An Investigation of the Quantized Matrix Algebra <img width="50" height="30" src="http://article.sciencepublishinggroup.com/journal/141/1411366/image001.png" /> from a Computational Viewpoint

Lina Niu; Rabigul Tuniyaz

doi:doi:10.11648/j.pamj.20231203.11

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An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint

Lina Niu, Rabigul Tuniyaz

Published in Pure and Applied Mathematics Journal (Volume 12, Issue 3)

Received: 13 April 2023 Accepted: 2 May 2023 Published: 11 August 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.

Published in	Pure and Applied Mathematics Journal (Volume 12, Issue 3)
DOI	10.11648/j.pamj.20231203.11
Page(s)	40-48
Creative Commons	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

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[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 GL_n. 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 U_q(A_2n−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 D_q(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 M_q(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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ACS 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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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://article.sciencepublishinggroup.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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TY  - JOUR
T1  - An Investigation of the Quantized Matrix Algebra  from a Computational Viewpoint
AU  - Lina Niu
AU  - Rabigul Tuniyaz
Y1  - 2023/08/11
PY  - 2023
N1  - https://doi.org/10.11648/j.pamj.20231203.11
DO  - 10.11648/j.pamj.20231203.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
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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.
VL  - 12
IS  - 3
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Author Information

Lina Niu

School of Science, Xinjiang Institute of Science and Technology, Akesu, China
Rabigul Tuniyaz

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

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APA Style

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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ACS 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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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://article.sciencepublishinggroup.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}
}

Copy | Download

TY  - JOUR
T1  - An Investigation of the Quantized Matrix Algebra  from a Computational Viewpoint
AU  - Lina Niu
AU  - Rabigul Tuniyaz
Y1  - 2023/08/11
PY  - 2023
N1  - https://doi.org/10.11648/j.pamj.20231203.11
DO  - 10.11648/j.pamj.20231203.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 40
EP  - 48
PB  - Science Publishing Group
SN  - 2326-9812
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.
VL  - 12
IS  - 3
ER  -

Copy | Download