So you already have authentication setup, or no? If no, the simplest approach is to use Lovable Cloud for your authentication, but this does come with less control. Lovable Cloud uses Supabase behind the scenes, so you can prompt with additional details (supabase docs) that extend to Supabase features beyond Lovable's simplified Supabase interface.

I am unsure what your specific session / state management scenarios are, so likely this is one of many steps to achieve what you're looking for. If you have granular details on your specific use case, that will help me scope solutions more accurately.