We present the results of mid-resolution spectroscopy in the \ion LiI 6708 Å spectral region of Asymptotic Giant Branch ( AGB ) stars belonging to young open clusters of the Large Magellanic Cloud .
Most stars belong to the clusters \object NGC 1866 and \object NGC 2031 , which have an age of \simeq 150 Myr .
Lithium lines of different strength are detected in the spectra of stars evolving along the AGB , in agreement with theoretical predictions .
According to stellar evolutionary models , at the start of the AGB the stars should all show a low residual lithium abundance as a consequence of dilution during the previous evolutionary phases .
The most luminous and cooler thermally pulsating AGB stars , if they are massive enough , once in the AGB go first through a phase of Li destruction , which is followed by a phase of strong lithium production and further destruction .
The production of lithium on the AGB is in particular explained by the onset of the “ Hot Bottom Burning ” ( HBB ) process .
Our most conclusive results are obtained for the populous cluster \object NGC 1866 where : the ‘ early–AGB ’ stars show a weak Li line , which can be attributed to the dilution of the initial abundance ; one of the two luminous stars seem to have completely depleted lithium , as no line is detected ; the second one shows a deep lithium line , whose strength can be explained by lithium production .
The bolometric magnitude of these stars are consistent with the values predicted by the theory , for the mass evolving on the AGB of \object NGC 1866 , at which lithium is first destroyed and then produced ( M _ { bol } \simeq - 6 ) .
We also analyze the infrared luminosities ( ISOCAM data ) of these stars , to discuss if their evolutionary phase precedes or follows the lithium production stage .
More intriguing and less clear results are obtained for the most luminous stars in \object NGC 2031 : the brightest star seems as well to have destroyed lithium , while the second one shows a strong lithium line .
However its bolometric luminosity –derived from the near–IR photometry , is much lower ( M _ { bol } \simeq - 5.2 \pm 0.15 ) than that expected from HBB models .
Although low luminosity lithium rich AGB stars are also known , whose appearance is attributed to non–canonical mixing processes , it is not clear why two almost coeval clusters show such a different behaviour .
It is also possible that this star does not belong to \object NGC 2031 .
Finally we suggest the observational tests that could shed further light on this matter .