No. The voltage of 4 AA cells is too low to run the lamp. In addition, the energy density of AA batteries would be low anyway: While the Li-ion pack contains about 49 Wh, four AA alkaline cells (or NiMH rechargeables) have about 10 Wh only - so you would need about 20 AA cells to have the same amount of light as with our Li-ion pack.
9V-batteries may be possibility...and the 9-Volt adapter is certainly a good backup solution. But only working with the 'speleo' battery system.