Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T22:31:28.669Z Has data issue: false hasContentIssue false
  • English
  • Français

SPECTRA AND CATENARITY OF MULTI-PARAMETER QUANTUM SCHUBERT CELLS*

Published online by Cambridge University Press:  01 October 2013

MILEN YAKIMOV
MILEN YAKIMOV*
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803USA email: yakimov@math.lsu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the ring theory of the multi-parameter deformations of the quantum Schubert cell algebras obtained from 2-cocycle twists. This is a large family, which extends the Artin–Schelter–Tate algebras of twisted quantum matrices. We classify set theoretically the spectra of all such multi-parameter quantum Schubert cell algebras, construct each of their prime ideals by contracting from explicit normal localizations and prove formulas for the dimensions of their Goodearl–Letzter strata for base fields of arbitrary characteristic and all deformation parameters that are not roots of unity. Furthermore, we prove that the spectra of these algebras are normally separated and that all such algebras are catenary.

Keywords

Primary 16W35Secondary 20G4214M15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue A: New developments in Noncommutative Algebra and its applications. A conference in honour of Kenny Brown's and Toby Stafford's 60th birthdays. , October 2013 , pp. 169 - 194
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

Footnotes

*

Dedicated to Kenny Brown and Toby Stafford on the occasion of their 60th birthdays

References

REFERENCES

1.Artin, M., Schelter, W. and Tate, J., Quantum deformations of GLn, Comm. Pure Appl. Math. 44 (1991), 879–895.Google Scholar
2.Bell, J. P. and Launois, S., On the dimension of H-strata in quantum algebras, Algebra Number Theory 4 (2010), 175–200.CrossRefGoogle Scholar
3.Bell, J., Casteels, K. and Launois, S., Primitive ideals in quantum Schubert cells: dimension of the strata, Forum Math. (in print) doi:10.1515/forum-2011-0155.CrossRefGoogle Scholar
4.Brown, K. A. and Goodearl, K. R., Lectures on algebraic quantum groups, Advanced Courses in Mathematics (CRM. Barcelona, Spain, 2002).Google Scholar
5.Brown, K. A., Goodearl, K. R. and Yakimov, M., Poisson structures of affine spaces and flag varieties. I. Matrix affine Poisson space, Adv. Math. 206 (2006), 567–629.CrossRefGoogle Scholar
6.Caldero, P., Étude des q-commutations dans l'algèbre U q(n+), J. Algebra 178 (1995), 444–457.Google Scholar
7.Cauchon, G., Effacement des dérivations et spectres premiers d'algébres quantiques, J. Algebra 260 (2003), 476–518.Google Scholar
8.Cauchon, G., Spectre premier de O q(M n(k)): image canonique et séparation normale, J. Algebra 260 (2003), 519–569.CrossRefGoogle Scholar
9.De Concini, C.Kac, V. and Procesi, C., Some quantum analogues of solvable Lie groups, in Geometry and analysis (Tata Inst. Fund. Res., Bombay, India, 1992), pp. 41–65.Google Scholar
10.Costantini, M. and Varagnolo, M., Quantum double and multiparameter quantum groups, Comm. Algebra 22 (1994), 6305–6321.CrossRefGoogle Scholar
11.Ekström, E. K., The Auslander condition on graded and filtered Noetherian rings, in Séminaire d'Algèbre Paul Dubreil et Marie-Paul Malliavin, Lecture Notes in Math., 1404 (Springer, Berlin, Germany, 1989), 220–245.Google Scholar
12.Geiß, C., Leclerc, B. and Schröer, J., Cluster structures on quantized coordinate rings, Selecta Math. 19 (2013), 337–397.Google Scholar
13.Goodearl, K. R., Prime spectra of quantized coordinate rings, in Interactions between ring theory and representations of algebras (Murcia), Lect. Notes in Pure and Appl. Math., 210 (Dekker, New York, 2000), 205–237.Google Scholar
14.Goodearl, K. R. and Lenagan, T. H., Catenarity in quantum algebras, J. Pure Appl. Algebra 111 (1996), 123–142.Google Scholar
15.Goodearl, K. R. and Lenagan, T. H., Quantum determinantal ideals, Duke Math. J. 103 (2000), 165–190.Google Scholar
16.Goodearl, K. R. and Lenagan, T. H., Winding-invariant prime ideals in quantum 3 × 3 matrices, J. Algebra 260 (2003), 657–687.Google Scholar
17.Goodearl, K. R. and Letzter, E. S., Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994), 1017–1025.CrossRefGoogle Scholar
18.Goodearl, K. R. and Letzter, E. S., The Dixmier–Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras, Trans. Amer. Math. Soc. 352 (2000) 1381–1403.Google Scholar
19.Goodearl, K. R. and Yakimov, M., Poisson structures of affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), 5753–5780.Google Scholar
20.Gorelik, M., The prime and the primitive spectra of a quantum Bruhat cell translate, J. Algebra 227 (2000), 211–253.Google Scholar
21.Hodges, T. J., Levasseur, T. and Toro, M., Algebraic structure of multiparameter quantum groups, Adv. Math. 126 (1997), 52–92.CrossRefGoogle Scholar
22.Heckenberger, I. and Schneider, H.-J., Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid, Israel J. Math. (in print) doi:10.1007/s11856-012-0180-3.Google Scholar
23.Horton, K. L., The prime and primitive spectra of multiparameter quantum symplectic and Euclidean spaces, Comm. Algebra 31 (2003), 4713–4743.CrossRefGoogle Scholar
24.Jantzen, J. C., Lectures on quantum groups, Grad. Studies in Math., 6 (American Mathematical Society, Providence, RI, 1996).Google Scholar
25.Johnson, G. and Nowlin, C., The FRT-construction via quantum affine algebras and smash products, J. Algebra 353 (2012), 158–173.Google Scholar
26.Joseph, A., On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), 441–511.CrossRefGoogle Scholar
27.Joseph, A., Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 (Springer-Verlag, Berlin, Germany, 1995).Google Scholar
28.Joseph, A., Sur les idéaux génériques de l'agèbre des fonctions sur un groupe quantique, C. R. Acad. Sci. Paris Sér. I. Math. 321 (1995), 135–140.Google Scholar
29.Levasseur, T. and Stafford, J. T., The quantum coordinate ring of the special linear group, J. Pure Appl. Algebra 86 (1993), 181–186.Google Scholar
30.Lusztig, G., Introduction to quantum groups, Progress in Mathematics, 110 (Birkhäuser, Boston, MA, 1993).Google Scholar
31.Malliavin, M.-P., La caténarité de la partie positive de l'algèbre enveloppante quantifiée de l'algèbre de Lie simple de type B 2, Beiträge Algebra Geom. 35 (1994), 73–83 (Festschrift on the occasion of the 65th birthday of Otto Krötenheerdt).Google Scholar
32.Marsh, R. J. and Rietsch, K., Parametrizations of flag varieties, Represent. Theory 8 (2004), 212–242.Google Scholar
33.Mériaux, A. and Cauchon, G., Admissible diagrams in U qw(${\mathfrak{g}$) and combinatoric properties of Weyl groups, Represent. Theory 14 (2010), 645–687.Google Scholar
34.Nowlin, C., Torus-invariant prime spectra of affine quantum nilpotent algebras, PhD thesis(University of California, Santa Barbara, CA, 2010).Google Scholar
35.Oh, S.-Q., Quantum and Poisson structures of multi-parameter symplectic and Euclidean spaces, J. Algebra 319 (2008), 4485–4535.Google Scholar
36.Tang, X., The prime ideal stratification and the automorphism group of U +r,s(B 2) (preprint) arXiv:1109.2640.Google Scholar
37.Yakimov, M., A classification of H-primes of quantum partial flag varieties, Proc. Amer. Math. Soc. 138 (2010), 1249–1261.Google Scholar
38.Yakimov, M., Invariant prime ideals in quantizations of nilpotent Lie algebras, Proc. London Math. Soc. 101 (2) (2010), 454–476.CrossRefGoogle Scholar
39.Yakimov, M., Strata of prime ideals of De Concini–Kac–Procesi algebras and Poisson geometry, in New trends in noncommutative algebra, Contem. Math. 562 (Ara, P., Brown, K. A., Lenagan, T. H., Letzter, E. S., Stafford, J. T. and Zhang, J. J., Editors) (American Mathematical Society, Providence, RI, 2012), 265–278.Google Scholar
40.Yakimov, M., On the spectra of quantum groups, Mem. Amer. Math. Soc. (in print), arXiv:1106.3821.Google Scholar
41.Yakimov, M., A proof of the Goodearl–Lenagan polynormality conjecture, Int. Math. Res. Not. 2013 (9) (2013), 2097–2132.Google Scholar