Can I leave the battery charger plugged in my Dyson dc 44 animal after it is fully charged?

Hi, I have a Dyson DC56 with a lithium battery for about 18 months and I leave it plugged into it's charger when not in use. I do notice the Dyson charger seems to switch off automatically when charged. I don't have any problems so far. It you have a Genuine Dyson Lithium battery and the genuine Dyson charger it should not be a problem. That is because Lithium batteries have very precise requirements for their charging cycles and therefore the correctly designed charger required to make them work safely is a somewhat intelligent design and needs to be of good quality in having to meet those requirements. So use the original Dyson Charger. Also most high current Lithium Batteries will also have a protection circuit built in further increasing safety and reliability. Just don't use cheap cloned Chinese made Lithium chargers, because you will never know if they have been designed or even will operate correctly.
Now if you have old technology Ni Cad (Nickel Cadmium), LA (Lead acid) or NiMH batteries with basic cheap chargers they should not really be left connected to a charger full time.