Notes on Enveloping Algebras (following Dixmier) Danny Crytser Summer 2019 Abstract The purpose of these notes is to provide a more-or-less self-contained proof of Theorem 5.1, which asserts that (borrowing some C∗-algebraic jargon) the universal enveloping algebra U = U(g) of a finite-dimensional Lie algebra g is residually finite-dimensional, in the sense that given any nonzero element u of U we can produce a finite-dimensional representation π of U that does not vanish on u. In fact, we prove something stronger: we can arrange for the intersection of ker π and U d to be trivial, where U d is the subspace spanned by symmetric homogeneous tensors of degree d. These notes follow [2, Ch. 2] very closely { mostly they were written to digest and provide additional background information. A reader who knows Lie's theorem on solvable Lie algebras and Engel's theorem on nilpotent Lie algebras will have adequate background knowledge. Contents 1 The Poincar´e-Birkhoff-Witt Theorem 1 2 Functorial properties of U 6 3 The symmetrization map 10 4 Existence of finite-dimensional representations 12 4.1 Nilpotency ideals . 14 5 Main result: residual finiteness of U(g) 19 1 The Poincar´e-Birkhoff-WittTheorem Definition 1.1. Let A be an associative algebra. By A− we denote the Lie algebra that has A as its underlying set and bracket given by [xy] = xy − yx. This is called the underlying Lie algebra of A. Remark 1.1. There is a faithful functor Alg ! LieAlg which is given by A 7! A− on objects and the identity on morphisms. It is kind of like a forgetful 1 functor, so we seek its left adjoint. (One could probably try to invoke some heavy duty Freyd theorem to prove that it is a right adjoint, but who has the energy?) The rough idea of a (universal) enveloping algebra is to reverse the construc- tion in Definition 1.1: we take a Lie algebra g and insert it in an associative algebra U in such a way that it generates the algebra (as an algebra! { not as a Lie algebra). The algebra representations of the algebra U ought to correspond precisely with the Lie algebra representations of g. The idea of the construction is pretty simple: just take the elements of g and treat them as elements in an associative algebra, while maintaining all relations coming from g. This corresponds with the tensor algebra of the vector space g. Definition 1.2. Let g be a Lie algebra. Let T 0 = C (or whatever ground field), let T 1 = g, and for n > 1 let T n = T n(g) = g ⊗ g ⊗ ::: ⊗ g : | {z } n-fold L1 n Set T = T (g) = n=0 T , which we refer to as the tensor algebra of g (here we are only treating g as a vector space). Remark 1.2. The tensor algebra provides a good example of an inclusion g ,! T which is not a homomorphism: x ⊗ y − y ⊗ x 6= [x; y]: Definition 1.3. Let g and T = T (g) be as above. Define J to be the two-sided ideal of T generated by all elements of the form x ⊗ y − y ⊗ x − [x; y] 2 T 2 ⊕ T 1 ⊂ T: Denote T=J by U = U(g). We refer to this algebra as the universal enveloping algebra of g. Remark 1.3. In view of remark 1.2, it is easy to see that U is the largest quotient of T so that the composite map g ,! T U(g) is a Lie algebra homomorphism. Definition 1.4. The map g ! T U(g) is denoted by σ and termed the canonical mapping of g into U(g). Remark 1.4. If g is abelian (trivial Lie bracket), then J is equal to the ideal generated by all x ⊗ y − y ⊗ x, and U is equal to the symmetric algebra of g. Remark 1.5. The ideal J in the definition of U is generated by elements belong- 1 2 ing to the two-sided ideal T+ = T ⊕ T ⊕ ::: be the subalgebra of T generated 0 0 by g, which has trivial intersection with T = C. Thus the map T U is injective. 2 Definition 1.5. We denote the image of T+ in U(g) by U+. (This is the 0 subalgebra of U generated by the image of σ.) We set U(g) = U ⊕ U+(g). The summand U 0 is identified with the ground field. For u 2 U we refer to its projection onto the ground field as the constant term of u (imagining u as a polynomial). Note that U is generated as an algebra by f1g [ σ(g). We are finally ready for our first result pertaining to U, which justifies its lofty title. Lemma 1.1 (cf. [2, Lemma 2.1.3]). Let σ be the canonical mapping of g into U(g) and let A be a unital algebra, and let τ : g ! A− be a Lie homomorphism. There exists a unique unital homomorphism τ 0 : U ! A such that τ 0 ◦ σ = τ. Proof. The uniqueness is evident from the fact that U is generated by f1g[σ(g). Let the inclusion g ,! T be denoted by ι. The universal property of the tensor algebra of g provides an algebra homomorphismτ ~ : T ! A satisfyingτ ~ ◦ ι = τ, which is unique if we require it to be unital. In particular,τ ~(x ⊗ y) = τ(x)τ(y). From this and the fact that τ is a Lie homomorphism we see thatτ ~ vanishes on the two-sided ideal J discussed above. It therefore factors through the quotient 0 0 q : T U to provide an algebra homomorphism τ : U ! A satisfyingτ ~ = τ ◦q. Then we have τ 0 ◦ σ = τ 0 ◦ q ◦ ι =τ ~ ◦ ι = τ: Remark 1.6. Lemma 1.1 shows that we have constructed a left adjoint to the \forgetful functor" A ! A−. If A is any associative algebra, we have a natural bijection ∼ HomLieAlg(g; A−) = HomAlg(U(g);A): Lemma 1.1 provides the left-to-right map and the right-to-left map is given by composition with σ (although we will shortly see that g embeds injectively in U via σ). Definition 1.6. Assume that g is finite-dimensional (although as in [4] its possible to make this work for countable-dimension Lie algebras as well) with p ordered basis (x1; x2; : : : ; xn). Set yi := σ(xi). If I = (i1; : : : ; ip) 2 f1; 2; : : : ; ng let yI = yi1 yi2 : : : yip . If i 2 Z, write i ≤ I if for all k we have i ≤ ik. If q : T U 0 1 d is the quotient we set Ud(g) to be q(T +T +:::+T ). We also allow an empty string I (of length 0) with y; = 1 2 U. We regard the Ud as the polynomials of degree ≤ d. The next Lemma shows that we can rearrange any monomial at the cost of introducing lower-order terms. Lemma 1.2 ([2, Lem. 2.1.5]). let a1; : : : ; ap 2 g and let σ : g ! U be canonical. If π is a permutation of the set f1; 2; : : : ; pg then σ(a1)σ(a2) : : : σ(ap) − σ(aπ(1))σ(aπ(2)) : : : σ(aπ(p)) 2 Up−1(g): 3 Proof. Every π can be written as the product of transpositions of the form (j j + 1), so we assume that π is of that form. Then σ(a1) : : : σ(ap) − σ(aπ(1)) : : : σ(aπ(p)) = σ(a1) : : : σ(aj)σ(aj+1) : : : σ(ap) − σ(a1) : : : σ(aj+1)σ(aj) : : : σ(ap) = σ(a1) : : : σ(aj−1)[σ(aj); σ(aj+1)]σ(aj+2) : : : σ(ap) = σ(a1) : : : σ(aj−1)σ([aj; aj+1])σ(aj+2) : : : σ(ap) 2 Up−1(g): Lemma 1.2 immediately implies the following. Lemma 1.3 ([2, Lem. 2.1.6]). The set of monomials fyI : I increasing of length pg spans the space Up(g). Definition 1.7. Let P = C[z1; : : : ; zn] be the algebra of complex polynomials in n indeterminates. Set Pi to be the subspace consisting of elements of P of p total degree ≤ i. For I 2 f1; : : : ; ng set zI = zi1 : : : zip . Remark 1.7. The following rather technical result is used to prove that the increasing monomials are linearly independent. Lemma 1.4. For every integer p ≥ 0 there exists a unique linear mapping fp : g ⊗ Pp ! P satisfying the following conditions: (Ap) fp(xi ⊗ zI ) = zizI if i ≤ I and zI 2 Pp; (Bp) fp(xi ⊗ zI ) − zizI 2 Pq for zI 2 Pq, q ≤ p; (Cp) fp(xi ⊗ fp(xj ⊗ zJ )) = fp(xj ⊗ fp(xi ⊗ zJ )) + fp([xi; xj] ⊗ zJ ). Moreover fpjg⊗Pp−1 = fp−1. Proof. The condition A0 implies that f0(xi ⊗ 1) = zi. This implies B0 and C0 both hold. The main issue is to extend fp−1 inductively to fp. Define fp(xi ⊗ zI ) = zizI as long as i ≤ I and zi 2 Pp. If i is not bounded by I then I = (j; J) where j > i and j ≤ J. (If i is less than the least element in I, its bounded by everything in I.) Note that zI = fp−1(xj ⊗ zJ ). Then fp(xi ⊗ zI ) = fp(xi ⊗ fp−1(xj ⊗ zJ )) = fp(xj ⊗ fp−1(xi ⊗ zJ )) + fp−1([xi; xj] ⊗ zJ ): Now fp−1(xi ⊗ zJ )) = zizJ + w for some w 2 Pp−1 by induction.