Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T09:32:02.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Canonical bases and standard monomials

Published online by Cambridge University Press:  20 January 2009

B. R. Marsh
Rights & Permissions [Opens in a new window]

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.

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 3 , October 1998 , pp. 611 - 623
Copyright
Copyright © Edinburgh Mathematical Society 1998

Footnotes

Last updated 14 April 2022

References

REFERENCES

1. Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4, 5 et 6 (Masson, Paris, 1981).Google Scholar
2. Chari, V. and Pressley, A., A Guide to Quantum Groups (C.U.P., Cambridge, 1995).Google Scholar
3. Drinfel'd, V. G., Hopf algebras and the Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258.Google Scholar
4. Jacobson, N., Lie Algebras (Dover Publications, New York, 1979).Google Scholar
5. Jimbo, M., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.CrossRefGoogle Scholar
6. Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63(2) (1991), 465–516.CrossRefGoogle Scholar
7. Lakshmibai, V., Bases for quantum Demazure modules, in Representations of groups, Banff, AB, 1994 (CMS Conf. Proc., 16, A.M.S., 1995), 199–216.Google Scholar
8. Lakshmibai, V., Bases for quantum Demazure modules I, in Geometry and Analysis, Bombay,1992 (Tata Inst. Fund. Res., Bombay, 1995), 197–224.Google Scholar
9. Lakshmibai, V., Bases for quantum Demazure modules II, Proc. Sympos. Pure Math. 56(2) (1994), 149–168.CrossRefGoogle Scholar
10. Lakshmibai, V. and Reshetikhin, N., Quantum deformations of flag and Schubert schemes. C. R. Acad. Sci. Paris, Sér. I Math. 313 (1991), 121–126.Google Scholar
11. Lakshmibai, V. and Reshetikhin, N., Quantum flag and Schubert schemes, in Deformation theory and quantum groups with applications to mathematical physics (A.M.S. Contemporary Mathematics 134, 1992), 145–181.CrossRefGoogle Scholar
12. Lakshmibai, V. and Seshadri, C. S., Geometry of G/P – V. J. Algebra 100 (1986), 462–557.CrossRefGoogle Scholar
13. Lakshmibai, V. and Seshadri, C. S., Standard monomial theory, in Proceedings of the Hyderabad Conference on Algebraic Groups (Manoj Prakashan, Madras, 1991), 279–323.Google Scholar
14. Littelmann, P., An algorithm to compute bases and representation matrices for SLn+1-representations, preprint, 1997.CrossRefGoogle Scholar
15. Littelmann, P., Cones, crystals, and patterns, preprint, 1996.Google Scholar
16. Lusztig, G., Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237–249.CrossRefGoogle Scholar
17. Lusztig, G., Quantum groups at roots of 1. Geom. Dedicata 35 (1990), 89–114.CrossRefGoogle Scholar
18. Lusztig, G., Introduction to Quantum Groups (Birkhäuser, Boston, 1993).Google Scholar
19. Xi, N., Root vectors in quantum groups, Comment. Math. Helv. 69(4) (1994), 612–639.CrossRefGoogle Scholar