A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity
Launois, Stéphane1; Lopes, Samuel A.2; Rogers, Alexandra3
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
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 P⊲A to a completely prime ideal Q⊲A ′ 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.

Received:
Revised:
Accepted:
Published online:
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 
@article{ART_2024__1_4_567_0,
     author = {Launois, St\'ephane and Lopes, Samuel A. and Rogers, Alexandra},
     title = {A {Deleting} {Derivations} {Algorithm} for {Quantum} {Nilpotent} {Algebras} at {Roots} of {Unity}},
     journal = {Annals of Representation Theory},
     pages = {567--594},
     publisher = {The Publishers of ART},
     volume = {1},
     number = {4},
     year = {2024},
     doi = {10.5802/art.19},
     language = {en},
     url = {https://art.centre-mersenne.org/articles/10.5802/art.19/}
}
TY  - JOUR
AU  - Launois, Stéphane
AU  - Lopes, Samuel A.
AU  - Rogers, Alexandra
TI  - A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity
JO  - Annals of Representation Theory
PY  - 2024
SP  - 567
EP  - 594
VL  - 1
IS  - 4
PB  - The Publishers of ART
UR  - https://art.centre-mersenne.org/articles/10.5802/art.19/
DO  - 10.5802/art.19
LA  - en
ID  - ART_2024__1_4_567_0
ER  - 
%0 Journal Article
%A Launois, Stéphane
%A Lopes, Samuel A.
%A Rogers, Alexandra
%T A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity
%J Annals of Representation Theory
%D 2024
%P 567-594
%V 1
%N 4
%I The Publishers of ART
%U https://art.centre-mersenne.org/articles/10.5802/art.19/
%R 10.5802/art.19
%G en
%F ART_2024__1_4_567_0
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/

[1] Bell, J. P.; Casteels, K.; Launois, S. Enumeration of ℋ-strata in quantum matrices with respect to dimension, J. Comb. Theory, Ser. A, Volume 119 (2012) no. 1, pp. 83-98 | DOI | Zbl

[2] Bell, J. P.; Launois, S.; Rogers, A. PI degree and irreducible representations of quantum determinants rings and their associated quantum Schubert varieties (2022) | arXiv

[3] Brown, K. A.; Goodearl, K. R. Lectures on algebraic quantum groups, Advanced Courses in Mathematics – CRM Barcelona, Birkhäuser, 2002 | DOI | Zbl

[4] Brown, K. A.; Yakimov, M. T. Azumaya loci and discriminant ideals of PI algebras, Adv. Math., Volume 340 (2018), pp. 1219-1255 | DOI | Zbl

[5] Casteels, K. Quantum matrices by paths, Algebra Number Theory, Volume 8 (2014) no. 8, pp. 1857-1912 | DOI | Zbl

[6] Cauchon, G. Effacement des dérivations et spectres premiers des algèbres quantiques, J. Algebra, Volume 260 (2003) no. 2, pp. 476-518 | DOI | Zbl

[7] Cauchon, G. Spectre premier de O q (M n (k)): image canonique et séparation normale, J. Algebra, Volume 260 (2003) no. 2, pp. 519-569 | DOI | Zbl

[8] Ceken, S.; Palmieri, J. H.; Wang, Y.-H.; Zhang, J. J. The discriminant controls automorphism groups of noncommutative algebras, Adv. Math., Volume 269 (2015), pp. 551-584 | DOI | Zbl

[9] Ceken, S.; Palmieri, J. H.; Wang, Y.-H.; Zhang, J. J. The discriminant criterion and automorphism groups of quantized algebras, Adv. Math., Volume 286 (2016), pp. 754-801 | DOI | Zbl

[10] De Concini, C.; Procesi, C. Quantum groups, D-modules, representation theory, and quantum groups (Lecture Notes in Mathematics), Volume 1565, Springer, 1993, pp. 31-140 | DOI | Zbl

[11] Geiger, J.; Yakimov, M. T. Quantum Schubert cells via representation theory and ring theory, Mich. Math. J., Volume 63 (2014) no. 1, pp. 125-157 | DOI | Zbl

[12] Goodearl, K. R.; Launois, S.; Lenagan, T. H. Tauvel’s height formula for quantum nilpotent algebras, Commun. Algebra, Volume 47 (2019) no. 10, pp. 4194-4209 | DOI | Zbl

[13] Goodearl, K. R.; Letzter, E. S. Prime and primitive spectra of multiparameter quantum affine spaces, Trends in ring theory (Miskolc, 1996) (CMS Conference Proceedings), Volume 22, American Mathematical Society; Canadian Mathematical Society, 1998, pp. 39-58 | Zbl

[14] Goodearl, K. R.; R. B. Warfield, Jr. An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, 61, Cambridge University Press, 2004 | DOI | Zbl

[15] Goodearl, K. R.; Yakimov, M. T. From quantum Ore extensions to quantum tori via noncommutative UFDs, Adv. Math., Volume 300 (2016), pp. 672-716 | DOI | Zbl

[16] Haynal, H. PI degree parity in q-skew polynomial rings, J. Algebra, Volume 319 (2008) no. 10, pp. 4199-4221 | DOI | Zbl

[17] Launois, S. On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras, From Lie Algebras to Quantum Groups. Coimbra, Portugal, 28-30 June, 2006, Volume 28, Centro Internacional de Matemática (2007), pp. 125-143

[18] Launois, S.; Lecoutre, C. Poisson deleting derivations algorithm and Poisson spectrum, Commun. Algebra, Volume 45 (2017) no. 3, pp. 1294-1313 | DOI | Zbl

[19] Launois, S.; Lenagan, T. H.; Nolan, B. Total positivity is a quantum phenomenon: the Grassmannian case, Memoirs of the American Mathematical Society, 1448, American Mathematical Society, 2023 | DOI

[20] Launois, S.; Lenagan, T. H.; Rigal, L. Quantum unique factorisation domains, J. Lond. Math. Soc., Volume 74 (2006) no. 2, pp. 321-340 | DOI | Zbl

[21] Leroy, A.; Matczuk, J. On q-skew iterated Ore extensions satisfying a polynomial identity, J. Algebra Appl., Volume 10 (2011) no. 4, pp. 771-781 | DOI | Zbl

[22] Newman, M. Integral matrices, Pure and Applied Mathematics, 45, Academic Press Inc., 1972 | Zbl

[23] Rogers, A. Irreducible Representations of Quantum Nilpotent Algebras at Roots of Unity, and Their Completely Prime Quotients, Ph. D. Thesis, University of Kent, Canterbury, England (2019)

Cited by Sources: