Depth Two for Infinite Index Subalgebras

In this paper, an algebra extension A |B is right depth two if its tensor-square is A-B-isomorphic to a direct summand of any (not necessarily finite) direct sum of A with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory [5] but extends the main theorem of depth two theory, as for example in [4]. That is, a right depth two extension has right bialgebroid T = (A⊗B A) B over its centralizer R = CA(B). The main theorem: an extension A |B is right depth two and right balanced iff A |B is T -Galois wrt. left projective, right R-bialgebroid T .