We report the detection of broad absorptions due to Si iv \lambda \lambda 4088–4116 in the Luminous Blue Variable ( LBV ) AG Carinae during its last hot phase ( 2001-2003 ) .
Our NLTE spectral analysis , with the radiative transfer code CMFGEN , revealed the photospheric nature of these lines predicting , however , much narrower and deeper absorption profiles than observed .
Using a recently-developed code to compute synthetic spectra in 2D geometry allowing for the effects of rotation , we could match the broad absorptions with a high projected rotational velocity of 190 \pm 30 km s ^ { -1 } on 2001 April .
Analysis of spectra obtained on 2002 March and 2003 January , when the star was cooling , yielded to a projected rotational velocity of 110 \pm 10 km s ^ { -1 } and 85 \pm 10 km s ^ { -1 } , respectively .
The derived rotational velocities are proportional to R _ { \star } ^ { -1 } , as expected from angular momentum conservation .
We discuss the effects of such high rotation on the spectral analysis of AG Car , and on the wind terminal velocity .
Our results show direct spectroscopic evidence , for the first time , that a LBV may rotate at a significant fraction of its break-up velocity .
Thus , AG Car ( and possibly other LBVs ) is indeed close to the \Gamma \Omega limit , as predicted by theoretical studies of LBVs .