Hostname: page-component-84b7d79bbc-7nlkj Total loading time: 0 Render date: 2024-07-29T03:09:21.285Z Has data issue: false hasContentIssue false
  • English
  • Français

An Algorithm to Compute the Canonical Basis of an Irreducible Module Over a Quantized Enveloping Algebra

Published online by Cambridge University Press:  01 February 2010

Willem A. de Graaf
Willem A. de Graaf
Affiliation:
Mathematical Institute, University of Utrecht, The Netherlandsdegraaf@math.uu.nl, http://www.math.uu.nl/people/graaf/

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 105 - 118
Copyright
Copyright © London Mathematical Society 2003

References

1Bourbaki, N., Groupes et algèbres de Lie (Hermann, Paris, 1968) Chapters 4, 5 and 6.Google Scholar
2THE GAP GROUP, ‘GAP – Groups, algorithms, and programming, Version 4.3’, 2002; http://www.gap-system.org.Google Scholar
3De Graaf, W.A., ‘QuaGroup, a GAP share package’, 2001; http://www.math.uu.nl/people/graaf/quagroup.html.Google Scholar
4De Graaf, W. A., ‘Constructing canonical bases of quantized enveloping algebras’, Experiment. Math. 11 (2002) 161–170.CrossRefGoogle Scholar
5De Graaf, W.A., ‘Five constructions of representations of quantum groups’, preprint, 2002.Google Scholar
6Jantzen, J.C., Lectures on quantum groups, Grad. Stud, in Math. 6 (Amer. Math. Soc, Providence, RI, 1996).Google Scholar
7Kang, S. and Misra, K.C., ‘Crystal bases and tensor product decompositions of Uq(G2)-modules’, J. Algebra 163 (1994) 675–691.CrossRefGoogle Scholar
8Kashiwara, M., ‘Similarity of crystal bases’, Lie algebras and their representations (Seoul, 1995) (ed. Kang, S., Kim, M. and Lee, I., Amer. Math. Soc, Providence, RI, 1996) 177–186.CrossRefGoogle Scholar
9Kashiwara, M. and Nakashima, T., ‘Crystal graphs for representations of the q-analogue of classical Lie algebras’, J. Algebra 165 (1994) 295–345.CrossRefGoogle Scholar
10Lakshmibai, V., ‘Bases for quantum Demazure modules’, Representations of groups (Banff, AB, 1994) (ed. Allison, B. and Cliff, G., Amer. Math. Soc., Providence, RI, 1995) 199–216.Google Scholar
11Leclerc, B. and Toffin, P., ‘A simple algorithm for computing the global crystal basis of an irreducible Uq(s1n)-module’, Internat. J. Algebra Comput. 10 (2000) 191–208.CrossRefGoogle Scholar
12Lecouvey, C., ‘An algorithm for computing the global basis of an irreducible -module’, Adv. in Appl. Math. 29 (2002) 46–78.CrossRefGoogle Scholar
13Littelmann, P., ‘A Littlewood–Richardson rule for symmetrizable Kac–Moody algebras’, Invent. Math. 116 (1994) 329–346.CrossRefGoogle Scholar
14Littelmann, P., ‘Paths and root operators in representation theory’, Ann. of Math. (2) 142 (1995) 499–525.CrossRefGoogle Scholar
15Littlemann, P., ‘Cones, crystals, and patterns’, Transform. Groups 3 (1998) 145–179.CrossRefGoogle Scholar
16Lusztig, G., ‘Canonical bases arising from quantized enveloping algebras II’, Common trends in mathematics and quantum field theories (Kyoto, 1990), Progr. Theoret. Phys. Suppl. 102 (1991) 75–201.Google Scholar
17Lusztig, G., Introduction to quantum groups (Birkhäuser Boston Inc., Boston, MA, 1993).Google Scholar