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

by

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 =

n≥0

V

n,

with multiplication defined by

a · b := a ⊗ b, a ∈ V

n,

b ∈ V

m.

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.