A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity
Annals of Representation Theory, Volume 1 (2024) no. 4, pp. 567-594.

This paper extends an algorithm and canonical embedding in [6] to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon’s original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space A from the original quantum algebra A via a series of change of variables within the division ring of fractions Frac(A). The canonical embedding takes a completely prime ideal PA to a completely prime ideal QA such that when A is a PI algebra, PI-deg(A/P)=PI-deg(A /Q). When the quantum parameter is a root of unity, combining our construction with results from [2] allows us to state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of A/P given a suitable irreducible representation of A /Q when A is PI.

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DOI: 10.5802/art.19
Classification: 16G99, 16R99, 16S36, 20G42
Keywords: Prime ideal, PI degree, Quantum Nilpotent Algebras

Launois, Stéphane 1; Lopes, Samuel A. 2; Rogers, Alexandra 3

1 Université de Caen Normandie UMR 6139 LMNO 14032 Caen France
2 CMUP Departamento de Matemática Faculdade de Ciências Universidade do Porto Rua do Campo Alegre s/n 4169–007 Porto Portugal
3 School of Mathematics, Statistics and Actuarial Science University of Kent Canterbury CT2 7FS UK
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Launois, Stéphane; Lopes, Samuel A.; Rogers, Alexandra. A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity. Annals of Representation Theory, Volume 1 (2024) no. 4, pp. 567-594. doi : 10.5802/art.19. https://art.centre-mersenne.org/articles/10.5802/art.19/

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