Fibers of characters in Gelfand-Tsetlin categories

Futorny, Vyacheslav; Ovsienko, Serge

doi:10.1090/S0002-9947-2014-05938-2

Fibers of characters in Gelfand-Tsetlin categories
HTML articles powered by AMS MathViewer

by Vyacheslav Futorny and Serge Ovsienko PDF

Trans. Amer. Math. Soc. 366 (2014), 4173-4208 Request permission

Abstract:

For a class of noncommutative rings, called Galois orders, we study the problem of an extension of characters from a commutative subalgebra. We show that for Galois orders this problem is always solvable in the sense that all characters can be extended, moreover, in finitely many ways, up to isomorphism. These results can be viewed as a noncommutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the representation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras (in particular $\mathrm {gl}_n$), Yangians and finite $W$-algebras. As an example we recover the theory of Gelfand-Tsetlin modules for $\mathrm {gl}_n$.

References

Jiří Adámek, Horst Herrlich, and George E. Strecker, Abstract and concrete categories, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. The joy of cats; A Wiley-Interscience Publication. MR 1051419
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4 (1992), no. 1, 75–97 (Russian); English transl., St. Petersburg Math. J. 4 (1993), no. 1, 71–92. MR 1171955
V. Bavula and F. van Oystaeyen, Simple modules of the Witten-Woronowicz algebra, J. Algebra 271 (2004), no. 2, 827–845. MR 2025552, DOI 10.1016/j.jalgebra.2003.06.013
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Yu. A. Drozd, Representations of Lie algebras ${\mathfrak {s}}{\mathfrak {l}}(2)$, Vīsnik Kiïv. Unīv. Ser. Mat. Mekh. 25 (1983), 70–77 (Ukrainian, with Russian summary). MR 746766
Yurij A. Drozd and Vladimir V. Kirichenko, Finite-dimensional algebras, Springer-Verlag, Berlin, 1994. Translated from the 1980 Russian original and with an appendix by Vlastimil Dlab. MR 1284468, DOI 10.1007/978-3-642-76244-4
Yu. A. Drozd, S. A. Ovsienko, and V. M. Futorny, On Gel′fand-Zetlin modules, Proceedings of the Winter School on Geometry and Physics (Srní, 1990), 1991, pp. 143–147. MR 1151899
Yu. A. Drozd, V. M. Futorny, and S. A. Ovsienko, Harish-Chandra subalgebras and Gel′fand-Zetlin modules, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 79–93. MR 1308982, DOI 10.1007/978-94-017-1556-0_{5}
S. L. Fernando, Lie algebra modules with finite-dimensional weight spaces. I, Trans. Amer. Math. Soc. 322 (1990), no. 2, 757–781. MR 1013330, DOI 10.1090/S0002-9947-1990-1013330-8
A. S. Miščenko and A. T. Fomenko, Euler equation on finite-dimensional Lie groups, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 2, 396–415, 471 (Russian). MR 0482832
Vyacheslav Futorny, Alexander Molev, and Serge Ovsienko, Harish-Chandra modules for Yangians, Represent. Theory 9 (2005), 426–454. MR 2142818, DOI 10.1090/S1088-4165-05-00195-0
Vyacheslav Futorny, Alexander Molev, and Serge Ovsienko, The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite $W$-algebras, Adv. Math. 223 (2010), no. 3, 773–796. MR 2565549, DOI 10.1016/j.aim.2009.08.018
Vyacheslav Futorny and Serge Ovsienko, Galois orders in skew monoid rings, J. Algebra 324 (2010), no. 4, 598–630. MR 2651560, DOI 10.1016/j.jalgebra.2010.05.006
Vyacheslav Futorny and Serge Ovsienko, Kostant’s theorem for special filtered algebras, Bull. London Math. Soc. 37 (2005), no. 2, 187–199. MR 2119018, DOI 10.1112/S0024609304003844
Vyacheslav Futorny and Serge Ovsienko, Harish-Chandra categories, Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 271–286. MR 2500363
Peter Gabriel, Finite representation type is open, Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974) Carleton Math. Lecture Notes, No. 9, Carleton Univ., Ottawa, Ont., 1974, pp. 23. MR 0376769
P. Gabriel and A. V. Roĭter, Representations of finite-dimensional algebras, Algebra, VIII, Encyclopaedia Math. Sci., vol. 73, Springer, Berlin, 1992, pp. 1–177. With a chapter by B. Keller. MR 1239447
I. M. Gel′fand and M. L. Cetlin, Finite-dimensional representations of groups of orthogonal matrices, Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 1017–1020 (Russian). MR 0034763
M. I. Graev, Infinite-dimensional representations of the Lie algebra $\mathfrak {gl}(n,{\Bbb C})$ related to complex analogs of the Gelfand-Tsetlin patterns and general hypergeometric functions on the Lie group $\textrm {GL}(n,{\Bbb C})$, Acta Appl. Math. 81 (2004), no. 1-3, 93–120. MR 2069333, DOI 10.1023/B:ACAP.0000024196.16709.f6
M. I. Graev, A continuous analogue of Gel′fand-Tsetlin schemes and a realization of the principal series of irreducible unitary representations of the group $\textrm {GL}(n,\Bbb C)$ in the space of functions on the manifold of these schemes, Dokl. Akad. Nauk 412 (2007), no. 2, 154–158 (Russian); English transl., Dokl. Math. 75 (2007), no. 1, 31–35. MR 2451955, DOI 10.1134/S1064562407010103
O. Khomenko, Some applications of Gelfand-Zetlin modules, Representations of algebras and related topics, Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 205–213. MR 2146251
Bertram Kostant and Nolan Wallach, Gelfand-Zeitlin theory from the perspective of classical mechanics. I, Studies in Lie theory, Progr. Math., vol. 243, Birkhäuser Boston, Boston, MA, 2006, pp. 319–364. MR 2214253, DOI 10.1007/0-8176-4478-4_{1}2
Bertram Kostant and Nolan Wallach, Gelfand-Zeitlin theory from the perspective of classical mechanics. II, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 387–420. MR 2181811, DOI 10.1007/0-8176-4467-9_{1}0
Olivier Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 537–592 (English, with English and French summaries). MR 1775361
H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1997.
Volodymyr Mazorchuk, Tableaux realization of generalized Verma modules, Canad. J. Math. 50 (1998), no. 4, 816–828. MR 1638623, DOI 10.4153/CJM-1998-043-x
Volodymyr Mazorchuk, Quantum deformation and tableaux realization of simple dense ${\mathfrak {gl}}(n,\Bbb C)$-modules, J. Algebra Appl. 2 (2003), no. 1, 1–20. MR 1964762, DOI 10.1142/S0219498803000325
Volodymyr Mazorchuk, Orthogonal Gelfand-Zetlin algebras. I, Beiträge Algebra Geom. 40 (1999), no. 2, 399–415. MR 1720114
V. S. Mazorchuk and S. A. Ovsienko, Submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules, Algebr. Represent. Theory 1 (1998), no. 1, 3–26. MR 1654598, DOI 10.1023/A:1009929615175
Volodymyr Mazorchuk and Catharina Stroppel, Cuspidal $\mathfrak {sl}_n$-modules and deformations of certain Brauer tree algebras, Adv. Math. 228 (2011), no. 2, 1008–1042. MR 2822216, DOI 10.1016/j.aim.2011.06.005
A. I. Molev, Gelfand-Tsetlin bases for classical Lie algebras, Handbook of algebra. Vol. 4, Handb. Algebr., vol. 4, Elsevier/North-Holland, Amsterdam, 2006, pp. 109–170. MR 2523420, DOI 10.1016/S1570-7954(06)80006-9
Andrei Okounkov and Anatoly Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 2 (1996), no. 4, 581–605. MR 1443185, DOI 10.1007/PL00001384
Serge Ovsienko, Finiteness statements for Gelfand-Zetlin modules, Third International Algebraic Conference in the Ukraine (Ukrainian), Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002, pp. 323–338. MR 2210503
Serge Ovsienko, Strongly nilpotent matrices and Gelfand-Zetlin modules, Linear Algebra Appl. 365 (2003), 349–367. Special issue on linear algebra methods in representation theory. MR 1987348, DOI 10.1016/S0024-3795(02)00675-4
Jean-Pierre Serre, Local algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author. MR 1771925, DOI 10.1007/978-3-662-04203-8
È. B. Vinberg, Some commutative subalgebras of a universal enveloping algebra, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 3–25, 221 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 1–22. MR 1044045, DOI 10.1070/IM1991v036n01ABEH001925
D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0473097