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Quantum deformations of imaginary Verma modules

Published online by Cambridge University Press:  01 January 1997

B Cox ,
V Futorny ,
S-J Kang  and
D Melville
B Cox
Affiliation:
Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812-1032, USA. Email: ma_blc@selway.umt.edu
V Futorny
Affiliation:
Department of Mathematics, Kiev University, Kiev 252617, Ukraine. Email: foutorny@qucdn.queensu.ca
S-J Kang
Affiliation:
Department of Mathematics, College of Natural Sciences, Seoul National University, Seoul 151-742, Republic of Korea. Email sjkang@math.snu.ac.kr
D Melville
Affiliation:
Department of Mathematics, St Lawrence University, Canton, NY 13617, USA. Email: dmel@music.stlawu.edu
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Abstract

We construct quantum deformations of imaginary Verma modules over $U(A_1^{(1)})$ and show that, for generic $q$, imaginary Verma modules over $U(A_1^{(1)})$ can be deformed to those over the quantum group $U_q(A_1^{(1)})$ in such a way that the dimensions of the weight spaces are invariant under the deformation. We also prove the PBW theorem for $U_q(A_1^{(1)})$ with respect to the triangular decomposition induced from the root partition corresponding to the imaginary Verma modules.

1991 Mathematics Subject Classification: 17B67, 17B65, 17B10.

Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 74 , Issue 1 , January 1997 , pp. 52 - 80
Copyright
London Mathematical Society 1997

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