DEDICATED TO GARRETT BIRKHOFF 1. NATURE OF THE PROBLEM AND RESULTS, EXAMPLES AND MOTIVATION
In the complex general linear group GL(n, d=), consisting of all n x n nonsingular complex matrices, each matrix A is an exponential. That is, there exists some complex matrix (log A) for which the corresponding l-parameter group exp(t log A) in GL(n, C) passes through A at t = 1. For the real general linear group GL(n, R) such a result must fail because GL(n, IR) is not a connected Lie group. However, even for the real special linear group S'L(n, R), the connected subgroup of GL(n, R) of all matrices with determinant + 1, not every matrix A E S'L(n, 02) lies on a 1 -parameter subgroup of SL(n, R). Yet, for each A E SL(n, R) the matrix A2 must lie on a l-parameter subgroup of SL(n, R), as is well known.
In this paper we consider the problem of passing a l-parameter subgroup through a matrix A within a prescribed matrix Lie group @ (or equivalently, the problem of obtaining a logarithm of A within the Lie algebra of a). Our main result asserts that for an algehaic matrix group 02 (as defined below) each matrix A E 6! has some positive integral power AP that lies on a l-parameter subgroup of GZ. The force of the statement is that the l-parameter subgroup through AP lies within G?!.
Thus, provided a is an algebraic Lie group, the range of the exponential map in G!! meets each discrete semigroup {A, A2, A3,...) generated by a matrix A E CZ. Of course this implies that some negative integral power {A-l, A-2, A-3,...} 1 1 a so ies on a l-parameter subgroup of GZ. We present some examples below to emphasize the nature of the hypotheses and conclusions of our theory. For each positive integer P > 1 we provide an example of a connected algebraic group a C GL(n, [w) and a matrix A E 67 such that no one of the matrices A, A2, A3,..., AP-1 lies on any l-parameter subgroup of a. Also we give an example of a 351