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Transactions of the American Mathematical Society

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Quantized enveloping algebras
for Borcherds superalgebras

Authors: Georgia Benkart, Seok-Jin Kang and Duncan Melville
Journal: Trans. Amer. Math. Soc. 350 (1998), 3297-3319
MSC (1991): Primary 17B37, 17B65, 17B67, 81R50
MathSciNet review: 1451594
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Abstract: We construct quantum deformations of enveloping algebras of Borcherds superalgebras, their Verma modules, and their irreducible highest weight modules.

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  • [BMPZ] Bahturin, Y.A., Mikhalev, A.A., Petrogradsky, V.M., Zaicev, M.V., Infinite Dimensional Lie Superalgebras, de Gruyter, Berlin, 1992. MR 94b:17001
  • [B1] Borcherds, R.E., Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501-512. MR 89g:17004
  • [B2] Borcherds, R.E., The monster Lie algebra, Adv. Math. 83 (1990), 30-47. MR 90k:17027
  • [B3] Borcherds, R.E., Central extensions of generalized Kac-Moody algebras, J. Algebra 140 (1991), 330-335. MR 92g:17031
  • [B4] Borcherds, R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444. MR 94f:11030
  • [B5] Borcherds, R.E., A characterization of generalized Kac-Moody algebras, J. Algebra 174 (1995), 1073-1079. MR 96e:17058
  • [CP] Chari, V., Pressley, A., A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. MR 95j:17010; MR 96h:17014
  • [D] Drinfel'd, V.G., Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254-258. MR 87h:58080
  • [FLV] Floreanini, R., Leites, D.A., Vinet, L., On the defining relations of quantum superalgebras, Lett. Math. Phys. 23 (1991), 127-131. MR 92m:17010
  • [G] Gebert, R.W., Introduction to vertex algebras, Borcherds algebras and the monster Lie algebra, Int. J. Mod. Phys. A 8 (1993), 5441-5503. MR 95a:17037
  • [H] Hungerford, T., Algebra, 5th ed., Springer-Verlag, New York, 1989. MR 82a:00006 (earlier ed.)
  • [Ja] Jantzen, J.C., Lectures on Quantum Groups, Amer. Math. Soc. Graduate Studies in Math., vol. 6, Providence, 1996. MR 96m:17029
  • [Ji] Jimbo, M., A $q$-difference analogue of $U({\mathfrak g})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. MR 86k:17008
  • [Kc1] Kac, V.G., Lie superalgebras, Adv. Math. 26 (1977), 8-96. MR 58:5803
  • [Kc2] Kac, V.G., Infinite-dimensional algebras, Dedekind's $\eta $-function, classical Möbius function and the very strange formula, Adv. Math. 30 (1978), 85-136; 35 (1980), 264-273. MR 83a:17014a,b
  • [Kc3] Kac, V.G., Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 92k:17038
  • [KW1] Kac, V.G., Wakimoto, M., Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), 4956-4960. MR 89j:17019
  • [KW2] Kac, V.G., Wakimoto, M., Integrable highest weight modules over affine superalgebras and number theory, Lie Theory and Geometry, Progress in Math., 123, Birkhäuser, Boston, 1994, pp. 415-456. MR 96j:11056
  • [Kn] Kang, S.-J., Quantum deformations of generalized Kac-Moody algebras and their modules, J. Algebra 175 (1995), 1041-1066. MR 96k:17023
  • [Ks] Kassel, C., Quantum Groups, Springer-Verlag, New York, 1995. MR 96e:17041
  • [KT] Khoroshkin, S.M., Tolstoy, V.N., Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), 599-617. MR 93a:16031
  • [L1] Lusztig, G., Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237-249. MR 89k:17029
  • [L2] Lusztig, G., Introduction to Quantum Groups, Birkhäuser, Boston, 1993. MR 94m:17016
  • [M] Miyamoto, M., A generalization of Borcherds algebra and denominator formula, J. Algebra 180 (1996), 631-651. MR 97a:17026
  • [O] Olshanski, G.I., Quantized universal enveloping superalgebra of type $Q$ and a super-extension of the Hecke algebra, Lett. in Math. Phys. 24 (1992), 93-102. MR 93i:17004
  • [Ra] Ray, U., A character formula for generalized Kac-Moody superalgebras, J. Algebra 177 (1995), 154-163. MR 97f:17008
  • [Ro] Rosso, M., Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), 581-593. MR 90k:17019
  • [S1] Scheunert, M., Generalized Lie algebras, J. Math. Phys. A 20 (1979), 712-720. MR 80f:17007
  • [S2] Scheunert, M., The presentation and $q$ deformation of special linear Lie superalgebras, J. Math. Phys. 34 (1993), 3780-3808. MR 94i:17007
  • [Y] Yamane, H., A Serre type theorem for affine Lie superalgebras and their quantized enveloping superalgebras, Proc. Japan Acad. Ser. A 70 (1994), 31-36. MR 95g:17021
  • [Z] Zou, Y.-M., Deformations of enveloping algebra of Lie superalgebra $sl(m,n)$, Comm. Math. Phys. 162 (1994), 467-479. MR 95h:17014

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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388

Seok-Jin Kang
Affiliation: Department of Mathematics, College of Natural Sciences, Seoul National University, Seoul 151-742, Korea

Duncan Melville
Affiliation: Department of Mathematics, St. Lawrence University, Canton, New York 13617

Received by editor(s): October 1, 1996
Additional Notes: The first author was supported in part by NSF Grant #DMS-9300523
The second author was supported in part by the Nondirected Research Fund, Korea Research Foundation, 1996
The third author was supported in part by a Faculty Research Grant from St. Lawrence University
Article copyright: © Copyright 1998 American Mathematical Society

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