4. The normal ordered product of differential operators 70
5.6. The case of an associative algebra
Now we can see Theorem 3.10 with Theorem 5.21 interact:
Theorem 5.22. Let A be a k-algebra. Thus, A becomes a left pre-Lie algebra (according to Proposition 3.5 (a)) and a right pre-Lie algebra (according to Proposition 3.5(b)). Consider the map η : U(A−) → SymA defined in The-orem 3.10 (f) (using the left pre-Lie algebra structure on A). Also, consider the map η0 : U(A−) → SymA defined in Theorem 5.21 (f) (using the right pre-Lie algebra structure on A).
Then,η =η0.
Before we prove this theorem, let us state a trivial lemma:
Lemma 5.23. Let A be a k-algebra. Thus, A becomes a left pre-Lie algebra (according to Proposition 3.5 (a)) and a right pre-Lie algebra (according to Proposition 3.5(b)).
(a)We have aBb =aCb for any a ∈ A andb ∈ A.
(b)We have aB(cCb) = (aBc)Cb for any a ∈ A,b ∈ Aand c ∈ A.
Proof of Lemma 5.23. (a) Let a ∈ A and b ∈ A. We have a B b = ab (by the definition of the left pre-Lie algebra structure on A). We also have a C b = ab (by the definition of the right pre-Lie algebra structure on A). Comparing this with aBb =ab, we obtain aBb =aCb. This proves Lemma 5.23(a).
(b)Let a ∈ A, b ∈ A and c ∈ A. We have c C b =cb (by the definition of the right pre-Lie algebra structure on A). We also have aBc =ac(by the definition of the left pre-Lie algebra structure on A). Now, the definition of the left pre-Lie algebra structure on Ashows that
aB (c Cb) = a(cCb)
| {z }
=cb
=acb.
On the other hand, the definition of the right pre-Lie algebra structure on A shows that
(aBc) Cb= (aB c)
| {z }
=ac
b =acb.
Comparing this with a B (cC b) = acb, we obtain a B (cCb) = (aBc) C b.
This proves Lemma 5.23(b).
Proof of Theorem 5.22. We shall use all notations introduced in Theorem 3.10 (in particular, the map K : A → Der(SymA) and the left A−-module structure on SymA). We shall also use all notations introduced in Theorem 5.21 (in particular, the mapK0 : A →Derop(SymA)and the right A−-module structure on SymA).
Our setup causes a notational ambiguity: When a and b are two elements of A, then the expression “ab” might mean the product of a andb in the k-algebra
A, but can also mean the product ab of the elements a and b of the symmetric algebra SymA. We resolve this ambiguity by agreeingnotto use the notationab with the former meaning. Thus, in this proof, a product such asab (with a ∈ A and b ∈ A) shall always mean the product of a and b regarded as elements of the symmetric algebra SymA, rather than the product of a and b regarded as elements of A.
For everya∈ A− andb ∈ A−, we have
K(a)◦K0(b) =K0(b)◦K(a) (117)
70.
70Proof of (117):Leta∈ A− andb∈A−.
Recall that we are regardingAas ak-submodule of SymAvia the injection ιSym,A : A→ SymA. Thus, A= ιSym,A(A) =Sym1A. Recall that the subset Sym1Aof SymAgenerates thek-algebra SymA. In other words, the subset Aof SymAgenerates the k-algebra SymA (sinceA=Sym1A). (since the Lie bracket used here
is that of Der(SymA))
(by (112), applied tobandaBc instead ofaandb)
In the proof of Theorem 3.10(e), we have shown that the following holds:
• The mapK: A− →Der(SymA) is a Lie algebra homomorphism.
• We have
ιSym,A([a,b]) =ιSym,A(a),ιSym,A(b)+ (K(a)) ιSym,A(b)−(K(b)) ιSym,A(a) for everya ∈ A− andb ∈ A− (where the Lie bracket
ιSym,A(a),ιSym,A(b) is computed in the Lie algebra(SymA)−).
• We can apply Theorem 1.15 tog= A−, C =SymAand f =ιSym,A.
• The A−-module structure on SymA defined in Theorem 3.10 (e) is pre-cisely the one that is constructed by Theorem 1.15(a) (applied tog = A−, C=SymA and f =ιSym,A).
• The mapη : U(A−) →SymA defined in Theorem 3.10 (f)is precisely the mapη :U(A−) →SymAthat is constructed by Theorem 1.15(b)(applied tog =A−,C =SymAand f =ιSym,A).
Similarly, in the proof of Theorem 5.21(e), we have shown that the following holds:
• The mapK0 : A− →Derop(SymA)is a Lie algebra homomorphism.
• We have
ιSym,A([a,b]) =ιSym,A(a),ιSym,A(b)+ K0(b) ιSym,A(a)− K0(a) ιSym,A(b) for everya ∈ A− andb ∈ A− (where the Lie bracket
ιSym,A(a),ιSym,A(b) is computed in the Lie algebra(SymA)−).
• We can apply Theorem 5.17 tog= A−, C =SymAand f0 =ιSym,A.
• The right A−-module structure on SymA defined in Theorem 5.21 (e) is precisely the one that is constructed by Theorem 5.17 (a) (applied to g = A−, C=SymA and f0 =ιSym,A).
• The mapη0 :U(A−) →SymAdefined in Theorem 5.21 (f)is precisely the mapη0 : U(A−) →SymAthat is constructed by Theorem 5.17(b)(applied tog =A−,C =SymAand f0 =ιSym,A).
Furthermore, we observe the following:
(applied to SymA,[K(a),K0(b)], 0 andAinstead ofA,d,eandS) shows that[K(a),K0(b)] = 0. Since[K(a),K0(b)] =K(a)◦K0(b)−K0(b)◦K(a), this rewrites asK(a)◦K0(b)−K0(b)◦ K(a) =0. In other words,K(a)◦K0(b) =K0(b)◦K(a). This proves (117).
• We have
K(a)◦K0(b)−K0(b)◦K(a)(u) = K0(b) ιSym,A(a)·u−u·(K(a)) ιSym,A(b) for all a∈ A−, b ∈ A− and u∈ SymA 71.
• We haveιSym,A=ιSym,A.
Combining all the observations in the bullet points made above, we conclude that we can apply Theorem 5.20 (c) to g = A−, C = SymA, f = ιSym,A and
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