Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-16T16:54:24.799Z Has data issue: false hasContentIssue false
  • English
  • Français

On Witt's dimension formula for free Lie algebras and a theorem of Klyachko

Published online by Cambridge University Press:  17 April 2009

D. Blessenohl  and
H. Laue
D. Blessenohl
Affiliation:
Mathematisches Seminar der UniversitätLudewig-Meyn-Str. 4D 2300 Kiel 1West Germany
H. Laue
Affiliation:
Dipartimento di MatematicaUniversità degli StudiVia Arnesano1–73100 Lecce, Italy.
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.

It is shown that Witt's basic dimension formula and a more recent result of Klyachko imply each other. Then Klyachko's identities between certain idempotents in the group ring of Sn are supplemented by identities involving Wever's classical idempotent. This leads to a direct proof of Klyachko's theorem (and hence Witt's formula), avoiding any commutator collecting process. Furthermore, this approach explains why the Witt dimensions are numbers which otherwise occur when “counting necklaces”.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 1 , August 1989 , pp. 49 - 57
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bahturin, Y.A., Identical Relations in Lie Algebras (VNU Science Press, Utrecht, 1987).Google Scholar
[2]Blessenohl, D. and Laue, H., ‘Generalised Jacobi identities’, Note Mat. (to appear).Google Scholar
[3]Hall, M., Combinatorial theory (Wiley-Interscience, New York, 1986).Google Scholar
[4]James, G.D. and Kerber, A., The representation theory of the symmetric group (Addison-Wesley, Reading, Mass., 1981).Google Scholar
[5]Klyachko, A.A., ‘Lie elements in the tensor algebra’, Siberian Math. J. 15 (1974), 914–920.Google Scholar
[6]Wever, F., ‘Über Invarianten in Lie'schen Ringen’, Math. Ann. 120 (1949), 563–580.Google Scholar