34 2. Basic notions of representation theory

and “right A-module” are particular cases of the notion of bimodules;

namely, a left A-module is the same as an (A, k)-bimodule, and a right

A-module is the same as a (k, A)-bimodule.

Let B be a k-algebra, W a left B-module, and V a right B-

module. We denote by V ⊗B W the k-vector space

(V ⊗k W ) / vb ⊗ w − v ⊗ bw | v ∈ V, w ∈ W, b ∈ B . We denote the

projection of a pure tensor v ⊗ w (with v ∈ V and w ∈ W ) onto the

space V ⊗B W by v ⊗B w. (Note that this tensor product V ⊗B W

is the one defined in Remark 2.11.4.)

If, additionally, A is another k-algebra and if the right B-module

structure on V is part of an (A, B)-bimodule structure, then V ⊗B W

becomes a left A-module by a (v ⊗B w) = av ⊗B w for any a ∈ A,

v ∈ V , and w ∈ W .

Similarly, if C is another k-algebra, and if the left B-module

structure on W is part of a (B, C)-bimodule structure, then V ⊗B W

becomes a right C-module by (v ⊗B w) c = v ⊗B wc for any c ∈ C,

v ∈ V , and w ∈ W .

If V is an (A, B)-bimodule and W is a (B, C)-bimodule, then

these two structures on V ⊗B W can be combined into one (A, C)-

bimodule structure on V ⊗B W .

(a) Let A, B, C, D be four algebras. Let V be an (A, B)-

bimodule, W a (B, C)-bimodule, and X a (C, D)-bimodule. Prove

that (V ⊗B W ) ⊗C X

∼

=

V ⊗B (W ⊗C X) as (A, D)-bimodules. The

isomorphism (from left to right) is given by the formula

(v ⊗B w) ⊗C x → v ⊗B (w ⊗C x)

for all v ∈ V , w ∈ W , and x ∈ X.

(b) If A, B, C are three algebras and if V is an (A, B)-bimodule

and W an (A, C)-bimodule, then the vector space HomA (V, W ) (the

space of all left A-linear homomorphisms from V to W ) canonically

becomes a (B, C)-bimodule by setting (bf) (v) = f (vb) for all b ∈ B,

f ∈ HomA (V, W ), and v ∈ V and setting (fc) (v) = f (v) c for all

c ∈ C, f ∈ HomA (V, W ) and v ∈ V .