AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
On the spectra of quantum groups
About this Title
Milen Yakimov, Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 229, Number 1078
ISBNs: 978-0-8218-9174-2 (print); 978-1-4704-1532-7 (online)
DOI: https://doi.org/10.1090/memo/1078
Published electronically: November 12, 2013
Keywords: Quantum groups,
prime spectra,
quantum nilpotent algebras,
separation of variables,
prime elements,
maximal ideals,
chain conditions
MSC: Primary 16T20; Secondary 20G42, 17B37, 53D17
Table of Contents
Chapters
- 1. Introduction
- 2. Previous results on spectra of quantum function algebras
- 3. A description of the centers of Joseph’s localizations
- 4. Primitive ideals of $R_q[G]$ and a Dixmier map for $R_q[G]$
- 5. Separation of variables for the algebras $S^\pm _w$
- 6. A classification of the normal and prime elements of the De Concini–Kac–Procesi algebras
- 7. Module structure of $R_{\mathbf {w}}$ over their subalgebras generated by Joseph’s normal elements
- 8. A classification of maximal ideals of $R_q[G]$ and a question of Goodearl and Zhang
- 9. Chain properties and homological applications
Abstract
Joseph and Hodges–Levasseur (in the A case) described the spectra of all quantum function algebras $R_q[G]$ on simple algebraic groups in terms of the centers of certain localizations of quotients of $R_q[G]$ by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. We determine the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it we deduce a more explicit description of all prime ideals of $R_q[G]$ than the previously known ones and an explicit parametrization of $\mathrm {Spec} R_q[G]$. We combine the latter with a result of Kogan and Zelevinsky to obtain in the complex case a torus equivariant Dixmier type map from the symplectic foliation of the group $G$ to the primitive spectrum of $R_q[G]$. Furthermore, under the general assumptions on the ground field and deformation parameter, we prove a theorem for separation of variables for the De Concini–Kac–Procesi algebras $\mathcal {U}^w_\pm$, and classify the sets of their homogeneous normal elements and primitive elements. We apply those results to obtain explicit formulas for the prime and especially the primitive ideals of $\mathcal {U}^w_\pm$ lying in the Goodearl–Letzter stratum over the $\{0\}$-ideal. This is in turn used to prove that all Joseph’s localizations of quotients of $R_q[G]$ by torus invariant prime ideals are free modules over their subalgebras generated by Joseph’s normal elements. From it we derive a classification of the maximal spectrum of $R_q[G]$ and use it to resolve a question of Goodearl and Zhang, showing that all maximal ideals of $R_q[G]$ have finite codimension. We prove that $R_q[G]$ possesses a stronger property than that of the classical catenarity: all maximal chains in $\mathrm {Spec} R_q[G]$ have the same length equal to $\mathrm {GK \, dim} R_q[G]= \dim G$, i.e., $R_q[G]$ satisfies the first chain condition for prime ideals in the terminology of Nagata.- Henning Haahr Andersen, Patrick Polo, and Ke Xin Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1–59. MR 1094046, DOI 10.1007/BF01245066
- Jason P. Bell and Stéphane Launois, On the dimension of $H$-strata in quantum algebras, Algebra Number Theory 4 (2010), no. 2, 175–200. MR 2592018, DOI 10.2140/ant.2010.4.175
- Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1–52. MR 2110627, DOI 10.1215/S0012-7094-04-12611-9
- Arkady Berenstein and Andrei Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), no. 2, 405–455. MR 2146350, DOI 10.1016/j.aim.2004.08.003
- K. A. Brown and K. R. Goodearl, Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2465–2502. MR 1348148, DOI 10.1090/S0002-9947-96-01597-8
- K. A. Brown and K. R. Goodearl, A Hilbert basis theorem for quantum groups, Bull. London Math. Soc. 29 (1997), no. 2, 150–158. MR 1425991, DOI 10.1112/S0024609396002263
- Ken A. Brown and Ken R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492
- Philippe Caldero, Sur le centre de $U_q({\mathfrak {n}}^+)$, Beiträge Algebra Geom. 35 (1994), no. 1, 13–24 (French, with English and French summaries). Festschrift on the occasion of the 65th birthday of Otto Krötenheerdt. MR 1287192
- Philippe Caldero, Étude des $q$-commutations dans l’algèbre $U_q({\mathfrak {n}}^+)$, J. Algebra 178 (1995), no. 2, 444–457 (French, with French summary). MR 1359896, DOI 10.1006/jabr.1995.1359
- Gérard Cauchon, Effacement des dérivations et spectres premiers des algèbres quantiques, J. Algebra 260 (2003), no. 2, 476–518 (French, with English summary). MR 1967309, DOI 10.1016/S0021-8693(02)00542-2
- A. W. Chatters, Non-commutative unique factorisation domains, Math. Proc. Cambridge Philos. Soc. 95 (1984), 49–54.
- A. W. Chatters and D. A. Jordan, Noncommutative unique factorisation rings, J. London Math. Soc. (2) 33 (1986), no. 1, 22–32. MR 829384, DOI 10.1112/jlms/s2-33.1.22
- C. De Concini, V. G. Kac, and C. Procesi, Some quantum analogues of solvable Lie groups, Geometry and analysis (Bombay, 1992) Tata Inst. Fund. Res., Bombay, 1995, pp. 41–65. MR 1351503
- C. Geiß, B. Leclerc, and J. Schröer, Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2013), no. 2, 337–397. MR 3090232, DOI 10.1007/s00029-012-0099-x
- K. R. Goodearl, Prime spectra of quantized coordinate rings, Interactions between ring theory and representations of algebras (Murcia), Lecture Notes in Pure and Appl. Math., vol. 210, Dekker, New York, 2000, pp. 205–237. MR 1759846
- K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra 111 (1996), no. 1-3, 123–142. MR 1394347, DOI 10.1016/0022-4049(95)00120-4
- K. R. Goodearl and T. H. Lenagan, Primitive ideals in quantum $SL_3$ and $GL_3$, New trends in noncommutative algebra, Contemp. Math., vol. 562, Amer. Math. Soc., Providence, RI, 2012, pp. 115–140. MR 2905557, DOI 10.1090/conm/562/11134
- K. R. Goodearl and E. S. Letzter, Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1017–1025. MR 1211579, DOI 10.1090/S0002-9939-1994-1211579-1
- K. R. Goodearl and E. S. Letzter, The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1381–1403. MR 1615971, DOI 10.1090/S0002-9947-99-02345-4
- K. R. Goodearl and J. J. Zhang, Homological properties of quantized coordinate rings of semisimple groups, Proc. Lond. Math. Soc. (3) 94 (2007), no. 3, 647–671. MR 2325315, DOI 10.1112/plms/pdl022
- Maria Gorelik, The prime and the primitive spectra of a quantum Bruhat cell translate, J. Algebra 227 (2000), no. 1, 211–253. MR 1754232, DOI 10.1006/jabr.1999.8235
- Timothy J. Hodges and Thierry Levasseur, Primitive ideals of $\textbf {C}_q[\textrm {SL}(3)]$, Comm. Math. Phys. 156 (1993), no. 3, 581–605. MR 1240587
- Timothy J. Hodges and Thierry Levasseur, Primitive ideals of $\textbf {C}_q[\textrm {SL}(n)]$, J. Algebra 168 (1994), no. 2, 455–468. MR 1292775, DOI 10.1006/jabr.1994.1239
- Timothy J. Hodges, Thierry Levasseur, and Margarita Toro, Algebraic structure of multiparameter quantum groups, Adv. Math. 126 (1997), no. 1, 52–92. MR 1440253, DOI 10.1006/aima.1996.1612
- Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532
- A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977), no. 2, 241–289. MR 453829, DOI 10.1016/0021-8693(77)90306-4
- Anthony Joseph, On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), no. 2, 441–511. MR 1297159, DOI 10.1006/jabr.1994.1294
- Anthony Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer-Verlag, Berlin, 1995. MR 1315966
- Anthony Joseph, Sur les idéaux génériques de l’algèbre des fonctions sur un groupe quantique, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 2, 135–140 (French, with English and French summaries). MR 1345435
- Anthony Joseph and Gail Letzter, Separation of variables for quantized enveloping algebras, Amer. J. Math. 116 (1994), no. 1, 127–177. MR 1262429, DOI 10.2307/2374984
- Mikhail Kogan and Andrei Zelevinsky, On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups, Int. Math. Res. Not. 32 (2002), 1685–1702. MR 1916837, DOI 10.1155/S1073792802203050
- Stefan Kolb, The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight, J. Algebra 319 (2008), no. 8, 3518–3534. MR 2408329, DOI 10.1016/j.jalgebra.2007.10.004
- Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
- B. Kostant, ${\mathrm {Cent}} \, U(n)$ and a construction of Lipsman–Wolf, preprint arXiv:1101.2459.
- Bertram Kostant, The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group, Mosc. Math. J. 12 (2012), no. 3, 605–620, 669 (English, with English and Russian summaries). MR 3024825, DOI 10.17323/1609-4514-2012-12-3-605-620
- S. Launois and T. H. Lenagan, Primitive ideals and automorphisms of quantum matrices, Algebr. Represent. Theory 10 (2007), no. 4, 339–365. MR 2333441, DOI 10.1007/s10468-007-9059-0
- S. Launois, T. H. Lenagan, and L. Rigal, Quantum unique factorisation domains, J. London Math. Soc. (2) 74 (2006), no. 2, 321–340. MR 2269632, DOI 10.1112/S0024610706022927
- T. H. Lenagan, private communication.
- T. Levasseur and J. T. Stafford, The quantum coordinate ring of the special linear group, J. Pure Appl. Algebra 86 (1993), no. 2, 181–186. MR 1215645, DOI 10.1016/0022-4049(93)90102-Y
- Serge Levendorskiĭ and Yan Soibelman, Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys. 139 (1991), no. 1, 141–170. MR 1116413
- Ronald L. Lipsman and Joseph A. Wolf, Canonical semi-invariants and the Plancherel formula for parabolic groups, Trans. Amer. Math. Soc. 269 (1982), no. 1, 111–131. MR 637031, DOI 10.1090/S0002-9947-1982-0637031-6
- Samuel A. Lopes, Separation of variables for $U_q(\mathfrak {sl}_{n+1})^+$, Canad. Math. Bull. 48 (2005), no. 4, 587–600. MR 2176155, DOI 10.4153/CMB-2005-054-8
- Samuel A. Lopes, Primitive ideals of $U_q(\mathfrak {sl}^+_n)$, Comm. Algebra 34 (2006), no. 12, 4523–4550. MR 2273722, DOI 10.1080/00927870600936682
- D.-M. Lu, Q.-S. Wu, and J. J. Zhang, Hopf algebras with rigid dualizing complexes, Israel J. Math. 169 (2009), 89–108. MR 2460900, DOI 10.1007/s11856-009-0005-1
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
- J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Revised edition, Graduate Studies in Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2001. With the cooperation of L. W. Small. MR 1811901
- Antoine Mériaux and Gérard Cauchon, Admissible diagrams in quantum nilpotent algebras and combinatoric properties of Weyl groups, Represent. Theory 14 (2010), 645–687. MR 2736313, DOI 10.1090/S1088-4165-2010-00382-9
- Masayoshi Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 51–64. MR 78974
- Nicolai Reshetikhin and Milen Yakimov, Quantum invariant measures, Comm. Math. Phys. 224 (2001), no. 2, 399–426. MR 1869872, DOI 10.1007/PL00005587
- Louis J. Ratliff Jr., Chain conjectures in ring theory, Lecture Notes in Mathematics, vol. 647, Springer, Berlin, 1978. An exposition of conjectures on catenary chains. MR 496884
- Ben Webster and Milen Yakimov, A Deodhar-type stratification on the double flag variety, Transform. Groups 12 (2007), no. 4, 769–785. MR 2365444, DOI 10.1007/s00031-007-0061-8
- Milen Yakimov, Invariant prime ideals in quantizations of nilpotent Lie algebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 454–476. MR 2679698, DOI 10.1112/plms/pdq006
- Milen Yakimov, A classification of $H$-primes of quantum partial flag varieties, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1249–1261. MR 2578519, DOI 10.1090/S0002-9939-09-10180-6
- Milen Yakimov, A proof of the Goodearl-Lenagan polynormality conjecture, Int. Math. Res. Not. IMRN 9 (2013), 2097–2132. MR 3053415, DOI 10.1093/imrn/rns111