Amitsur–Levitzki Theorem

In algebra, the Amitsur–Levitzki theorem states that the algebra of ''n'' × ''n'' matrices over a commutative ring satisfies a certain identity of degree 2''n''. It was proved by . In particular matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2''n''.

Statement

The standard polynomial of degree ''n'' is
:$S_n(x_1,\dots,x_n) = \sum_\text(\sigma)x_ \cdots x_$
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''_2''n'' whose entries are taken from a commutative ring then

:$S_(A_1,\dots,A_) = 0.$

Proofs

gave the first proof.

deduced the Amitsur–Levitzki theorem from the Koszul–Samelson theorem about primitive cohomology of Lie algebras.

and 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.

gave a proof related to the Cayley–Hamilton theorem.

gave a short proof using the exterior algebra of a vector space of dimension 2''n''.

gave another proof, showing that the Amitsur–Levitzki theorem is the Cayley–Hamilton identity for the generic Grassman matrix.

References

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{{DEFAULTSORT:Amitsur-Levitzki theorem
Linear algebra
Theorems in algebra
Matrix theory