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Transactions of the American Mathematical Society

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A class of algebras similar to the enveloping algebra of $ {\rm sl}(2)$


Author: S. P. Smith
Journal: Trans. Amer. Math. Soc. 322 (1990), 285-314
MSC: Primary 17B35; Secondary 16S30
DOI: https://doi.org/10.1090/S0002-9947-1990-0972706-5
MathSciNet review: 972706
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Abstract: Fix $ f \in {\mathbf{C}}[X]$. Define $ R = {\mathbf{C}}[A,B,H]$ subject to the relations

$\displaystyle HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H)$

. We study these algebras (for different $ f$) and in particular show how they are similar to (and different from) $ U({\text{sl}}(2))$, the enveloping algebra of $ {\text{sl}}(2,{\mathbf{C}})$. There is a notion of highest weight modules and a category $ \mathcal{O}$ for such $ R$. For each $ n > 0$, if $ f(x) = {(x + 1)^{n + 1}} - {x^{n + 1}}$, then $ R$ has precisely $ n$ simple modules in each finite dimension, and every finite-dimensional $ R$-module is semisimple.

References [Enhancements On Off] (What's this?)

  • [AHV] J. Alev, T. J. Hodges, and J. D. Velez, Fixed rings of the Weyl algebra $ {A_1}({\mathbf{C}})$, J. Algebra (to appear).
  • [AS] M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), 171-216. MR 917738 (88k:16003)
  • [BS] A. Bell and S. P. Smith, Some $ 3$-dimensional skew polynomial rings, in preparation.
  • [B] G. M. Bergman, The Diamond lemma for ring theory, Adv. in Math. 29 (1978), 178-218. MR 506890 (81b:16001)
  • [BGG] J. Bernstein, I. M. Gelfand, and S. I. Gelfand, A category of $ \mathfrak{g}$-modules, Functional Anal. Appl. 10 (1976), 87-92. MR 0407097 (53:10880)
  • [C] P. M. Cohn, Algebra, Vol. 2, Wiley, New York, 1977. MR 0530404 (58:26625)
  • [D] J. Dixmier, Enveloping algebras, North-Holland, Amsterdam, 1977. MR 0498740 (58:16803b)
  • [H1] T. J. Hodges, Letter, November 1987.
  • [H2] -, Letter, August 1988.
  • [HS] T. J. Hodges and S. P. Smith, Sheaves of non-commutative algebras and the Beilinson-Bernstein equivalence of categories, Proc. Amer. Math. Soc. 93 (1985), 379-386. MR 773985 (86j:17015)
  • [I] R. S. Irving, BGG algebras and the BGG reciprocity principle, J. Algebra (to appear). MR 1080852 (92d:17011)
  • [J] A. Joseph, Rings which are modules in the Bernstein-Gelfand-Gelfand $ \mathcal{O}$-category, J. Algebra 113 (1988), 110-126. MR 928058 (89e:17012)
  • [KL] G. Krause and T. H. Lenagan, Gelfand-Kirillov dimension and growth of algebras, Pitman, London, 1984.
  • [L.] S. Lang, Diophantine geometry, Interscience, New York, 1962. MR 0142550 (26:119)
  • [LSS] T. Levasseur, S. P. Smith, and J. T. Stafford, The minimal nilpotent orbit, the Joseph ideal and differential operators, J. Algebra 116 (1988), 480-501. MR 953165 (89k:17028)
  • [LS] T. Levasseur and J. T. Stafford, Differential operators on classical rings of invariants, Mem. Amer. Math. Soc. No. 412 (1989). MR 988083 (90i:17018)
  • [MR] J. C. McConnell and J. C. Robson, Gelfand-Kirillov dimension, Hilbert-Samuel polynomials, and rings of differential operators, preprint, Univ. of Leeds, 1987. MR 1048411 (91c:16021)
  • [MS] J. C. McConnell and J. T. Stafford, Gelfand-Kirillov dimension, and associated graded modules, J. Algebra 125 (1989), 197-214. MR 1012671 (90i:16002)
  • [SW] L. Small and R. Warfield, Prime affine algebras of Gelfand-Kirillov dimension 1, J. Algebra 91 (1984), 386-389. MR 769581 (86h:16006)
  • [Sm] S. P. Smith, Overrings of primitive factor rings of $ U({\text{sl}}(2,{\mathbf{C}}))$, J. Pure Appl. Algebra (to appear). MR 1043751 (91k:17008)
  • [St] J. T. Stafford, Homological properties of $ U({\text{sl}}(2))$, Proc. Cambridge Philos. Soc. 91 (1982), 29-37. MR 633253 (83h:17013)

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DOI: https://doi.org/10.1090/S0002-9947-1990-0972706-5
Article copyright: © Copyright 1990 American Mathematical Society

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