Gelfand–Kirillov dimensions of simple modules over twisted group algebras $k \ast A$
- 作者: Gupta A.1, Arunachalam U.2
-
隶属关系:
- Ramakrishna Mission Vivekananda Educational and Research Institute
- Harish-Chandra Research Institute
- 期: 卷 86, 编号 4 (2022)
- 页面: 103-115
- 栏目: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133877
- DOI: https://doi.org/10.4213/im9182
- ID: 133877
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详细
For the $n$-dimensional multi-parameter quantum torus algebra $\Lambda_{\mathfrak q}$ over a field $k$ defined by a multiplicativelyantisymmetric matrix $\mathfrak q = (q_{ij})$ we show that, in the case whenthe torsion-free rank of the subgroup of $k^\times$ generated by the $q_{ij}$is large enough, there is a characteristic set of values (possibly with gaps)from $0$ to $n$ that can occur as the Gelfand–Kirillov dimensions of simplemodules. The special case when $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = n - 1$and $\Lambda_{\mathfrak q}$ is simple, studied in A. Gupta, $\mathrm{GK}$-dimensions of simple modules over $K[X^{\pm 1},\sigma]$, Comm. Algebra, 41(7) (2013), 2593–2597, is considered withoutassuming the simplicity, and it is shown that a dichotomy still holds for theGK dimension of simple modules.
作者简介
Ashish Gupta
Ramakrishna Mission Vivekananda Educational and Research Institute
Email: a0gupt@gmail.com
Umamaheswaran Arunachalam
Harish-Chandra Research Institute
Email: ruthreswaran@gmail.com
参考
- Y. Manin, Topics in non-commutative geometry, M. B. Porter Lectures, Princeton Univ. Press, Princeton, NJ, 1991, viii+164 pp.
- C. J. B. Brookes, “Crossed products and finitely presented groups”, J. Group Theory, 3:4 (2000), 433–444
- В. А. Артамонов, “Строение модулей над квантовыми полиномами”, УМН, 50:6(306) (1995), 167–168
- В. А. Артамонов, “Проективные модули над квантовыми алгебрами полиномов”, Матем. сб., 185:7 (1994), 3–12
- В. А. Артамонов, “Неприводимые модули над квантовыми полиномами”, УМН, 51:6(312) (1996), 189–190
- В. А. Артамонов, “Модули над квантовыми полиномами”, Матем. заметки, 59:4 (1996), 497–503
- В. А. Артамонов, “Квантовая проблема Серра”, УМН, 53:4(322) (1998), 3–76
- В. А. Артамонов, “Общие квантовые многочлены: неприводимые модули и Морита-эквивалентность”, Изв. РАН. Сер. матем., 63:5 (1999), 3–36
- V. A. Artamonov, “On projective modules over quantum polynomials”, J. Math. Sci. (N.Y.), 93:2 (1999), 135–148
- J. Alev, M. Chamarie, “Derivations et automorphismes de quelques algèbres quantiques”, Comm. Algebra, 20:6 (1992), 1787–1802
- В. А. Артамонов, “Алгебры квантовых многочленов”, Алгебра – 4, Итоги науки и техн. Сер. Соврем. мат. и ее прил. Темат. обз., 26, ВИНИТИ, М., 2002, 5–34
- V. A. Artamonov, R. Wisbauer, “Homological properties of quantum polynomials”, Algebr. Represent. Theory, 4:3 (2001), 219–247
- K.-H. Neeb, “On the classification of rational quantum tori and the structure of their automorphism groups”, Canad. Math. Bull., 51:2 (2008), 261–282
- J. M. Osborn, D. S. Passman, “Derivations of skew polynomial rings”, J. Algebra, 176:2 (1995), 417–448
- E. Aljadeff, Y. Ginosar, “On the global dimension of multiplicative Weyl algebras”, Arch. Math. (Basel), 62:5 (1994), 401–407
- J. C. McConnell, J. J. Pettit, “Crossed products and multiplicative analogues of Weyl algebras”, J. London Math. Soc. (2), 38:1 (1988), 47–55
- A. Gupta, “The Krull and global dimension of the tensor product of quantum tori”, J. Algebra Appl., 15:9 (2016), 1650174, 19 pp.
- V. A. Artamonov, “Quantum polynomials”, Advances in algebra and combinatorics, World Sci. Publ., Hackensack, NJ, 2008, 19–34
- J. C. McConnell, “Quantum groups, filtered rings and Gelfand–Kirillov dimension”, Noncommutative ring theory (Athens, OH, 1989), Lecture Notes in Math., 1448, Springer, Berlin, 1990, 139–147
- G. R. Krause, T. H. Lenagan, Growth of algebras and Gelfand–Kirillov dimension, Grad. Stud. Math., 22, Rev. ed., Amer. Math. Soc., Providence, RI, 2000, x+212 pp.
- J. C. McConnell, J. C. Robson, Noncommutative Noetherian rings, Grad. Stud. Math., 30, Rev. ed., Amer. Math. Soc., Providence, RI, 2001, xx+636 pp.
- S. C. Coutinho, A primer of algebraic $D$-modules, London Math. Soc. Stud. Texts, 33, Cambridge Univ. Press, Cambridge, 1995, xii+207 pp.
- A. Gupta, A. D. Sarkar, “A dichotomy for the Gelfand–Kirillov dimensions of simple modules over simple differential rings”, Algebr. Represent. Theory, 21:3 (2018), 579–587
- J. C. McConnell, “Representations of solvable Lie algebras. V. On the Gelfand–Kirillov dimension of simple modules”, J. Algebra, 76:2 (1982), 489–493
- J. T. Stafford, “Non-holonomic modules over Weyl algebras and enveloping algebras”, Invent. Math., 79:3 (1985), 619–638
- J. Bernstein, V. Lunts, “On non-holonomic irreducible $D$-modules”, Invent. Math., 94:2 (1988), 223–243
- S. C. Coutinho, “On involutive homogeneous varieties and representations of Weyl algebras”, J. Algebra, 227:1 (2000), 195–210
- A. Gupta, “GK dimensions of simple modules over $K[X^{pm 1}, sigma]$”, Comm. Algebra, 41:7 (2013), 2593–2597
- A. Gupta, “Representations of the $n$-dimensional quantum torus”, Comm. Algebra, 44:7 (2016), 3077–3087
- D. S. Passman, The algebraic structure of group rings, Corr. reprint of the 1977 original, R. E. Krieger Publishing Co., Inc., Melbourne, FL, 1985, xiv+734 pp.
- C. J. B. Brookes, J. R. J. Groves, “Modules over crossed products of a division ring by a free Abelian group. I”, J. Algebra, 229:1 (2000), 25–54
- D. S. Passman, Infinite crossed products, Pure Appl. Math., 135, Academic Press, Inc., Boston, MA, 1989, xii+468 pp.
- L. H. Rowen, Graduate algebra: commutative view, Grad. Stud. Math., 73, Amer. Math. Soc., Providence, RI, 2006, xviii+438 pp.
- Quanshui Wu, “Gelfand–Kirillov dimension under base field extension”, Israel J. Math., 73:3 (1991), 289–296
- L. H. Rowen, Graduate algebra: noncommutative view, Grad. Stud. Math., 91, Amer. Math. Soc., Providence, RI, 2008, xxvi+648 pp.
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