An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint:: Science Publishing Group

An Investigation of the Quantized Matrix Algebra from a Computational Viewpoint

Lina Niu, Rabigul Tuniyaz

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

Pure and Applied Mathematics Journal, Volume 12, Issue 3, June 2023

Pages: 40-48

Received: Apr. 13, 2023

Accepted: May 02, 2023

Published: Aug. 11, 2023

DOI: 10.11648/j.pamj.20231203.11

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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.

Keywords

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

To cite this article

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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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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