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States that the algebra of n by n matrices satisfies a certain identity of degree 2n
In
algebra , the Amitsur–Levitzki theorem states that the algebra of n × n
matrices over a
commutative ring satisfies a certain identity of
degree 2n . It was
proved by
Amitsur and
Levitsky (
1950 ). In particular
matrix rings are
polynomial identity rings such that the smallest identity they satisfy has degree exactly 2n .
Statement
The standard polynomial of degree n is
S
n
(
x
1
,
…
,
x
n
)
=
∑
σ
∈
S
n
sgn
(
σ
)
x
σ
(
1
)
⋯
x
σ
(
n
)
{\displaystyle S_{n}(x_{1},\dots ,x_{n})=\sum _{\sigma \in S_{n}}{\text{sgn}}(\sigma )x_{\sigma (1)}\cdots x_{\sigma (n)}}
in non-commuting variables x 1 , ..., x n , where the sum is taken over all n ! elements of the
symmetric group S n .
The Amitsur–Levitzki theorem states that for n × n matrices A 1 , ..., A 2n whose entries are taken from a commutative ring then
S
2
n
(
A
1
,
…
,
A
2
n
)
=
0.
{\displaystyle S_{2n}(A_{1},\dots ,A_{2n})=0.}
Proofs
Amitsur and Levitzki (
1950 ) gave the first proof.
Kostant (1958) deduced the Amitsur–Levitzki theorem from the
Koszul–Samelson theorem about primitive cohomology of
Lie algebras .
Swan (1963) and
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 2n 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 2n edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2n , thus proving the Amitsur–Levitzki theorem.
Razmyslov (1974) gave a proof related to the
Cayley–Hamilton theorem .
Rosset (1976) gave a short proof using the
exterior algebra of a
vector space of
dimension 2n .
Procesi (2015) gave another proof, showing that the Amitsur–Levitzki theorem is the Cayley–Hamilton identity for the generic Grassman matrix.
References
Amitsur, A. S. ;
Levitzki, Jakob (1950),
"Minimal identities for algebras" (PDF) ,
Proceedings of the American Mathematical Society , 1 (4): 449–463,
doi :
10.1090/S0002-9939-1950-0036751-9 ,
ISSN
0002-9939 ,
JSTOR
2032312 ,
MR
0036751
Amitsur, A. S.; Levitzki, Jakob (1951),
"Remarks on Minimal identities for algebras" (PDF) ,
Proceedings of the American Mathematical Society , 2 (2): 320–327,
doi :
10.2307/2032509 ,
ISSN
0002-9939 ,
JSTOR
2032509
Formanek, E. (2001) [1994],
"Amitsur–Levitzki theorem" ,
Encyclopedia of Mathematics ,
EMS Press
Formanek, Edward (1991), The polynomial identities and invariants of n× n matrices , Regional Conference Series in Mathematics, vol. 78, Providence, RI:
American Mathematical Society ,
ISBN
0-8218-0730-7 ,
Zbl
0714.16001
Kostant, Bertram (1958), "A theorem of Frobenius, a theorem of Amitsur–Levitski and cohomology theory", J. Math. Mech. , 7 (2): 237–264,
doi :
10.1512/iumj.1958.7.07019 ,
MR
0092755
Razmyslov, Ju. P. (1974), "Identities with trace in full matrix algebras over a field of characteristic zero", Mathematics of the USSR-Izvestiya , 8 (4): 727,
doi :
10.1070/IM1974v008n04ABEH002126 ,
ISSN
0373-2436 ,
MR
0506414
Rosset, Shmuel (1976), "A new proof of the Amitsur–Levitski identity",
Israel Journal of Mathematics , 23 (2): 187–188,
doi :
10.1007/BF02756797 ,
ISSN
0021-2172 ,
MR
0401804 ,
S2CID
121625182
Swan, Richard G. (1963),
"An application of graph theory to algebra" (PDF) ,
Proceedings of the American Mathematical Society , 14 (3): 367–373,
doi :
10.2307/2033801 ,
ISSN
0002-9939 ,
JSTOR
2033801 ,
MR
0149468
Swan, Richard G. (1969),
"Correction to "An application of graph theory to algebra" " (PDF) ,
Proceedings of the American Mathematical Society , 21 (2): 379–380,
doi :
10.2307/2037008 ,
ISSN
0002-9939 ,
JSTOR
2037008 ,
MR
0255439
Procesi, Claudio (2015), "On the theorem of Amitsur—Levitzki",
Israel Journal of Mathematics , 207 : 151–154,
arXiv :
1308.2421 ,
Bibcode :
2013arXiv1308.2421P ,
doi :
10.1007/s11856-014-1118-8