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An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint

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.

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

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

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

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

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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