. J. Symbolic Comput., 9 (1990), 1-26. Also available as: Technical Report University of Passau, MIP-8807, March 1988.
[7] G.R. Krause and T.H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension. Graduate Studies in Mathematics. American Mathematical Society, 1991.
[8] V.LevandovskyyandH.Schönemann, Plural: acomputer algebra system for noncommutative polynomial algebras. In: Proc. Symbolic and Algebraic Computation, International Symposium ISSAC 2003, Philadelphia, USA, 2003, 176-183.
[9] T. Levasseur, Some properties of noncommutative regular graded rings. Glasgow Math. J., 34 (1992), 277-300.
[10] H. Li, Noncommutative Gröbner Bases and Filtered- graded Transfer. Lecture Notes in Mathematics, Vol. 1795, Springer, 2002. https://doi.org/10.1007/b84211
[11] H. Li, Gröbner Bases in Ring Theory. World Scientific Publishing Co., 2011. https://doi.org/10.1142/8223
[12] H. Li, An elimination lemma for algebras with PBW bases. Communications in Algebra, 46 (8) (2018), 3520- 3532. https://doi.org/10.1080/00927872.2018.1424863
[13] H. Li, Noncommutative Polynomial Algebras of Solvable Type and Their Modules: Basic Constructive- Computational Theory and Methods. Chapman and Hall/CRC Press, 2021.
[14] H. Li, F. Van Oystaeyen, (1996, 2003). Zariskian Filtrations. K-Monograph in Mathematics, Vol. 2. Kluwer Academic Publishers, Berlin Heidelberg: Springer-Verlag.
[15] Lina Niu, Rabigul Tuniyaz, Structural properties of the quantized matrix algebra Dq(n) established by means of Gröbner-Shirshov basis theory. arXiv: 2201.00631v2 [math. RA]. https://doi.org/10.48550/arXiv.2201.00631
[16] Rabigul Tuniyaz, On the standard quantized matrix algebra Mq(n): from a constructive- computational viewpoint. Journal of Algebra and Its Applications. Published online: 18 January 2023. https://doi.org/10.1142/S0219498824500920
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    Lina Niu, Rabigul Tuniyaz. (2023). An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint. Pure and Applied Mathematics Journal, 12(3), 40-48. https://doi.org/10.11648/j.pamj.20231203.11

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    Lina Niu; Rabigul Tuniyaz. An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint. Pure Appl. Math. J. 2023, 12(3), 40-48. doi: 10.11648/j.pamj.20231203.11

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    AMA Style

    Lina Niu, Rabigul Tuniyaz. An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint. Pure Appl Math J. 2023;12(3):40-48. doi: 10.11648/j.pamj.20231203.11

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  • @article{10.11648/j.pamj.20231203.11,
      author = {Lina Niu and Rabigul Tuniyaz},
      title = {An Investigation of the Quantized Matrix Algebra  from a Computational Viewpoint},
      journal = {Pure and Applied Mathematics Journal},
      volume = {12},
      number = {3},
      pages = {40-48},
      doi = {10.11648/j.pamj.20231203.11},
      url = {https://doi.org/10.11648/j.pamj.20231203.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20231203.11},
      abstract = {In the study of quantum groups, quantized matrix algebras have been widely investigated from the viewpoints of representation theory and noncommutative geometry. This paper addresses a computational approach to the investigation of quantized matrix algebra , namely, by employing the Shirshov algorithmic method, it is shown that the defining relations of  constitute a Gröbner-Shirshov basis; by constructing an appropriate monomial ordering on , it is shown that  is a solvable polynomial algebra. Consequently, it is shown that several further structural properties of , such as  being a Noetherian domain, having Hilbert series , having GK dimension n2, having global homological dimension n2, and being a classical quadratic Koszul algebra, may be derived in a constructive-computational way. Moreover, applying the foregoing structural properties in turn to investigate several structural properties of modules over , such as constructing finite free resolutions of finitely generated modules, establishing the stability of finitely generated projective modules, establishing the K0-groups of , computing minimal graded generating sets of finitely generated graded modules, and establishing the elimination property of one-sided ideals (finitely generated modules), it is shown that all of those properties may be obtained and realized in a computational way.},
     year = {2023}
    }
    

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    AU  - Lina Niu
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    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    UR  - https://doi.org/10.11648/j.pamj.20231203.11
    AB  - In the study of quantum groups, quantized matrix algebras have been widely investigated from the viewpoints of representation theory and noncommutative geometry. This paper addresses a computational approach to the investigation of quantized matrix algebra , namely, by employing the Shirshov algorithmic method, it is shown that the defining relations of  constitute a Gröbner-Shirshov basis; by constructing an appropriate monomial ordering on , it is shown that  is a solvable polynomial algebra. Consequently, it is shown that several further structural properties of , such as  being a Noetherian domain, having Hilbert series , having GK dimension n2, having global homological dimension n2, and being a classical quadratic Koszul algebra, may be derived in a constructive-computational way. Moreover, applying the foregoing structural properties in turn to investigate several structural properties of modules over , such as constructing finite free resolutions of finitely generated modules, establishing the stability of finitely generated projective modules, establishing the K0-groups of , computing minimal graded generating sets of finitely generated graded modules, and establishing the elimination property of one-sided ideals (finitely generated modules), it is shown that all of those properties may be obtained and realized in a computational way.
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Author Information
  • School of Science, Xinjiang Institute of Science and Technology, Akesu, China

  • School of Science, Xinjiang Institute of Science and Technology, Akesu, China

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