The project has a layer with several groups of polygons with different weights. Each group is associated to a weight. How can I get the sum of the area of each group?
You can use the GroupStats plugin to achieve your goals. Just make sure to add the area column to your data table. Here's a link to a tutorial for the plugin GroupStats
In case I understand correctly, the common attribute for your groups is a weight attribute?
Then you could use any of the dissolve functions in the toolbox, using that weight-field for dissolving (receiving probably multipart polygons). Next step would be to add a new area-field to your layer in the field calculator, using the
$area expression to calculate the area for each group (now one polygon).