6 1. INTRODUCTION g ❍ ❍ ❍ ι U(g) ✟ ✟ ✟ (φ1 ⊗ φ2) ◦ Δ End(V1 ⊗ V2) ✲ ρ1 ⊗ 1 + 1 ⊗ ρ2 We call Δ the coproduct of U(g). Proof. We set A = U(g) ⊗ U(g) in the commutative diagram to define the universality of U(g), and consider g → A given by X → ι(X) ⊗ 1 + 1 ⊗ ι(X). Then the existence and uniqueness of Δ follows from the universality of U(g) and the fact that ι ([X, Y ]) ⊗ 1 + 1 ⊗ ι ([X, Y ]) equals [ι(X) ⊗ 1 + 1 ⊗ ι(X), ι(Y ) ⊗ 1 + 1 ⊗ ι(Y )] . Next set A = End(V1 ⊗ V2). Then we can check that the required map φ for the tensor product representation is given by (φ1 ⊗ φ2) ◦ Δ. In fact, the commutativity of the diagram is verified by (φ1 ⊗ φ2) ◦ Δ ◦ ι(X) = (φ1 ⊗ φ2) (ι(X) ⊗ 1 + 1 ⊗ ι(X)) = φ1 ◦ ι(X) ⊗ 1 + 1 ⊗ φ2 ◦ ι(X) = ρ1(X) ⊗ 1 + 1 ⊗ ρ2(X) = ρ(X). Hence the uniqueness of φ implies that φ1 ⊗ φ2 is the map for the tensor product representation ρ1 ⊗ ρ2. For enveloping algebras, the following result, the PBW(Poincare-Birkhoff-Witt) theorem, is fundamental. Proposition 1.13. If {Xi}i∈I is a basis of a Lie algebra g, then the following set is a basis of U(g). { Xi 1 · · · Xi m | i1 ≤ · · · ≤ im,m = 0, 1,... } Proof. We first show that U(g) is spanned by these elements. If we con- sider those elements Xi 1 · · · Xi m whose indices i1,...,im are not necessarily non- decreasing, it is obvious that these span U(g). Choose the minimal index among i1,...,im, and move this to the left end using the relation XjXk = XkXj+[Xj, Xk]. Next choose the minimal index among the remaining indices and move this to the second to the leftmost position. Continue this procedure to reorder the in- dices i1,...,im in non-decreasing order. Since newly appearing terms have smaller length, we apply the same procedure to these terms and after a finite number of steps, we can rewrite the original monomial as a linear combination of monomials with non-decreasing indices. Next we show that these monomials are linearly independent. To do this, we consider indeterminates {zi}i∈I which are in bijection with {Xi}i∈I, and denote the polynomial ring generated by these indeterminates by S. Since { zi 1 · · · zi m | i1 ≤ · · · ≤ im} is a basis of S, we can define an action of g on S as follows. (If m = 0, we set Xi1 = zi.) Xizi 1 · · · zi m = zizi 1 · · · zi m (i ≤ i1) Xi 1 (Xizi 2 · · · zi m ) + [ Xi, Xi 1 ]zi 2 · · · zi m (i i1)

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