Opposite algebras

Next, we shall define some concepts which boil down to “turning around the product” in ak-algebra, in a Lie algebra and in a pre-Lie algebra:

Definition 5.6. Let A be ak-algebra. Then, we define A^op to be thek-module A, equipped with the multiplication∗ : A×A→ A(written in infix notation) which is defined by

(a∗b =ba for every a∈ Aand b ∈ A).

It is easy to see that A^op is again a k-algebra, with the same unity as A.

(Roughly speaking, A^op is the same k-algebra as A, except that the order of the factors in its multiplication has been turned around.) We call A^op the opposite algebraof A.

Definition 5.7. Letgbe a Lie algebra. Then, we defineg^op to be thek-module g, equipped with the Lie bracketλ :g×g→g which is defined by

(λ(a,b) = [b,a] for every a∈ gand b ∈g).

It is easy to see that g^op is again a Lie algebra. (Roughly speaking, g^op is the same Lie algebra asg, except that the order of the arguments in its Lie bracket has been turned around.) We call g^op theopposite Lie algebraofg.

Definition 5.8. Let A be a left pre-Lie algebra. Then, we define A^op to be the k-module A, equipped with the mapC: A×A→ A(written in infix notation)

which is defined by

(aCb =bB a for all a,b ∈ A).

Proposition 3.3 shows that A^op is a right pre-Lie algebra. We call A^op the opposite pre-Lie algebra of A.

Definition 5.9. Let Abe a right pre-Lie algebra. Then, we define A^op to be the k-module A, equipped with the mapB: A×A→ A(written in infix notation) which is defined by

(aBb =bC a for all a,b ∈ A).

Proposition 3.4 shows thatA^opis a left pre-Lie algebra. We callA^optheopposite pre-Lie algebraof A.

The four definitions that we just made are similar to each other: Each of them transforms an algebraic structure by switching the order of arguments in its binary operation. (Similar operations exist for groups, monoids, etc..) However, their properties differ. For example, every Lie algebra g is isomorphic to its opposite Lie algebrag^op:

Remark 5.10. Let gbe a Lie algebra. Then, the map g→g^op, x 7→ −x is a Lie algebra isomorphism.

However, in general, a k-algebra A is not isomorphic to A^op, and for pre-Lie algebras, such a statement would not even make sense. Also, the opposite Lie algebra g^op of a Lie algebra g can be simply obtained by multiplying the Lie bracket ofgby −1.

Remark 5.11. Let Abe a k-algebra. Then,(A^op)⁻ = (A⁻)^op.

Definition 5.12. LetVbe ak-module. We let End^opVbe thek-module EndV, equipped with the multiplication ∗ _: (EndV)×(EndV) → _EndV (written in infix notation) which is defined by

(a∗b=b◦a for every a∈ EndV andb ∈ EndV).

This End^opV is a k-algebra. Actually, End^opV = (EndV)^op ask-algebras.

We have introduced the special notation End^opV (instead of using (EndV)^op) because we want to stress that End^opV is an analogue of EndV (rather than just the opposite algebra of EndV). More precisely, End^opV is defined in the same

way as EndV, with the only difference that the multiplication now corresponds to the composition of maps “in the opposite order”. The k-algebra End^opV plays the same role in regard to right modules as the k-algebra EndV plays in regard to left modules; more precisely, we have the following two analogous propositions:

Proposition 5.13. Let V be a k-module. Let A be a k-algebra. Left A-module structures on V are in a 1-to-1 correspondence with k-algebra ho-momorphisms A → EndV; this correspondence is defined as follows: Given a left A-module structure on V, we can define a k-algebra homomorphism Φ : A→EndV by setting

((_Φ(a)) (v) = av for everya∈ A andv ∈V).

Conversely, given a k-algebra homomorphismΦ : A→EndV, we can define a left A-module structure on V by setting

(av= (Φ(a)) (v) for every a∈ _{Aand v}∈ _V).

Proposition 5.14. LetVbe ak-module. Let Abe ak-algebra. RightA-module structures on V are in a 1-to-1 correspondence with k-algebra homomor-phisms A → _End^opV; this correspondence is defined as follows: Given a right A-module structure on V, we can define a k-algebra homomorphism Φ : A→End^opV by setting

((_Φ(a)) (v) = va for everya∈ A andv ∈V).

Conversely, given a k-algebra homomorphism Φ : A → End^opV, we can define a right A-module structure on V by setting

(va= (_Φ(a)) (v) for every a∈ Aand v∈ V).

The analogy between Proposition 5.13 and Proposition 5.14 would become even more obvious if we would write the action ofΦ(a) on the right of thev in Proposition 5.14 (that is, if we would writev(_Φ(a)) instead of (_Φ(a)) (v)); but we prefer to keep to the more standard notations.

We can now state the analogue of Definition 1.9 for rightg-modules:

Definition 5.15. Let gbe a Lie algebra.

(a) Every right U(g)-module M canonically becomes a right g-module by setting

(m (a =mιU,g(a) for all a ∈g andm ∈ M).

Moreover, any right U(g)-module homomorphism between two right U(g) -modules becomes a right g-module homomorphism if we regard these right

U(g)-modules as right g-modules. Thus, we obtain a functor from the cate-gory of rightU(g)-modules to the category of right g-modules.

(b) Every rightg-module M canonically becomes a right U(g)-module. To define the right U(g)-module structure on M, we proceed as follows: Define a map ϕ: g→End^opM by

((ϕ(a)) (m) = m( a for all a∈ gand m ∈ M).

It is easy to see that this map ϕ is a Lie algebra homomorphism from g to (End^opM)⁻. (Indeed, this is a restatement of the axioms of a right g-module;

the fact thatϕ([a,b]) = [ϕ(a),ϕ(b)]for alla,b ∈ gis equivalent to the relation (101).) Now, Theorem 1.8 (applied to A = End^opM and f = ϕ) shows that there exists a unique k-algebra homomorphism F : U(g) → End^opM such that ϕ = F◦_ιU,g. Consider this F. Now, we define a right U(g)-module structure on M by

(mp= (F(p)) (m) for all p ∈U(g) and m ∈ M).

Thus, every rightg-module canonically becomes a rightU(g)-module. More-over, any right g-module homomorphism between two right g-modules be-comes a right U(g)-module homomorphism if we regard these right g -modules as right U(g)-modules. Hence, we obtain a functor from the cat-egory of right g-modules to the category of rightU(g)-modules.

(c) In Definition 5.15 (a), we have constructed a functor from the category of right U(g)-modules to the category of right g-modules. In Definition 5.15 (b), we have constructed a functor from the category of rightg-modules to the category of rightU(g)-modules. These two functors are mutually inverse. In particular, if Mis a right g-module, then the right U(g)-module structure on M obtained according to Definition 5.15(b)satisfies

mιU,g(a) = m( a for everya ∈ gand m∈ _M.

(d) According to Definition 5.15 (a), every right U(g)-module canonically becomes a rightg-module. In particular,U(g)itself becomes a rightg-module (because U(g) is a right U(g)-module). This is the right g-module structure onU(g)“given by right multiplication” (because it satisfiesu (x =uι_U,g(x) for every x ∈ g and u ∈ U(g)). Other canonical right g-module structures on U(g)exist as well, but we shall not use them for the time being.

We furthermore define a “right analogue” of DerC:

Definition 5.16. Let C be a k-algebra. We can regard DerC not only as a subset of EndC, but also as a subset of End^opC (since End^opC = _EndC as sets). Then, DerC is a Lie subalgebra of (End^opC)⁻. (This is an analogue of Proposition 1.11 (a).) We define Der^opC to be the Lie subalgebra DerC of

(End^opC)⁻. (Thus, Der^opC =_DerC as sets and ask-modules, but Der^opC = (DerC)^op as Lie algebras.)