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 .
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