subalgebra of invariants: By DeMeyer's criterion A is G-Galois over B if there are elements in A satisfying: whence also Then A is a Frobenius extension of B with E: A → B defined by which satisfies (Furthermore, an example of a separable algebra extension since is a separability element satisfying ea = ae for all a in A as well as . Also an example of a depth two subring (B in A) since where for each g in G and a in A.) Frobenius extensions have a well-developed theory of induced representations investigated in papers by Kasch and Pareigis, Nakayama and Tzuzuku in the 1950s and 1960s. For example, for each B-module M, the induced module A ⊗B M (if M is a left module) and co-induced module HomB(A, M) are naturally isomorphic as A-modules (as an exercise one defines the isomorphism given E and dual bases). The endomorphism ring theorem of Kasch from 1960 states that if A | B is a Frobenius extension, then so is A → End(AB) where the mapping is given by a ↦ λa(x) and λa(x) = ax for each a,x ∈ A. Endomorphism ring theorems and converses were investigated later by Mueller, Morita, Onodera and others. Frobenius adjunctions As already hinted at in the previous paragraph, Frobenius extensions have an equivalent categorical formulation. Namely, given a ring extension , the induced induction functor from the category of, say, left S-modules to the category of left R-modules has both a left and a right adjoint, called co-restriction and restriction, respecitvely. The ring extension is then called Frobenius if and only if the left and the right adjoint are naturally isomorphic. This leads to the obvious abstraction to ordinary category theory: An adjunction is called a Frobenius adjunction iff also . A functor F is a Frobenius functor if it is part of a Frobenius adjunction, i.e. if it has isomorphic left and right adjoints. See also Bialgebra Frobenius category Frobenius norm Frobenius inner product Hopf algebra Quasi-Frobenius Lie algebra Dagger compact category References External links Algebras Module theory Monoidal categories