First, there is a big difference between wood ashes and potash. Potash is the term given to mined or manufactured mineral salts that contain potassium in a soluble form. Potassium chloride (KCl) is the most common form. In a few cases, potash can be extracted from plant remains - like ashes. Wood ashes generally contain a very small amount of potash - less than 10%. They are primarily calcium carbonate, a common liming material used to increase soil pH. And a little wood ash goes a long way - the recommended application rate is 15-20lb per 1000sf or about the equivalent of a 5 gallon bucket. Wood ashes should only be applied directly to the garden if your soil pH is below 7. Better to use as a component in your compost operation.

Potassium tends to move through most soils very slowly so unless you are doing a lot of heavy cropping, there is typically sufficient levels available in the existing soil. It is always a good idea to test your soil before applying, perhaps unnecessarily, and certainly if you are uncertain about the soil pH and contemplate adding wood ashes.

Lye is a very corrosive alkaline compound that is known as 'caustic soda' or sodium hydroxide (NaOH). Historically, it used to be made from potassium hydroxide (KOH), but that process proved to be inefficient and is no longer in practice. To my knowledge, lye has NO use in the garden and using it for any purpose should be undertaken with great care.