v
|{z}
=(grnζ)(w)
= Ωn((grnζ) (w)) =
Ωn◦grnζ
| {z }
=grn(η◦ϕ)◦grnζ
(w) = (grn(η◦ϕ)◦grnζ) (w)
= (grn(η◦ϕ))
(grnζ) (w)
| {z }
=v
= (grn(η◦ϕ)) (v).
Since this holds for every v ∈ FnFn−1, we thus have proven that Ωn = grn(η◦ϕ).
This proves (51).
From (51), we see that every k-module homomorphism Ωn : FnFn−1 → grn(⊗n) for which the diagram (27) commutes must be equal to grn(η◦ϕ). Hence, there exists at most one k-module homomorphism Ωn : FnFn−1 →grn(⊗n) for which the diagram (27) commutes. But since we also know that there exists at least onek-module homomorphism Ωn : FnFn−1 → grn(⊗n) for which the diagram (27) commutes (namely, the homomorphism grn(η◦ϕ), because of (50)), we thus conclude that there exists one and only onek-module homomorphism Ωn:FnFn−1 →grn(⊗n) for which the diagram (27) commutes. This proves part of Theorem 2.1 (c).
According to Theorem 2.1 (c), we define ωn as the k-module homomorphism Ωn : FnFn−1 →grn(⊗n) for which the diagram (27) commutes (the existence and unique-ness of this homomorphism Ωn was already proven above). This definition immediately yields that the diagram (28) commutes. In other words, the diagram (27) commutes if Ωn=ωn. Thus, (51) (applied to Ωn =ωn) yields thatωn= grn(η◦ϕ). Thus,ωn is an h-module isomorphism (because we know that grn(η◦ϕ) is anh-module isomorphism).
Now, all nontrivial statements in Theorem 2.1 (c) are proven. This completes the proof of Theorem 2.1.
2.10. Independency of the splitting
As a bonus from the above proof of Theorem 2.1, we obtain the following strengthening of this theorem:
Proposition 2.21. In the context of Theorem 2.1(b), for everyn∈N, there exists anh-module isomorphismFnFn−1 →n⊗n which is independent of the choice ofN. Proof of Proposition 2.21. According to Theorem 2.1(c), the map grad−1n,n◦ωn(where gradn,nandωnare defined in Theorem 2.1(c)) is anh-module isomorphismFnFn−1 → n⊗n. This isomorphism is clearly independent of the choice ofN (since the definitions of gradn,n and ωn are independent of the choice of N). This proves Proposition 2.21.
3. (g, h)-semimodules
Before we prove some more interesting results, we are going to introduce a notion -that of a (g,h)-semimodule. This notion will be defined for every commutative ringk, every k-Lie algebra gand every Lie subalgebra hof g. It will be a kind of intermediate link between the notion of ag-module and that of anh-module. Here is the definition:
3.1. (g, h)-semimodules: the definition
Definition 3.1. Letk be a commutative ring. Let gbe a k-Lie algebra. Let h be a Lie subalgebra of g. Let V be a k-module. Let µ:g×V →V be a k-bilinear map.
We say that (V, µ) is a (g,h)-semimodule if and only if
(µ([a, b], v) = µ(a, µ(b, v))−µ(b, µ(a, v)) for every a∈h, b∈g and v ∈V). (52) If (V, µ) is a (g,h)-semimodule, then thek-bilinear map µ:g×V →V is called the Lie action of the (g,h)-semimoduleV.
Often, when the mapµ is obvious from the context, we abbreviate the termµ(a, v) bya * v for anya∈g andv ∈V. Using this notation, the relation (52) rewrites as ([a, b]* v =a *(b * v)−b *(a * v) for everya ∈h, b∈g and v ∈V). (53)
Also, an abuse of notation allows us to write ”V is a (g,h)-semimodule” instead of
”(V, µ) is a (g,h)-semimodule” if the mapµis clear from the context or has not been introduced yet.
Besides, when (V, µ) is a (g,h)-semimodule, we will say thatµis a (g,h)-semimodule structure on V. In other words, ifV is ak-module, then a (g,h)-semimodule structure on V means a map µ: g×V → V such that (V, µ) is a (g,h)-semimodule. (Thus, in order to make a k-module into a (g,h)-semimodule, we must define a (g, h)-semimodule structure on it.)
This definition is very similar to the Definition 1.9. We will see that this similarity is not just superficial, and that most properties of g-modules have their analogues concerning (g,h)-semimodules.
But first let us notice that:
Proposition 3.2. Letk be a commutative ring. Letg be a k-Lie algebra. Leth be a Lie subalgebra of g. Then, every g-module is a (g,h)-semimodule.
In fact, this proposition follows trivially from comparing Definition 1.9 with Defi-nition 3.1. The converse of this proposition does not hold. However, a g-module is exactly the same as a (g,g)-semimodule, i. e., we have:
Proposition 3.3. Letk be a commutative ring. Letgbe ak-Lie algebra. Let V be a k-module. Let µ:g×V →V be a map. Then, (V, µ) is ag-module if and only if (V, µ) is a (g,g)-semimodule.
This is again clear from comparing Definition 1.9 with Definition 3.1.
Proposition 3.3 shows that the notion of a (g,h)-semimodule is a generalization of the notion of ag-module. Much of this Section 3 will be devoted to formulating some properties of (g,h)-semimodules which are analogous to the well-known properties of g-modules which we collected in Section 1. We are not going to prove all of these properties anew, because the proofs will be identical to the corresponding proofs for g-modules done in Section 1, up to changing ”g-module” to ”(g,h)-semimodule” (and,
similarly, ”g-algebra” to ”(g,h)-semialgebra”) and changing the references to results about g-modules to the corresponding results about (g,h)-semimodules.
Along with the theorems, most definitions from the theory of g-modules can be generalized to (g,h)-semimodules. For example, we can define the tensor product of two (g,h)-semimodules in the same way as we have defined the tensor product of two g-modules in Definition 1.31. Of course, this new notion of the tensor product of two (g,h)-semimodules will not conflict with the old notion of the tensor product of two g-modules (because the definition is the same, so that, when we have two g-modules, their tensor product does not depend on whether we consider them asg-modules or as (g,h)-semimodules).
We begin with a convention:
Convention 3.4. We are going to use the notation a * v as a universal notation for the Lie action of a (g,h)-semimodule. This means that whenever we have some Lie algebra g, some Lie subalgebra h ofg and some (g,h)-semimodule V (they need not be actually called g, h and V; I only refer to them as g, h and V here in this Convention), and we are given two elements a ∈ g and v ∈ V (they need not be actually called a and v; I only refer to them by a and v here in this Convention), we will denote by a * v the Lie action of V applied to (a, v) (unless we explicitly stated that the notation a * v means something different).
This convention is, of course, just the extension of Convention 1.10 to (g,h)-semimodules.
Warning 3.5. We know from Proposition 3.2 that every g-module is a (g, h)-semimodule. Thus, when k is a commutative ring, g is a k-Lie algebra, h is a Lie subalgebra of g, andV is some g-module, thenV is a (g,h)-semimodule as well, and thus, the notation a * v (where a ∈ g and v ∈ V) is overloaded: It can be interpreted according to Convention 1.10, but it can also be interpreted according to Convention 3.4. However, fortunately these two conventions give the same definition for a * v, and thus they do not conflict. So when we have a g-module V, then we do not have to worry about Convention 1.10 and Convention 3.4 leading to different interpretations a * v; they don’t.
However,Convention 3.4 can conflict with Convention 1.10 in two other cases: The first case is when we have a g-module structure and a different (g,h)-semimodule structure defined on one and the same k-module; the second case is when we have an h-module structure and a (g,h)-semimodule structure defined on one and the same k-module. The first of these cases will not appear in our studies; however, the second will appear. In such a case, we will not be allowed to use Convention 3.4 until we verify that, for every a∈h and v ∈ V, the meaning of a * v according to Convention 1.10 (that is, the Lie action of the h-module V applied to (a, v)) equals to the meaning of a * v according to Convention 3.4 (that is, the Lie action of the (g,h)-semimodule V applied to (a, v)), so that the value of a * v does not depend on which of the two Conventions we are using.
Fortunately, within this paper, this will always be fulfilled and easy to verify. In fact, within this paper, each time when we have an h-module structure and a (g, h)-semimodule structure defined on one and the same k-module, the h-module will be the restriction of the (g,h)-semimodule to h (see Definition 3.8 for the definition
of ”restriction”), and thus we will be allowed to use Convention 3.4 (according to Remark 3.10).
Now that we have defined a (g,h)-semimodule, let us do the next logical step and define a (g,h)-semimodule homomorphism:
Definition 3.6. Letk be a commutative ring. Let gbe a k-Lie algebra. Let h be a Lie subalgebra of g. Let V and W be two (g,h)-semimodules. Let f :V →W be a k-linear map. Then,f is said to be a (g,h)-semimodule homomorphism if and only if
(f(a * v) =a *(f(v)) for every a∈g and v ∈V).
Often, we will use the words ”(g,h)-semimodule map” or the words ”homomorphism of (g,h)-semimodules” or the words ”(g,h)-semilinear map” as synonyms for ”(g, h)-semimodule homomorphism”.
This Definition 3.6 is the analogue of Definition 1.12 for (g,h)-semimodules. There-fore, we have:
Proposition 3.7. Let k be a commutative ring. Let g be a k-Lie algebra. Let h be a Lie subalgebra of g. Let V and W be two g-modules. Let f : V → W be a map. Then, f is ag-module homomorphism if and only if f is a (g,h)-semimodule homomorphism. (Here, it makes sense to say that ”f is ag-module homomorphism”
since V andW are (g,h)-semimodules (which is because Proposition 3.2 yields that every g-module is a (g,h)-semimodule).)
It is easy to see that for every commutative ringk, for everyk-Lie algebrag, and for every Lie subalgebra h of g, there is a category whose objects are (g,h)-semimodules and whose morphisms are (g,h)-semimodule homomorphisms. We define the notion of a (g,h)-semimodule isomorphism as an isomorphism in this category; in other words, we define the notion of a (g,h)-semimodule isomorphism by analogy to the notion of a g-module isomorphism (which we defined in Definition 1.9). The obvious analogue of Proposition 1.14 holds.