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

\( S_n(x_1,\ldots,x_n) = \sum_{\sigma\in S_{n}}\text{sgn}(\sigma)x_{\sigma 1}\cdots x_{\sigma n} \ \)

in non-commutative polynomials x₁,...,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 by n matrices A₁,...,A_2n then

\( S_{2n}(A_1,\ldots,A_{2n}) = 0 \ . \)

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.
References

Amitsur, A. S.; Levitzki, Jakob (1950), "Minimal identities for algebras" (PDF), Proceedings of the American Mathematical Society 1: 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: 320–327, ISSN 0002-9939, JSTOR 2032509, MR ?
Formanek, E. (2001), "Amitsur–Levitzki theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Formanek, Edward (1991), The polynomial identities and invariants of n×n matrices, Regional Conference Series in Mathematics 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: 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
Swan, Richard G. (1963), "An application of graph theory to algebra" (PDF), Proceedings of the American Mathematical Society 14: 367–373, 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: 379–380, doi:10.2307/2037008, ISSN 0002-9939, JSTOR 2037008, MR 0255439

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