Fictitious domain method - Revision history

en>Bgwhite: Do general fixes and cleanup. - using AWB (9838)

2014-01-10T22:17:34Z

Do general fixes and cleanup. - using AWB (9838)

← Older revision		Revision as of 00:17, 11 January 2014
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I enjoyed to surf and required all the most recent fancy dress costumes.		In algebra, the '''Amitsur–Levitzki theorem''' states that the algebra of ''n'' by ''n'' matrices satisfies a certain identity of degree 2''n''. It was proved by {{harvs\|txt\|last=Amitsur\|author1-link=Shimshon Amitsur\|last2=Levitsky\|author2-link=Jacob Levitzki\|year=1950}}. In particular matrix rings are [[PI ring]]s such that the smallest identity they satisfy has degree exactly 2''n''.

			==Statement==
			If ''A''<sub>1</sub>,...,''A''<sub>2''n''</sub> are ''n'' by ''n'' matrices then
			:<math>\sum_{\sigma\in S_{2n}}\text{sgn}(\sigma)A_{\sigma 1}\cdots A_{\sigma 2n}=0,</math>
			where the sum is over all (2''n'')! elements of the [[symmetric group]] ''S''<sub>2''n''</sub>. (This polynomial is called the standard polynomial of degree 2''n''.)

			==Proofs==
			{{harvs\|txt\|last=Amitsur\|last2=Levitzki\|year=1950}} gave the first proof.

			{{harvtxt\|Kostant\|1958}} deduced the Amitsur–Levitzki theorem from the [[Koszul–Samelson theorem]] about primitive cohomology of Lie algebras.

			{{harvtxt\|Swan\|1963}} and {{harvtxt\|Swan\|1969}} gave a simple combinatorial proof as follows. By linearity it is enough to prove the theorem when each matrix has only one nonzero entry, which is 1. In this case each matrix can be encoded as a directed edge of a graph with ''n'' vertices. So all matrices together give a graph on ''n'' vertices with 2''n'' directed edges. The identity holds provided that for any two vertices ''A'' and ''B'' of the graph, the number of odd Eulerian paths from ''A'' to ''B'' is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2''n'' edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2''n'', thus proving the Amitsur–Levitzki theorem.

			{{harvtxt\|Razmyslov\|1974}} gave a proof related to the [[Cayley–Hamilton theorem]].

			{{harvtxt\|Rosset\|1976}} gave a short proof using the exterior algebra of a vector space of dimension 2''n''.

			==References==

			*{{Citation\|author1-link=Shimshon Amitsur\|last1=Amitsur \| first1=A. S. \| last2=Levitzki \| first2=Jakob \| author2-link=Jacob Levitzki \| title=Minimal identities for algebras \| jstor=2032312 \| mr=0036751 \| year=1950 \| journal=[[Proceedings of the American Mathematical Society]] \| issn=0002-9939 \| volume=1 \| pages=449–463 \| url=http://www.ams.org/journals/proc/1950-001-04/S0002-9939-1950-0036751-9/S0002-9939-1950-0036751-9.pdf \| doi=10.1090/S0002-9939-1950-0036751-9}}
			*{{Citation \| last1=Amitsur \| first1=A. S. \| last2=Levitzki \| first2=Jakob\|title=Remarks on Minimal identities for algebras \| jstor=2032509 \| mr=? \| year=1951 \| journal=[[Proceedings of the American Mathematical Society]] \| issn=0002-9939 \| volume=2 \| pages=320–327 \| url=http://www.ams.org/journals/proc/1951-002-02/S0002-9939-1951-0040285-6/S0002-9939-1951-0040285-6.pdf}}
			*{{eom\|id=a/a110570\|title=Amitsur–Levitzki theorem \|first=E.\|last= Formanek}}
			*{{Citation\|author1-link=Bertram Kostant\|last1=Kostant \| first1=Bertram \| title=A theorem of Frobenius, a theorem of Amitsur–Levitski and cohomology theory \| doi=10.1512/iumj.1958.7.07019 \| mr=0092755 \| year=1958 \| journal=J. Math. Mech. \| volume=7 \| pages=237–264}}
			*{{Citation \| last1=Razmyslov \| first1=Ju. P. \| title=Identities with trace in full matrix algebras over a field of characteristic zero \| doi=10.1070/IM1974v008n04ABEH002126 \| mr=0506414 \| year=1974 \| journal= Mathematics of the USSR-Izvestiya \| issn=0373-2436 \| volume=8 \| issue=4 \| pages=727}}
			*{{Citation\| last1=Rosset \| first1=Shmuel \| title=A new proof of the Amitsur–Levitski identity \| doi=10.1007/BF02756797 \| mr=0401804 \| year=1976 \| journal=Israel Journal of Mathematics \| issn=0021-2172 \| volume=23 \| issue=2 \| pages=187–188}}
			*{{Citation\|author1-link=Richard Swan\|last1=Swan \| first1=Richard G. \| title=An application of graph theory to algebra \| jstor=2033801 \| mr=0149468 \| year=1963 \| journal=[[Proceedings of the American Mathematical Society]] \| issn=0002-9939 \| volume=14 \| pages=367–373 \| url=http://www.ams.org/journals/proc/1963-014-03/S0002-9939-1963-0149468-6/S0002-9939-1963-0149468-6.pdf}}
			*{{Citation \| last1=Swan \| first1=Richard G. \| title=Correction to "An application of graph theory to algebra" \| jstor=? \| mr=0255439 \| year=1969 \| journal=[[Proceedings of the American Mathematical Society]] \| issn=0002-9939 \| volume=21 \| pages=379–380 \| url=http://www.ams.org/journals/proc/1969-021-02/S0002-9939-1969-0255439-7/S0002-9939-1969-0255439-7.pdf}}

			{{DEFAULTSORT:Amitsur-Levitzki theorem}}
			[[Category:Linear algebra]]
			[[Category:Theorems in algebra]]
			[[Category:Matrix theory]]

en>Gaius Cornelius: Fix indefinite article, replaced: an method → a method using AWB

2011-11-04T06:21:09Z

Fix indefinite article, replaced: an method → a method using AWB

New page

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