A class of algebras similar to the enveloping algebra of 𝑠𝑙(2)

Smith, S. P.

doi:10.1090/S0002-9947-1990-0972706-5

A class of algebras similar to the enveloping algebra of $\textrm {sl}(2)$
HTML articles powered by AMS MathViewer

by S. P. Smith PDF

Trans. Amer. Math. Soc. 322 (1990), 285-314 Request permission

Abstract:

Fix $f \in {\mathbf {C}}[X]$. Define $R = {\mathbf {C}}[A,B,H]$ subject to the relations \[ HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H)\]. 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

Fixed rings of the Weyl algebra

Michael Artin and William F. Schelter, Graded algebras of global dimension $3$, Adv. in Math. 66 (1987), no. 2, 171–216. MR 917738, DOI 10.1016/0001-8708(87)90034-X

Some

-dimensional skew polynomial rings

George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218. MR 506890, DOI 10.1016/0001-8708(78)90010-5
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak {g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
Paul Moritz Cohn, Algebra. Vol. 2, John Wiley & Sons, London-New York-Sydney, 1977. With errata to Vol. I. MR 0530404
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
T. J. Hodges and S. P. Smith, Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories, Proc. Amer. Math. Soc. 93 (1985), no. 3, 379–386. MR 773985, DOI 10.1090/S0002-9939-1985-0773985-3
Ronald S. Irving, BGG algebras and the BGG reciprocity principle, J. Algebra 135 (1990), no. 2, 363–380. MR 1080852, DOI 10.1016/0021-8693(90)90294-X
Anthony Joseph, Rings which are modules in the Bernstein-Gel′fand-Gel′fand ${\scr O}$ category, J. Algebra 113 (1988), no. 1, 110–126. MR 928058, DOI 10.1016/0021-8693(88)90186-X

Gelfand-Kirillov dimension and growth of algebras

Serge Lang, Diophantine geometry, Interscience Tracts in Pure and Applied Mathematics, No. 11, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0142550
T. Levasseur, S. P. Smith, and J. T. Stafford, The minimal nilpotent orbit, the Joseph ideal, and differential operators, J. Algebra 116 (1988), no. 2, 480–501. MR 953165, DOI 10.1016/0021-8693(88)90231-1
T. Levasseur and J. T. Stafford, Rings of differential operators on classical rings of invariants, Mem. Amer. Math. Soc. 81 (1989), no. 412, vi+117. MR 988083, DOI 10.1090/memo/0412
J. C. McConnell and J. C. Robson, Gel′fand-Kirillov dimension, Hilbert-Samuel polynomials and rings of differential operators, Perspectives in ring theory (Antwerp, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 233–238. MR 1048411
J. C. McConnell and J. T. Stafford, Gel′fand-Kirillov dimension and associated graded modules, J. Algebra 125 (1989), no. 1, 197–214. MR 1012671, DOI 10.1016/0021-8693(89)90301-3
L. W. Small and R. B. Warfield Jr., Prime affine algebras of Gel′fand-Kirillov dimension one, J. Algebra 91 (1984), no. 2, 386–389. MR 769581, DOI 10.1016/0021-8693(84)90110-8
S. P. Smith, Overrings of primitive factor rings of $U(\textrm {sl}(2,\textbf {C}))$, J. Pure Appl. Algebra 63 (1990), no. 2, 207–218. MR 1043751, DOI 10.1016/0022-4049(90)90027-F
J. T. Stafford, Homological properties of the enveloping algebra $U(\textrm {Sl}_{2})$, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 29–37. MR 633253, DOI 10.1017/S0305004100059089