  Understanding the origin and mechanism of the transverse polarization of
hyperons produced in unpolarized proton-proton collision, $pp\to
\Lambda^\uparrow X$, has been one of the long-standing issues in high-energy
spin physics. In the framework of the collinear factorization applicable to
large-$p_T$ hadron productions, this phenomenon is a twist-3 observable which
is caused by multi-parton correlations either in the initial protons or in the
process of fragmentation into the hyperon. We derive the twist-3 gluon
fragmentation function (FF) contribution to this process in the leading order
(LO) with respect to the QCD coupling constant. Combined with the known results
for the contribution from the twist-3 distribution function and the twist-3
quark FF, this completes the LO twist-3 cross section. We also found that the
model independent relations among the twist-3 gluon FFs based on the QCD
equation of motion and the Lorentz invariance property of the correlation
functions guarantee the color gauge invariance and the frame-independence of
the cross section.
