2.12. The tensor algebra 35
Let A, B, C, D be four algebras. Let V be a (B, A)-bimodule,
W a (C, B)-bimodule, and X a (C, D)-bimodule. Prove that
HomB (V, HomC (W, X))
= HomC (W ⊗B V, X)
as (A, D)-bimodules. The isomorphism (from left to right) is given
f → (w ⊗B v → f (v) w)
for all v ∈ V , w ∈ W and f ∈ HomB (V, HomC (W, X)).
Exercise 2.11.7. Show that if M and N are modules over a commu-
tative ring A, then M ⊗A N has a natural structure of an A-module.
2.12. The tensor algebra
The notion of tensor product allows us to give more conceptual (i.e.,
coordinate-free) definitions of the free algebra, polynomial algebra,
exterior algebra, and universal enveloping algebra of a Lie algebra.
Namely, given a vector space V , define its tensor algebra TV
over a field k to be TV =
with multiplication defined by
a · b := a ⊗ b, a ∈ V
b ∈ V
Observe that a choice of a basis
x1,...,xN in V defines an isomorphism of TV with the free algebra
k x1,...,xn .
Also, one can make the following definition.
Definition 2.12.1. (i) The symmetric algebra SV of V is the
quotient of TV by the ideal generated by v ⊗ w − w ⊗ v, v, w ∈ V .
(ii) The exterior algebra ∧V of V is the quotient of TV by the
ideal generated by v ⊗ v, v ∈ V .
(iii) If V is a Lie algebra, the universal enveloping alge-
bra U(V ) of V is the quotient of TV by the ideal generated by
v ⊗ w − w ⊗ v − [v, w], v, w ∈ V .
It is easy to see that a choice of a basis x1,...,xN in V identifies
SV with the polynomial algebra k[x1,...,xN ], ∧V with the exterior
algebra ∧k(x1,...,xN ), and the universal enveloping algebra U(V )
with one defined previously.