Description

\[\operatorname{Tr}(XY-YX)=\operatorname{Tr}(XY)-\operatorname{Tr}(YX)=\operatorname{Tr}(XY)-\operatorname{Tr}(XY)=0.\]

We may now wonder, out of all matrices with trace zero, which ones are of the form \(XY-YX\), for some \(X\) and \(Y\)? This simple question has a surprisingly long history, culminating in problems which still await their solutions.

A matrix of the form \(XY-YX\) is called a commutator and is usually denoted by \([X,Y]\).

• In 1937, Shoda proved that all trace zero matrices over a field of characteristic zero (in particular \(\mathbb{C}\) or \(\mathbb{R}\)) is a commutator. His proof does not work over fields of positive characteristic (e.g. finite fields). Exercise: prove this result for \(2\times2\) matrices over \(\mathbb{C}\)!


• In 1957, Albert and Muckenhoupt proved that all trace zero matrices over any field are commutators. Their proof does not work over rings which are not fields (e.g. \(\mathbb{Z}\)).


• In 1961, Lissner proved that all trace zero \(2\times2\) matrices over any Principal Ideal Domain (PID) are commutators. He also gave examples that if \(R\) is a ring that is ``sufficiently far'' from being a PID, then there exist trace zero matrices over \(R\) which are not commutators.


• In 1989, Vaserstein posed the problem of whether every trace zero matrix over \(\mathbb{Z}\) is a commutator. (Since \(\mathbb{Z}\) is a PID, this was known to hold for \(2\times2\) matrices).


• In 1994, Laffey and Reams settled Vaserstein's problem in the affirmative. Their proof introduced several new substantial ideas but had the drawback that it relied on Dirichlet's theorem on primes in arithmetic progressions, which holds in \(\mathbb{Z}\) but not over several other PIDs like the ring of polynomials \(\mathbb{C}[x]\).


• The obvious question at this point, already implicit in Lissner's result above, is whether every trace zero matrix over a PID \(R\) is a commutator. this was proved in 2015-16 by adding several new ideas to Laffey and Reams, and in particular freeing their argument from the use of Dirichlet's theorem.


• Let \(\mathfrak{sl}_{n}(R)\) denote the ring of trace zero \(n\times n\) matrices over a commutative ring \(R\). When \(R\) is a field, \(\mathfrak{sl}_{n}(R)\) is a so-called Lie algebra, which is a ring whose multiplication is non-associative and given precisely by commutators: \(x*y:=xy-yx\) (but \(x*y\) is usually written \([x,y]\)). It is then natural to ask whether every element in \(\mathfrak{sl}_{n}(R)\), with \(R\) a PID, say, is a ``bracket'' \([x,y]\) in \(\mathfrak{sl}_{n}(R)\). In other words, is every trace zero matrix the commutator of two trace zero matrices? It is easy to see that this cannot be true when \(n=2\), but for \(n \geq 3\) this was proved in 2018 and required a completely new approach.

• Is every trace zero matrix over a Dedekind domain a commutator? (Important examples of Dedekind domains are rings of integers in algebraic number fields; they are not always PIDs, although every PID is a Dedekind domain.)


• Is every element in a simple Lie algebra over a PID a Lie bracket (assuming we ignore finitely many small counter-examples)?

Prerequisites

Resources

1. Albert and Muckenhoupt, On matrices of trace zero, Michigan Math. J. 4 (1957), 1–3.
2. Laffey and Reams, Integral similarity and commutators of integral matrices, Linear Algebra Appl. 197-198, 671-689 (1994).
3. Lissner, Matrices over polynomial rings, Trans. Amer. Math. Soc. 98 (1961), 285–305.
4. Stasinski, Similarity and commutators of matrices over principal ideal rings,
   Trans. Math. Soc., 368 (2016), 2333-2354.
5. Stasinski, Commutators of trace zero matrices over principal ideal rings,
   Israel J. Math. 228(1) (2018), 211–227.