Amitsur–Levitzki theorem
Encyclopedia
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 . In particular matrix rings are PI rings such that the smallest identity they satisfy has degree exactly 2n.
where the sum is over all (2n)! elements of the symmetric group S2n. (This polynomial is called the standard polynomial of degree 2n.)
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 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.
gave a proof related to the Cayley–Hamilton theorem
.
gave a short proof using the exterior algebra of a vector space of dimension 2n.
Statement
If A1,...,A2n are n by n matrices thenwhere the sum is over all (2n)! elements of the symmetric group S2n. (This polynomial is called the standard polynomial of degree 2n.)
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 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.
gave a proof related to the Cayley–Hamilton theorem
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
.
gave a short proof using the exterior algebra of a vector space of dimension 2n.