Commutators of Skew-Symmetric Matrices

In this paper we develop a theory for analysing the size of a Lie bracket or commutator in a matrix Lie algebra. Complete details are given for the Lie algebra so(n) of skew symmetric matrices. 1 Norms and commutators in Mn[R] and so(n) This paper is concerned with the following question. Let g be a Lie algebra (Carter, Segal & Macdonald 1995, Humphreys 1978, Varadarajan 1984). Given X,Y ∈ g and a norm ‖ · ‖ : g → R+, what is the size of ‖[X,Y ‖ in comparison with ‖X‖ · ‖Y ‖? On the face of this, this question has little merit since the elementary inequality ‖[X,Y ]‖ ≤ 2‖X‖ · ‖Y ‖ (1.1) always holds for X,Y ∈ Mn[R], the set of all n × n real matrices and an arbitrary norm ‖ · ‖. Moreover, it is easy to prove that the bound (1.1) can be attained for most norms of practical interest. In particular, this is the case for two types of norms closely associated with a remarkable paper of von Neumann (1937). We recall that a symmetric gauge is a vector norm · which is both symmetric and positive. In other words, for every x ∈ R it is true that xπ = x and |x| = x , where π is a permutation of {1, 2, . . . , n}, x> π = [xπ1 , xπ2 , . . . , xπn ] and |x|> = [|x1|, |x2|, . . . , |xn|]. We consider two norms, firstly the operator norm X = max v 6=0 Xv v and secondly the norm ‖X‖ = σ(X) , (1.2) where σ(X) are the singular values of X , arbitrarily ordered. While it is easy to see that (1.2) is a unitary norm (i.e,, invariant under multiplication by a unitary matrix), von Neumann proved that all unitary norms are of this form. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, email [email protected], supported in part by the National Science Foundation. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, England, email [email protected]. 1 We consider just the case n = 2, since it can be imbedded in Mn[R] for any n ≥ 2. Let X = [ 1 0 0 −1 ]