×

zbMATH — the first resource for mathematics

Down-up algebras. (English) Zbl 0922.17006
J. Algebra 209, No. 1, 305-344 (1998); addendum ibid. 213, No. 1, 378 (1999).
A down-up algebra \(A(\alpha, \beta, \gamma)\) is the associative unital algebra (say over \(\mathbf C\)) presented by generators \(d\) and \(u\) and defining relations \(d^2u=\alpha dud+\beta ud^2+\gamma d\), \(du^2=\alpha udu+\beta u^2d+\gamma u\); where \(\alpha,\beta,\gamma\) are arbitrary complex numbers. The authors motivation for this definition (and terminology) comes from the down and up operators on \((q,r)\)-differential posets [see R. P. Stanley, J. Am. Math. Soc. 4, 919-961 (1988; Zbl 0658.05006), S. V. Fomin, Zap. Nauchn. Semin. LOMI Steklov 155, 156-175 (1986; Zbl 0698.05003)]. An extensive list of examples of (homomorphic images of) down-up algebras is given; including \(q\)-Weyl algebras, quantum planes, universal enveloping algebras of certain 3-dimensional Lie (super-)algebras, Witten’s deformations of \(U(sl_2)\) [E. Witten, Commun. Math. Phys. 137, 29-66 (1991; Zbl 0717.53074)], conformal \(sl_2\)-algebras [L. Le Bruyn, Commun. Algebra 23, 1325-1362 (1995; Zbl 0824.17012)] and certain subalgebras of quantum enveloping algebras.
In the paper under review, the structure and representation theory of down-up algebras is studied. A PBW theorem is proved. “Highest weight” and “lowest weight” modules are explicitly constructed; the simple ones are determined and the exact expressions for all the weights are calculated. Two categories \(\mathcal O\) and \(\mathcal O'\), in the spirit of the Bernstein-Gelfand-Gelfand categories for representations of simple Lie algebras, are investigated.

MSC:
17B05 Structure theory for Lie algebras and superalgebras
06A06 Partial orders, general
17B35 Universal enveloping (super)algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] G. Benkart, Down – up algebras and Witten’s deformations of the universal enveloping algebra of \(s\)\(l\)_2, Contemp. Math, Amer. Math. Soc. · Zbl 0922.17007
[2] Bergman, G.M., The diamond lemma for ring theory, Adv. math., 29, 178-218, (1978) · Zbl 0326.16019
[3] Bernstein, J.; Gelfand, I.M.; Gelfand, S.I., A category of \(g\)-modules, Funct. anal. appl., 10, 87-92, (1976)
[4] Brualdi, R.A., Introduction combinatorics, (1992), North-Holland New York
[5] Fomin, S.V., The generalized Robinson Schensted Knuth correspondence, Zap. nauchn. sem. leningrad otdel mat. inst. Steklov. LOMI, 155, 156-175, (1986) · Zbl 0661.05004
[6] Fomin, S.V., Duality of graded graphs, J. algebraic combin., 3, 357-404, (1994) · Zbl 0810.05005
[7] Hodges, T.J., Noncommutative deformations of type-A, J. algebra, 161, 271-290, (1993) · Zbl 0807.16029
[8] Jacobson, N., Lie algebras, (1962), Wiley-Interscience New York · Zbl 0121.27504
[9] Jordan, D.A., Finite-dimensional simple modules over certain iterated skew polynomial rings, J. pure appl. algebra, 98, 45-55, (1995) · Zbl 0829.16017
[10] E. Kirkman, I. Musson, D. Passman
[11] R. Kulkarni, Irreducible representations of Witten’s deformations ofUsl2, J. Algebra · Zbl 0920.17008
[12] R. Kulkarni, Down – up algebras and their representations
[13] Le Bruyn, L., Two remarks on Witten’s quantum enveloping algebra, Comm. algebra, 22, 865-876, (1994) · Zbl 0832.17012
[14] Le Bruyn, L., Conformal \(s\)\(l\)_2, Comm. algebra, 23, 1325-1362, (1995) · Zbl 0824.17012
[15] Le Bruyn, L.; Smith, S.P., Homogenized \(s\)\(l\)_2, Proc. amer. math. soc., 118, 725-730, (1993) · Zbl 0795.16029
[16] Manin, Yu., Some remarks on Koszul algebras and quantum groups, Ann. inst. Fourier (Grenoble), 37, 191-205, (1987) · Zbl 0625.58040
[17] Rosenberg, A.L., Noncommutative algebraic geometry and representations of quantized algebras, (1995), Kluwer Academic Dordrecht · Zbl 0839.16002
[18] Smith, S.P., A class of algebras similar to the enveloping algebra ofsl, Trans. amer. math. soc., 322, 285-314, (1990) · Zbl 0732.16019
[19] Stanley, R.P., Differential posets, J. amer. math. soc., 4, 919-961, (1988) · Zbl 0658.05006
[20] Stanley, R.P., Variations on differential posets, (), 145-165
[21] Terwilliger, P., The incidence algebra of a uniform poset, (), 193-212 · Zbl 0737.05032
[22] Witten, E., Gauge theories, vertex models, and quantum groups, Nuclear phys. B, 330, 285-346, (1990)
[23] Witten, E., Quantization of chern – simons gauge theory with complex gauge group, Comm. math. phys., 137, 29-66, (1991) · Zbl 0717.53074
[24] Woronowicz, S.L., TwistedSU, Publ. res. inst. math. sci., 23, 117-181, (1987) · Zbl 0676.46050
[25] K. Zhao, Centers of down – up algebras, J. Algebra · Zbl 0932.17007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.