Prime spectra of quantum semisimple groups

Brown, K.; Goodearl, K.

doi:10.1090/S0002-9947-96-01597-8

Prime spectra of quantum semisimple groups
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by K. A. Brown and K. R. Goodearl

Trans. Amer. Math. Soc. 348 (1996), 2465-2502

DOI: https://doi.org/10.1090/S0002-9947-96-01597-8

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Abstract:

We study the prime ideal spaces of the quantized function algebras $R_{q}[G]$, for $G$ a semisimple Lie group and $q$ an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras $R_{q}[G]$ satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar’s strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature – it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of $R_{q}[G]$. In the final section the results are specialized to the case $G= SL_{n}(\mathbb {C})$, where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine $n$-space satisfy our axiom scheme when the group generated by the parameters is torsionfree.

References

Walter Borho, Peter Gabriel, and Rudolf Rentschler, Primideale in Einhüllenden auflösbarer Lie-Algebren (Beschreibung durch Bahnenräume), Lecture Notes in Mathematics, Vol. 357, Springer-Verlag, Berlin-New York, 1973 (German). MR 0376790, DOI 10.1007/BFb0069765
Kenneth A. Brown, On the representation theory of solvable Lie algebras. II. The abelian group attached to a prime ideal, J. London Math. Soc. (2) 43 (1991), no. 1, 49–60. MR 1099085, DOI 10.1112/jlms/s2-43.1.49
Kenneth A. Brown and Fokko du Cloux, On the representation theory of solvable Lie algebras, Proc. London Math. Soc. (3) 57 (1988), no. 2, 284–300. MR 950592, DOI 10.1112/plms/s3-57.2.284
Kenneth A. Brown and R. B. Warfield Jr., The influence of ideal structure on representation theory, J. Algebra 116 (1988), no. 2, 294–315. MR 953153, DOI 10.1016/0021-8693(88)90219-0
William Chin and Ian M. Musson, Hopf algebra duality, injective modules and quantum groups, Comm. Algebra 22 (1994), no. 12, 4661–4692. MR 1285700, DOI 10.1080/00927879408825095
C. De Concini and C. Procesi, Quantum groups, $D$-modules, representation theory, and quantum groups (Venice, 1992) Lecture Notes in Math., vol. 1565, Springer, Berlin, 1993, pp. 31–140. MR 1288995, DOI 10.1007/BFb0073466
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Alfred Goldie and Gerhard Michler, Ore extensions and polycyclic group rings, J. London Math. Soc. (2) 9 (1974/75), 337–345. MR 357500, DOI 10.1112/jlms/s2-9.2.337
K. R. Goodearl, Classical localizability in solvable enveloping algebras and Poincaré-Birkhoff-Witt extensions, J. Algebra 132 (1990), no. 1, 243–262. MR 1060846, DOI 10.1016/0021-8693(90)90266-Q
K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra (to appear).
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 R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1020298
Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965, DOI 10.1090/surv/014
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, DOI 10.1007/BF02096864
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
T. J. Hodges, T. Levasseur, and M. Toro, Algebraic structure of multi-parameter quantum groups, Advances in Math. (to appear).
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
A. V. Jategaonkar, Localization in Noetherian rings, London Mathematical Society Lecture Note Series, vol. 98, Cambridge University Press, Cambridge, 1986. MR 839644, DOI 10.1017/CBO9780511661938
Anthony Joseph, Idéaux premiers et primitifs de l’algèbre des fonctions sur un groupe quantique, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 11, 1139–1142 (French, with English and French summaries). MR 1221638
—, On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), 441-511.
—, Quantum Groups and Their Primitive Ideals, Ergeb. der Math. und ihrer Grenz- geb. (3) 29, Springer-Verlag, Berlin, 1995.
—, Sur les ideaux génériques de l’algèbre des fonctions sur un groupe quantique, C. R. Acad. Sci. Paris, Sér. I 321 (1995), 135-140.
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
A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR 0412321, DOI 10.1007/978-3-642-66243-0
T. H. Lenagan and Edward S. Letzter, The fundamental prime ideals of a Noetherian prime PI ring, Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 1, 113–121. MR 1038770, DOI 10.1017/S0013091500028935
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
Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. MR 1037412
Cornelius Greither, Construction of a Hopf algebra of $\textbf {Z}$-rank $n^{\phi (n)+1}$ whose antipode has order $2n$, Comm. Algebra 17 (1989), no. 5, 1147–1155. MR 993393, DOI 10.1080/00927878908823778
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
I. M. Musson, Links between cofinite prime ideals in quantum function algebras, Preprint (1995).
Masatoshi Noumi, Hirofumi Yamada, and Katsuhisa Mimachi, Finite-dimensional representations of the quantum group $\textrm {GL}_q(n;\textbf {C})$ and the zonal spherical functions on $\textrm {U}_q(n-1)\backslash \textrm {U}_q(n)$, Japan. J. Math. (N.S.) 19 (1993), no. 1, 31–80. MR 1231510, DOI 10.4099/math1924.19.31
Brian Parshall and Jian Pan Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439, vi+157. MR 1048073, DOI 10.1090/memo/0439
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
M. A. Semenov-Tyan-Shanskiĭ, What a classical $r$-matrix is, Funktsional. Anal. i Prilozhen. 17 (1983), no. 4, 17–33 (Russian). MR 725413
S. P. Smith, Quantum groups: an introduction and survey for ring theorists, Noncommutative rings (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 24, Springer, New York, 1992, pp. 131–178. MR 1230220, DOI 10.1007/978-1-4613-9736-6_{6}
J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
R. B. Warfield, Jr., Review of “Localization in Noetherian Rings” by A. V. Jategaonkar, Bull. Amer. Math. Soc. 17 (1987), 396-400.