Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T06:47:52.898Z Has data issue: false hasContentIssue false
  • English
  • Français

ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  01 February 2009

JONATHAN BROWN  and
JONATHAN BRUNDAN
JONATHAN BROWN
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA (email: [email protected])
JONATHAN BRUNDAN*
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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.

We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

Keywords

nilpotent matricescentralizerssymmetric invariants

MSC classification

Secondary: 17B35: Universal enveloping (super)algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 1 - 15
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

Research supported in part by NSF grant no. DMS-0139019.

References

[1]Brundan, J. and Kleshchev, A., ‘Parabolic presentations of the Yangian ’, Comm. Math. Phys. 254 (2005), 191–220.CrossRefGoogle Scholar
[2]Brundan, J. and Kleshchev, A., ‘Shifted Yangians and finite W-algebras’, Adv. Math. 200 (2006), 136–195.CrossRefGoogle Scholar
[3]Brundan, J. and Kleshchev, A., ‘Representations of shifted Yangians and finite W-algebras’, Mem. Amer. Math. Soc. 196 (2008), 1–107.Google Scholar
[4]Brundan, J. and Kleshchev, A., ‘Schur–Weyl duality for higher levels’, Selecta Math., to appear; math.RT/0605217.Google Scholar
[5]Dixmier, J., Enveloping Algebras, Graduate Studies in Math., 11 (American Mathematical Society, Providence, RI, 1996).Google Scholar
[6]Molev, A., ‘Casimir elements for certain polynomial current Lie algebras’, in: Physical Applications and Mathematical Aspects of Geometry, Groups and Algebras (eds. H.-D. Doebner, W. Scherer and P. Nattermann) (World Scientific, Singapore, 1997), pp. 172–176.Google Scholar
[7]Molev, A., Nazarov, M. and Olshanskii, G., ‘Yangians and classical Lie algebras’, Russian Math. Surveys 51 (1996), 205–282.CrossRefGoogle Scholar
[8]Panyushev, D., Premet, A. and Yakimova, O., ‘On symmetric invariants of centralisers in reductive Lie algebras’, J. Algebra 313 (2007), 343–391.CrossRefGoogle Scholar
[9]Premet, A., ‘Enveloping algebras of Slodowy slices and the Joseph ideal’, J. Eur. Math. Soc. 9 (2007), 487–543.CrossRefGoogle Scholar
[10]Rais, M. and Tauvel, P., ‘Indice et polynômes invariants pour certaines algèbres de Lie’, J. Reine Angew. Math. 425 (1992), 123–140.Google Scholar
[11]Springer, T., Linear Algebraic Groups, 2nd edn (Birkhäuser, Basel, 1998).CrossRefGoogle Scholar