Is it permissible to lie to a foreign embassy to obtain an entry visa for the purpose of study and work, by agreement with someone to sponsor me nominally?

It is not permissible to lie to the embassy or violate its conditions to obtain a visa, especially if you have pledged to abide by those conditions. This is based on Allah Almighty's saying: "O you who have believed, fulfill [your] contracts," and the Prophet Muhammad's (peace be upon him) saying: "Muslims are bound by their conditions."