I play the Renaissance lute. Here's the solutions lute players use for the same problem (either one of the following or combining two solutions could be the answer):
a. Use non-slippy fabric on your leg underneath the instrument, such as a chamois (correct spelling?) leather - slippy leather's no good, of course! Some lutenists use the rubbery webbing sometimes used under carpets.
b. Wear a strap such that the instrument is suspended on your front and positioned slightly angled when viewed from the front. You'd have to experiment to find what suits you best. It might look a bit like this
c. Attach a strap or fabric to the strap button on the tail end of the instrument only (not the head stock), then tuck or fold the strap or fabric under your right leg (if you're right handed) so that the instrument becomes secure. This way the instrument becomes permanently suspended at the angle you like as long as you don't shuffle in your seat!
d. Try various ways of crossing your legs, perhaps like this. I must admit I find this gives me pins and needles after a while, but it works for some.
e. Use a foldable footstool.

I hope some of this helps. Of course, the smaller size of the mandolin may make many of these suggestions impractical. Good luck!