We were trying to guess the left adjoint to restriction

Let and be algebras.

Note that a right -module is just a left -module.

An -bimodule is an abelian group which is both a left -module and right -module such that

for all , , .

If are subalgebras, then is an -bimodule.

If and and , where as the subgroup that fixes , then as an -bimodule,

We can check that both of these submodules are -stable. (c.f. double cosets.)

Let be an -bimodule and be a -bimodule. The output will be will be an -bimodule.

is the abelian group generated by:

subject to relations:

The left -action is and extending linearly. The right -action is and extending linearly.

Let be a field. Then in we have

So the pure tensors are . A sum of is still a . So is at most a -dimensional vector space (since the set of all is a -dimensional subspace.) To show that is not , we can map out of it. Let us construct this map

in other words,

One can check that is well-defined. It is a surjection by inspection. So is not , i.e.,

Consider . We have

Subtracting, . So .

For prime,

since the -action factors through . This is also true for all (even non-prime).

. The maps are

Consider the short exact sequence

Now, we apply , and we find that

which is not exact. It turns out that is right but not left-exact.

Let be a field, and and are vector spaces over . Let be a basis of and be a basis of .

is a basis of the vector space .

This follows fro the -dimensional case and the fact that

and similarly on the nd factor.

Let be an -bimodule. Let be a left -module and be a left -module. Then there is a natural isomorphism

i.e., and are an adjoint pair.

Let and . Then the bijection is

The above adjunction is for the covariant , yielding the adjoint pair

The contravariant version is more complicated.

If , we define . We need to check that this is a left -module.

Let be a subalgebra. Let . Now if is a left -module,

since is an -module homomorphism and . This is the same action as the -action on . So we define to be the induction functor. By Tensor-Hom adjunctionTensor-Hom adjunction, is the left-adjoint to .