1.3. ENVELOPING ALGEBRAS 3 deform the structure constants of the Serre relations by introducing a parameter, the resulting algebras manifest hidden symmetries of certain spin models, and we can compute physically important quantities of the models. These algebras are the quantum algebras of Drinfeld and Jimbo. In this brief history, we can see a natural evolution of mathematical ideas and the discovery of new mathematical concepts. The quantum algebra so discovered is the subject of these lectures. 1.3. Enveloping algebras We define the enveloping algebra of a Lie algebra. The word “algebra”always means a unital associative C-algebra in this chapter. Definition 1.4. Let g be a Lie algebra. Among pairs (A, ρ) of an algebra A and a C-linear map ρ : g → A satisfying ρ([X, Y ]) = ρ(X)ρ(Y ) − ρ(Y )ρ(X), we consider the universal one: namely, the pair (U(g),ι) having the property that for any pair (A, ρ), there exists a unique algebra homomorphism φ : U(g) → A which satisfies ρ = φ ◦ ι. We call the algebra U(g) the (universal) enveloping algebra of g. Perhaps we should denote the enveloping algebra by (U(g),ι), but it is conven- tion to drop ι. This is harmless since, if U is isomorphic to U(g) as an algebra, we can define a map ι to make (U, ι) a universal pair in the above sense. It is important to note that we consider not only finite dimensional algebras but also infinite dimensional algebras A. The universality of U(g) is usually visualized by a commutative diagram: g ❍ ❍ ❍ ι U(g) ✟ ✟ ✟ ∃1φ A ✲ ρ Since it is not at all clear from the definition that U(g) exists, it is necessary to show its existence. Lemma 1.5. The enveloping algebra of a Lie algebra always exists. It is unique up to an isomorphism of algebras. This is a standard fact, and we leave it to the reader in the exercises below. Since the second exercise is a bit diﬃcult for beginners, I recommend consulting a textbook on Lie algebras. Exercise 1.6. Show that there is at most one enveloping algebra up to an isomorphism of algebras. Exercise 1.7. Let T (g) = ⊕n≥0 g⊗n be the tensor algebra over g, and let I be the two-sided ideal generated by X ⊗ Y − Y ⊗ X − [X, Y ] (X, Y ∈ g). Show that U(g) T (g)/I, and ι : g → U(g) is injective. Here, ι is the map naturally induced by the inclusion g ⊂ T (g).

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