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.

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