  We discuss a conjecture that the twisted transfer matrix of the six-vertex
model at roots of unity with some discrete twist angles should have the sl(2)
loop algebra symmetry. As an evidence of this conjecture, we show the following
mathematical result on a subalgebra of the sl(2) loop algebra, which we call a
Borel subalgebra: any given finite-dimensional highest weight representation of
the Borel subalgebra is extended into that of the sl(2) loop algebra, if the
parameters associated with it are nonzero. Thus, if operators commuting or
anti-commuting with the twisted transfer matrix of the six-vertex model at
roots of unity generate the Borel subalgebra, then they also generate the sl(2)
loop algebra. The result should be useful for studying the connection of the
sl(2) loop algebra symmetry to the Onsager algebra symmetry of the
superintegrable chiral Potts model.
