John Stillwell - Naive Lie Theory.pdf - Index of

128 6 Structure of Lie algebras
Our strategy for proving that so(n) is simple is like that used in Section 6.4
to prove that sl(n,C) is simple. It involves two stages:
• First we suppose that X is a nonzero member of some ideal I and
take Lie brackets of X with suitable basis vectors until we obtain a
nonzero multiple of some basis vector in I.
• Then, by further Lie bracketing, we show that all basis vectors are
in fact in I, soI = so(n).
The first stage, as with sl(n,C), selectively nullifies rows and columns until
only a nonzero multiple of a basis vector remains. It is a little trickier to
do this for so(n), because multiplying by E ij leaves intact two columns
(or rows, if one multiplies on the left), rather than one. To nullify all but
two, symmetrically positioned, entries we need n > 4, which is no surprise
because so(4) is not simple.
In the first stage we need to keep track of matrix entries as columns
and rows change position, so we introduce a notation that provides number
labels to the left of rows and above columns. For example, we write
⎛
i
j
⎞
E ij =
i
⎜
j ⎝
−1
1
⎟
⎠
to indicate that E ij has 1 in the (i, j)-position, −1inthe( j,i)-position, and
zeros elsewhere.
Now suppose X is the n × n matrix with (i, j)-entry x ij . Multiplying X
on the right by E ij and on the left by −E ij ,wefindthat
XE ij =
⎛
⎜
⎝
i
⎞
−x 1 j x 1i
−x 2 j x 2i
⎟
. . ⎠
−x nj x ni
j