Can you tell me if the £130,000 mortgage will be the only mortgage on the house and that you don't own another property.
You have taken on responsibility for half the mortgage and your name is ***** ***** to the title deeds of the property. The "cash" transaction which triggers stamp duty is the mortgage. However, as half the mortgage is less than the £125,000 at which stamp duty starts, you should have none to pay. Take a look here for more information.
I hope this helps but let me know if you have any further questions.
If you had paid a sum of cash for your share of the equity, that would be taken into account as one of the examples shows.