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 from the original quantum algebra via a series of change of variables within the division ring of fractions . The canonical embedding takes a completely prime ideal to a completely prime ideal such that when is a PI algebra, . 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 given a suitable irreducible representation of when is PI.
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Keywords: Prime ideal, PI degree, Quantum Nilpotent Algebras
Launois, Stéphane 1; Lopes, Samuel A. 2; Rogers, Alexandra 3
@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] Enumeration of -strata in quantum matrices with respect to dimension, J. Comb. Theory, Ser. A, Volume 119 (2012) no. 1, pp. 83-98 | DOI | Zbl
[2] PI degree and irreducible representations of quantum determinants rings and their associated quantum Schubert varieties (2022) | arXiv
[3] Lectures on algebraic quantum groups, Advanced Courses in Mathematics – CRM Barcelona, Birkhäuser, 2002 | DOI | Zbl
[4] Azumaya loci and discriminant ideals of PI algebras, Adv. Math., Volume 340 (2018), pp. 1219-1255 | DOI | Zbl
[5] Quantum matrices by paths, Algebra Number Theory, Volume 8 (2014) no. 8, pp. 1857-1912 | DOI | Zbl
[6] Effacement des dérivations et spectres premiers des algèbres quantiques, J. Algebra, Volume 260 (2003) no. 2, pp. 476-518 | DOI | Zbl
[7] Spectre premier de : image canonique et séparation normale, J. Algebra, Volume 260 (2003) no. 2, pp. 519-569 | DOI | Zbl
[8] The discriminant controls automorphism groups of noncommutative algebras, Adv. Math., Volume 269 (2015), pp. 551-584 | DOI | Zbl
[9] The discriminant criterion and automorphism groups of quantized algebras, Adv. Math., Volume 286 (2016), pp. 754-801 | DOI | Zbl
[10] Quantum groups, D-modules, representation theory, and quantum groups (Lecture Notes in Mathematics), Volume 1565, Springer, 1993, pp. 31-140 | DOI | Zbl
[11] Quantum Schubert cells via representation theory and ring theory, Mich. Math. J., Volume 63 (2014) no. 1, pp. 125-157 | DOI | Zbl
[12] Tauvel’s height formula for quantum nilpotent algebras, Commun. Algebra, Volume 47 (2019) no. 10, pp. 4194-4209 | DOI | Zbl
[13] 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] An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, 61, Cambridge University Press, 2004 | DOI | Zbl
[15] From quantum Ore extensions to quantum tori via noncommutative UFDs, Adv. Math., Volume 300 (2016), pp. 672-716 | DOI | Zbl
[16] PI degree parity in -skew polynomial rings, J. Algebra, Volume 319 (2008) no. 10, pp. 4199-4221 | DOI | Zbl
[17] On the automorphism groups of -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] Poisson deleting derivations algorithm and Poisson spectrum, Commun. Algebra, Volume 45 (2017) no. 3, pp. 1294-1313 | DOI | Zbl
[19] Total positivity is a quantum phenomenon: the Grassmannian case, Memoirs of the American Mathematical Society, 1448, American Mathematical Society, 2023 | DOI
[20] Quantum unique factorisation domains, J. Lond. Math. Soc., Volume 74 (2006) no. 2, pp. 321-340 | DOI | Zbl
[21] On -skew iterated Ore extensions satisfying a polynomial identity, J. Algebra Appl., Volume 10 (2011) no. 4, pp. 771-781 | DOI | Zbl
[22] Integral matrices, Pure and Applied Mathematics, 45, Academic Press Inc., 1972 | Zbl
[23] Irreducible Representations of Quantum Nilpotent Algebras at Roots of Unity, and Their Completely Prime Quotients, Ph. D. Thesis, University of Kent, Canterbury, England (2019)
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