what is the chemical equation for electrolysis?

Your question is not that clear... there are numerous equations, depending on what you're electrolysing...

Electrolysis is just a method of forcing redox (oxidation-reduction) reactions to go one way or another. For example, Aluminum is reduced from bauxite ore to the metal:

Al^{+3} + 3e^- \rightleftharpoons Al(s)

This would be accompanied by another half-reaction.

There are many, many other electrolysis reactions. A very common one is the production of chlorine and sodium hydroxide (caustic-chlorine) from seawater (NaCl).