Amitsur–Levitzki theorem

In algebra, the Amitsur–Levitzki theorem states that the algebra of n by n matrices 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

${\displaystyle S_{n}(x_{1},\ldots ,x_{n})=\sum _{\sigma \in S_{n}}{\text{sgn}}(\sigma )x_{\sigma (1)}\cdots x_{\sigma (n)}\ }$

in non-commutative variables x1,...,xn, where the sum is taken over all n! elements of the symmetric group Sn.

The Amitsur–Levitzki theorem states that for n by n matrices A1,...,A2n then

${\displaystyle S_{2n}(A_{1},\ldots ,A_{2n})=0\ .}$

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.