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 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

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

in non-commuting variables 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 × n matrices A₁, ..., A_2n whose entries are taken from a commutative ring then

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

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