Journal of Algebra, Vol. 58, No. 1, May 1979, pp. 37-50
William H. Wilson
N. R. Wallach has constructed, and studied properties of, a type of induced module for Lie algebras with decompositions like those of semisimple complex Lie algebras. In this paper, we study properties of induced-module functors like Wallach's, for Lie algebras, groups acting on sets, and coalgebras. It is shown that the property of being a weak double adjoint functor is responsible for some of the usefulness of Wallach's functor. Such functors are characterized in terms of natural transformations from, (in the Lie algebra case), – ⊗_{Uh} Ug to Hom_{Uh}(Ug, –).
1. DESCRIPTION OF WALLACH'S FUNCTOR AND OF WEAK ADJOINTS
1.1 Let g be a Lie algebra over a field k. As in [7], [8], we say that g has a decomposition if there are subalgebras n_{1}, n_{2} and h of g such that g = n_{1} ⊕ h ⊕ n_{2}, (vector space direct sum), and [h, n_{i}] ⊆ n_{i} for i = 1, 2. Let Uh, Ug denote the universal enveloping algebras of h and g respectively. We shall denote by Mod-h and Mod-g the categories of right Uh-modules and right Ug-modules, respectively.
Wallach, in [7], constructs, for each W ∈ Mod-h, an "induced" module W* ∈ Mod-g. This construction is functorial, and it will be more convenient for us to denote functors by letters, so we shall write I_{ω}W instead of W*. We shall denote by F: Mod-g → Mod-h the obvious forgetful functor.
It is convenient to outline the construction of Wallach's induced module, as it will be needed in Section 3. Set t = n_{1} ⊕ h, and make W into a Ut-module by having n_{1} act trivially on W. By the Poincare-Birkhoff-Witt theorem (cf. Humphreys [2, page 92]), we can write Ug = Ut ⊕ Un_{2} . n_{2} . Ut, and this is a left Uh-module, right Ut-module direct sum. Let V: Ug → Ut be the corresponding projection. Define j_hat_{W}: W → Hom_{Ut}(Ug, W) ⊆ Hom_{Uh}(Ug, W) by
[j_hat_{W}(w)](s) = w . γ(s) for w ∈ W, s ∈ Ug.
j_hat_{W} is a Uh-module monomorphism. Set I_{ω}W = [im j_hat_{W}] . Ug. If ψ ∈ Hom_{Uh}(W_{1}, W_{2}), define I_{ω}ψ by (I_{ω}ψ)(f) = ψ ° f for f ∈ I_{ω}W_{1} ⊆ Hom_{Uh}(Ug, W_{1}): it turns out (see [7]) that (I_{ω}ψ)(f) ∈ I_{ω}W_{2}, and that I_{ω} is a functor.
Wallach shows that his induced modules have various interesting properties, e.g.:
(0) I_{ω} is a functor.
(1) There is an injection of Uh-modules j_{W}: W → FI_{ω}W, and im j_{W}, generates I_{ω}W as Ug-module.
(2) There is a natural injection Hom_{Ug}(V, I_{ω}W) → Hom_{Uh}(FV, W).
(3) I_{ω}W is a submodule of Hom_{Uh}(Ug, W). If W is a simple Uh-module of finite dimension over k, and Hom_{Uh}(Ug, W) contains a finite-dimensional simple Ug-module V, then V = I_{ω}W, (whereas W ⊗_{Uh} Ug and Hom_{Uh}(Ug, W) are not finite-dimensional unless h = g or W = {0}).
1.2 Property (1) above (respectively property (2)) is a weakening of the condition that I_{ω} be a left (respectively right) adjoint to F. Let us formalize these properties with definitions.
1.3 DEFINITION. Let H and G be categories and let F: G → H, I: H → G be functors. We say that I is an injective weak left adjoint to F if for all W ∈ H, V ∈ G, there is an injection
θ_{WV}: G(IW, V) → H(W, FV) | (1) |
natural in W and V. We say that I is an injective weak right adjoint to F if for all W ∈ H and V ∈ G there is an injection
η_{VW}: G(V, IW) → H(FV, W) | (2) |
natural in V and W.
Notation. Given natural injections (1) and (2) and W ∈ H, we denote by j_{W} the morphism θ_{W,IW}(1_{IW}) ∈ H(W, FIW), and we denote by d_{W} the morphism η_{IW,W}(1_{IW}) ∈ H(FIW, W). The naturality of (1) and (2) implies that j: 1_{H} → FI and d: FI → 1_{H} are natural transformations.
DEFINITION. We shall say that I is an injective weak double adjoint to F if (1) and (2) are satisfied, and also
∀ W ∈ H d_{W} ° j_{W} = 1_{W}. | (3) |
1.4 In the terminology defined in Section 1.3,
PROPOSITION. Wallach's functor I_{ω}: Mod-h → Mod-g is an injective weak double adjoint to the restriction functor F: Mod-g → Mod-h.
A direct proof of this proposition could be given, but a shorter proof will be possible when we have characterized injective weak double adjoints in Section 2. In Sections 4, 5, 6, we shall describe other examples of injective weak double adjoints to forgetful functors.
2. INJECTIVE WEAK ADJOINTS
2.1 Let H and G be categories, and let F: G → H be a functor. Suppose that F has a left adjoint L: H → G and a right adjoint R: H → G. Let i: 1_{H} → FL denote the unit of the adjoint pair (L, F), and let e: FR → 1_{H} denote the counit of the adjoint pair (F, R), in the terminology of MacLane [3]. We shall refer to the 7-tuple (H, G, F, L, i, R, e) as a double adjoint situation.
Examples will be described in subsequent sections. In most cases F will be a forgetful functor, L will be a tensor-type functor such as – ⊗_{Uh} Ug and R will be a Hom-type functor such as Hom_{Uh}(Ug, –).
For the statement of Theorem 2. 1, below, we need the notions of coequalizers and epimorphic images (defined in Mitchell [4, pages 8, 12]). Module categories, and the category Set-G of Section 5.3, have epimorphic images; furthermore, in these categories, epimorphisms are coequalizers.
THEOREM 2.1 ON INJECTIVE WEAK DOUBLE ADJOINTS. Let (H, G, F, L, i, R, e) be a double adjoint situation, and let G be a category with epimorphic images, in which epimorphisms are coequalizers. If φ: L → R is a natural transformation with the property that
∀ W ∈ H, e_{W} ° Fφ ° i_{W} = 1_{W} | (4) |
then φ determines an injective weak double adjoint to F. Conversely, an injective weak double adjoint to F determines a natural transformation φ: L → R satisfying (4).
For the proof of this theorem, we need a Lemma.
LEMMA. Suppose
is a commutative diagram in a category in which epimorphisms are coequalizers, and that α is epi, β is monic. Then there is a unique ε: B → C such that the following diagram commutes:
(5)
Proof. Suppose α is the coequalizer of f, g as shown in the next diagram:
αf = αg ⇒ δαf = δαg ⇒ βγf = βγg ⇒ γf = γg since β is monic. Thus, by the universal property of coequalizers, there is a unique morphism ε: B →i>C such that εα = γ. Since εα = γ, βεα = βγ = δα, hence βε = δ since α is epi. Thus diagram (5) commutes.
Proof of theorem. Let (H, G, F, L, i, R, e) be as in the statement of the theorem.
First, suppose φ: L → R is a natural transformation satisfying condition (4), and let W ∈ H. Since G has epimorphic images, φ_{W} factorizes as φ_{W} = ν_{W} ° μ_{W}, with μ_{W} epi, ν_{W} monic:
Define an object function I: H → G by IW = im φ_{W}. Suppose W_{1}, W_{2} ∈ H and φ ∈ H(W_{1}, W_{2}) and consider the commutative diagram (6); this diagram can be regrouped as in diagram (7).
In this form, it can be seen that our lemma applies. Thus there exists a unique morphism, which we denote by Iψ: IW_{1} → IW_{2}, satisfying
μ_{W2} ° Lψ = Iψ ° μ_{W1} ν_{W2} ° Iψ = Rψ ° ν_{W1} | (8) |
The uniqueness property of Iψ makes it easy to verify that I is a functor from H to G, and then by (8), μ: L → I and ν: I → R are natural transformations.
Define j = Fμ ° i: 1_{H} → FI and d = e ° Fν: FI → 1_{H} . Then, for W ∈ H the following diagram commutes:
Next, for W ∈ H and V ∈ G, we define η_{VW}: G(V, IW) → H(FV, W) by η_{VW}(α) = d_{W} ° Fα for α ∈ G(V, IW) and θ_{WV}: G(IW, V) → H(W, FV) by θ_{WV}(β) = Fβ ° j_{W} for β ∈ G(IW, V). We claim that η_{VW} and θ_{WV} are injective, and natural in V and W. In fact, the naturality follows from that of j_{W} and d_{W}, in a trivial way.
Suppose that α_{1}, α_{2} ∈ G(V, IW) and that η_{VW}(α_{1}) = η_{VW}(α_{2}). That is, d_{W} ° Fα_{1} = d_{W} ° Fα_{2}. Since d_{W} = e_{W} ° Fν_{W}, it follows that
e_{W}
°
Fν_{W},
°
Fα_{1}
=
e_{W}
°
Fν_{W},
°
Fα_{2}
i.e.
e_{W}
°
F(ν_{W},
°
α_{1})
=
e_{W}
°
F(ν_{W},
°
α_{2})
Since, according to MacLane [3, page 80, theorem 1, part (ii)], the bijection G(V, RW) → H(FV, W) is given by χ → e_{W} ° Fχ the last equation implies that
ν_{W} ° α_{1} = ν_{W} ° α_{2}
But, by its definition, ν_{W} is monic. Thus α_{1} = α_{2}. Hence η_{VW} is injective. A similar argument shows that θ_{WV} is injective.
Finally, for W ∈ H one calculates that
η_{IW,W}(1_{IW}) = d_{W}
and
θ_{W,IW}(1_{IW}) = j_{W}
Inspection of the commutative diagram (9) shows that
d_{W} ° j_{W} = 1_{W}
that is, condition (3) is satisfied. Thus I is an injective weak double adjoint to F.
Now we prove the converse. Suppose that there exist natural injections
η_{VW}:
G(V, IW) → H(FV, W)
and
θ_{WV}:
G(IW, V) → H(W, FV)
for all W ∈ H and V ∈ G. Set d_{W} = η_{IW,W}(1_{IW}) and j_{W} = θ_{W,IW}(1_{IW}) and suppose, further, that for all W ∈ H, d_{W} ° j_{W} = 1_{W}. (Thus we are supposing that I is an injective weak double adjoint to F.)
F has left and right adjoints L and R respectively; let us denote the adjunction bijections by
λ_{WV}: H(W, FV)
→ G(LW, V)
and
ρ_{VW}: H(FV, W)
→ G(V, RW)
respectively.
By MacLane [3, page 80, theorem 1], for α ∈ G(LW, V), and β ∈ G(V, RW),
λ_{WV}^{-1} = Fα ° i_{W} | (10a) |
ρ_{VW}^{-1} = e_{W} ° Fβ | (10b) |
Now λ_{WV} ° θ_{VW}: G(IW, V) → H(W, FV) → G(LW, V) is injective. Set μ_{W} = λ_{W,IW}(θ_{W,IW}(1_{IW})). Using Yoneda's lemma (MacLane [3, page 61]), one can deduce that μ_{W} is natural in W, and that λ_{WV} ° θ_{VW}: = G(μ_{W}, V). Thus, by a result in MacLane [3, page 89, lemma], μ_{W} is epi.
By the definitions of μ_{W} and j_{W}, and Eq. (10a),
j_{W} = θ_{W,IW}(1_{IW}) = λ_{W,IW}^{-1}(μ_{W}) = Fμ_{W} ° i_{W}. | (11a) |
Similarly, setting ν_{W} = ρ_{IW,W}(η_{IW,W}(1_{IW})), we find that ν_{W} is monic, natural in W, and satisfies
d_{W} = η_{IW,W}(1_{IW}) = ρ_{IW,W}^{-1}(ν_{W}) = e_{W} ° Fν_{W}. | (11b) |
Thus φ = ν ° μ is a natural transformation from L to R, and, for all W ∈ H,
e_{W} ° Fφ_{W} ° i_{W} | = | (e_{W} ° Fν_{W}) ° (Fμ_{W} ° i_{W}) |
= | d_{W} ° j_{W} | |
= | 1_{W}. |
So φ: L → R satisfies condition (4). The theorem is proved.
2.2 INJECTIVE WEAK LEFT AND RIGHT ADJOINTS.
THEOREM 2.2. Let G and H be module categories and let F: G → H be a functor. Then a junctor L: H → G is an injective weak left adjoint to F if and only if there exists a natural transformation j: 1_{H} → FI such that
for all W ∈ H, V ∈ G,
χ ∈ G(IW, V), im j_{W} ⊆ ker Fχ ⇒ χ = 0 | (12) |
THEOREM 2.2 DUAL. Let G and H be module categories and let F: G → H be a functor. Then a functor L: H → G is an injective weak right adjoint to F if and only if there exists a natural transformation d: FI → 1_{H} Such that
for all W ∈ H, V ∈ G,
χ ∈ G(V, IW), im Fχ ⊆ ker d_{W} ⇒ χ = 0. | (12') |
Remarks. (12) can be restated as: im j_{W} generates FIW as G-object, while (12)' can be restated as: ker d_{W} contains no subobjects which are F-images of nonzero G-objects. Theorem 2.2 can be reformulated for non-preadditive categories by replacing condition (12) by
for all W ∈ H, V ∈ G, χ, χ' ∈ G(IW, V),
im j_{W} is a subobject of the equalizer of Fχ, Fχ' ⇒ χ = χ',
provided H has equalizers and images; and dually for Theorem 2.2 dual.
Proof of Theorem 2.2. Suppose I: H → G is an injective weak left adjoint to F, so that there is a natural injection θ_{WV}: G(IW, V) → H(W, FV) for all W ∈ H, V ∈ G. In the notation of Section 1.3, j_{W} = θ_{I,IW}(1_{IW}) is a natural transformation 1_{H} → FI. We must show that j_{W} satisfies condition (12). Let χ ∈ G(IW, V). By naturality of θ_{WV}, the following diagram commutes:
H(W,Fχ)(θ_{W,IW}(1_{IW})) = θ_{WV}(G(IW,χ)(1_{IW})),
or
Fχ ° j_{W} = θ_{WV}(χ).
Now, im j_{W} ⊆ ker Fχ ⇒ Fχ ° j_{W} = 0 ⇒ θ_{WV}(χ) = 0 ⇒ χ = 0 since θ_{WV} is injective. So (12) holds.
Conversely, suppose there is a natural transformation j: 1_{H} → FI satisfying condition (12). Define θ_{WV}: G(IW, V) → H(W, FV) by θ_{WV}(χ) = Fχ ° j_{W}, for W ∈ H, V ∈ G, χ ∈ G(IW, V). It is routine to check that θ_{WV} is natural in W and V. θ_{WV}(χ) = 0 ⇔ Fχ ° j_{W} = 0 ⇒ im j_{W} ⊆ ker Fχ ⇒ χ = 0 by condition (12).
The proof of Theorem 2.2 dual is similar to that of Theorem 2.2.
3. THE FACTORIZATION OF WALLACH'S FUNCTOR
3.1 As promised in Section 1.4, we now show that Wallach's functor I_{ω} described in Section 1.1, is an injective weak double adjoint to the forgetful functor F: Mod-g → Mod-h, where h ≤ g are Lie algebras with the decomposition g = n_{1} ⊕ h ⊕ n_{2} (see sect. 1. 1).
3.2 In the notation of Theorem 2.1, we shall set H = Mod-h, G = Mod-g, F as above, L = – ⊗_{Uh} Ug and R = Hom_{Uh}(Ug, –). To use Theorem 2. 1, we must define a natural transformation φ: L → R and show that φ satisfies condition (4), and that the functor I induced by φ is equivalent to Wallach's functor, I_{ω}. Let W ∈ Mod-h = H.
Define a map φ¯_{W}: W ⊗ Ug → Hom_{Uh}(Ug, W) by φ¯(w, g)(s) = w . γ(gs), where w ∈ W, g, s ∈ Ug, and γ: Ug → Ut are as in Section 1. 1. It is easy to check that φ¯_{W} is bilinear, and, for all h ∈ Uh, φ¯_{W}(wh, g)(s) = wh . γ(gs) = w . γ(hgs) = φ¯_{W}(w, hg)(s), so φ¯_{W} induces a map phi;_{W}: W ⊗_{Uh} Ug → Hom_{Uh}(Ug, W), given by φ_{W}(w ⊗ g)(s) = w . γ(gs). It is easy to check that φ_{W} is a Ug-homomorphism, and natural in W. We claim that φ: L → R satisfies condition (4) with respect to the double adjoint situation (Mod-h, Mod-g, F, – ⊗_{Uh}, Ug, i, Hom_{Uh}(Ug, –), e), where i and e are specified below.
The unit i_{W}: W → W ⊗_{Uh} Ug is given by i_{W}(w) = w ⊗ 1_{Ug}, (w ∈ W), and the counit e_{W}: Hom_{Uh}(Ug, W) → W is given by e_{W}(f) = f(1_{Ug}), for f ∈ Hom_{Uh}(Ug, W).
Thus, for w ∈ W, (e_{W} ° Fφ_{W} ° i_{W})(w) = e_{W}(Fφ_{W}(w ⊗ 1)) = [Fφ_{W}(w ⊗ 1)](1_{Ug}) = w . γ(1_{Ug}) = w, so condition (4) holds.
It now remains to show that the functor I, induced by φ according to the proof of Theorem 2. 1, is equivalent to I_{ω}. IW is defined to be im φw for W ∈ Mod-h. But, by inspection of the definition of φ_{W} above, and of j_hat_{W} in Section 1.1, we see that im φ_{W} = im j_hat_{W} . Ug. Thus IW = im j_hat_{W}. Ug = I_{ω}W, so the object functions of I and I_{ω} coincide.
By the lemma of Section 2 and the construction of I in the proof of Theorem 2.1, the morphism function of I is uniquely determined by the fact that if W_{1}, W_{2}, ∈ Mod-h, and ψ ∈ Hom_{Uh}(W_{1}, W_{2}), then Iψ is the unique Ug-homomorphism making the following diagram commute:
where μ_{W1}, μ_{W2}, ν_{W1}, ν_{W2} are maps defined as in the proof of Theorem 2.1.
It is routine to check that the diagram above actually does commute with I_{ω}ψ in place of Iψ, hence I_{ω}ψ = Iψ, and so I_{ω} = I. Thus I_{ω} is an injective weak double adjoint to F: Mod-g → Mod-h.
3.3 It is possible to calculate that j_{W} and d_{W}, the maps arising from φ_{W} in the proof of Theorem 2. 1, are given by j_{W}(w)(g) = w . γ(g) and d_{W}(f) = f(1_{Ug}) with w ∈ W, g ∈ Ug, f ∈ FIW.
It is also possible to generalize from the situation of universal enveloping algebras Uh, Ug with a projection γ: Ug → Uh, to the case of any ring R with 1 and subring S such that 1 ∈ S, with a ring retraction γ: R → S. Details are to be found in Wilson [9, Appendix]; for general associative algebras the process is discussed from another viewpoint by Trushin [6, Corollary 2.31].
4. ANOTHER INJECTIVE WEAK DOUBLE ADJOINT TO F: MOD-g → MOD-h
4.1 The functor J to be described in this section is in a sense dual to Wallach's functor. By the Poincare-Birkhoff-Witt theorem (Humphreys 12, page 92]), Ug = Ut ⊕ Ut . n_{2} . Un_{2} - a left Ut-, right Uh-module direct sum. Let γ': Ug → Ut denote the projection onto the first summand. Note that γ ≠ γ' of Section 1.1.
4.2 Construction of J. Define the Uh-homomorphism d_hat_{W}: W ⊗_{Ut} Ug → W by d_hat_{W}(w ⊗ g) = w . γ'(g). where W ∈ Mod-h, w ∈ W, g ∈ Ug. Consider the Uh-submodule ker d_hat_{W} of W ⊗_{Ut} Ug. Ker d_{W} contains a unique largest Ug-module, namely the sum of all Ug-modules contained in ker d_{W}. Call this largest Ug-module Y(W), and set J_{W} = (W ⊗_{Ut} Ug)/Y(W). We wish to make J into a functor from Mod-h to Mod-g, and show that JW can be embedded as a Ug-submodule of Hom_{Uh}(Ug, W).
To make J into a functor, we must define its action on Uh-homomorphisms. Let W, W¯ ∈ Mod-h, and let ψ ∈ Hom_{Uh}(W, W¯). Suppose w ∈ W, g ∈ Ug, so that w ⊗ g + Y(W) ∈ JW. We set (Jψ)(w ⊗ g + Y(W)) = ψ(w) ⊗ g + Y(W¯) and extend this definition to all of JW by linearity.
We must check that Jψ is well-defined. Note firstly that the map w ⊗ g → ψ(w) ⊗ g; W ⊗_{Ut} Ug → W¯ ⊗_{Ut} Ug is well-defined by functoriality of – ⊗_{Ut} Ug. Suppose that w_{1}, …, w_{n} ∈ W and g_{1}, …, g_{n} ∈ Ug, and that ∑_{i=1}^{n} w_{i} ⊗ g_{i} ∈ Y(W). Then, since Y(W) is a Ug-module contained in ker d_hat_{W}, for all x ∈ Ug,
0 | = | d_hat_{W}(∑ w_{i} ⊗ g_{i}x) | |
= | ∑w_{i} . γ'(g_{i}x) by definition of j_{W}. | ||
Hence | |||
0 | = | ψ(∑ w_{i} . γ'(g_{i}x)) | |
= | ∑(ψ(w_{i}) . γ'(g_{i}x)) | ||
= | d_hat_{W}(∑(ψ(w_{i}) ⊗ g_{i}x). |
That is, for all x ∈ Ug
[∑ ψ(w_{i}) ⊗ g_{i}] . x
∈ ker d_hat_{W}.
In other words, [∑ ψ(w_{i}) ⊗ g_{i}]
. Ug ⊆ ker d_hat_{W¯}
But this forces
∑_{i=1}^{n} ψ(w_{i})
⊗ g_{i} ∈ Y(W¯), so Jψ is
well-defined.
It is easy to check that J has the multiplicative property of a functor.
4.3 Embedding of JW in Hom_{Uh}(Ug, W). Define a map ν_{W}: JW → Hom_{Uh}(Ug, W) by ν_{W}(w ⊗ g + Y(W))(u) = w . γ'(gu) for w ∈ W, u, g ∈ Ug. If h ∈ Uh, then ν_{W}(w ⊗ g + Y(W))(uh) = w . γ'(guh) = w . γ'(gu) . h = [ν_{W}(w ⊗ g + Y(W))(u)] . h so im ν_{W} does consist of Uh-homomorphisms. Also, if x ∈ Ug, then ν_{W}(w ⊗ g + Y(W))x(u) = w . γ'(gxu) = ν_{W}(w ⊗ gx + Y(W))(u), so ν_{W} is a Ug-homomorphism. Finally, if ψ: W → W¯ is a Uh-homomorphism, then Hom_{Uh}(Ug, ψ) [ν_{W}(w ⊗ g + Y(W))](u) = ψ(w) . γ'(gu) = ν_{W}(Jψ(w ⊗ g + Y(W))), so ν_{W} is natural in W.
4.4 Proof that J is an injective weak double adjoint. We define μ_{W}: W ⊗_{Uh} Ug → JW by μ_{W}(w ⊗ g) = w ⊗ g + Y(W) for w ∈ W, g ∈ Ug. Again, this is easily seen to be a natural transformation. Set φ_{W} = ν_{W} ° μ_{W}.
φ: – ⊗_{Uh} Ug → Hom_{Uh}(Ug, –) is a natural transformation, and, for w ∈ W, e_{W}(Fφ_{W}(i_{W}(w))) = e_{W}(Fφ_{W}(w ⊗ 1_{Ug})) = w . γ'(1) = w, where e, i are as in Section 3.3, so e_{W} ° Fφ_{W} ° i_{W} = 1_{W}. By Theorem 2.1, φ induces an injective weak double adjoint to F: Mod-g → Mod-h, and easy calculations show that J is equivalent to the functor I constructed in the theorem. (Cf. sect. 3.1) Thus J is an injective weak double adjoint.5. AN APPLICATION TO PERMUTATION REPRESENTATIONS OF FINITE GROUPS
5.1 If H ≤ G are finite groups, k a field, and M a finite-dimensional right kH-module, then M ⊗_{kH} kG ≅ Hom_{kH}(kG, M) as kG-modules. However, this convenient property does not hold when we consider permutation representations of H and G. In this section we describe an injective weak double adjoint functor which serves as an alternative induction functor for permutation representations, provided that H is a retraction of G.
5.2 G-sets and induced G-sets. Let G be a group. A right G-set T is a pair (T, m), where T is a set and m: T × G → T is a map such that g → (t → m(t, g)) is a group homomorphism from G to the group of all permutations of T. We shall write t . g for m(t, g).
Let H < G. Dress [1, p. 43], has described induction for H-sets. Let Set-H, Set-G be the categories of all right H-sets and right G-sets respectively (with appropriate structure-preserving maps for morphisms). Let S be a right H-set. Define S ×_{H} G to be the set {s × g: s ∈ S, g ∈ G}, where s × g = {(s . h_{-1}, hg): h ∈ H}. We give S ×_{H} G the structure of a right G-set by setting (s × g) . g¯ = s × g.g¯ for g ∈ G. Define Hom_{H}(G, S) to be the set {f: G → S | ∀h ∈ H ∀g ∈ G f(gh) = f(g) . h}. We give Hom_{H}(G, S) the structure of a right G-set by setting f^{g¯}(g) = f(g.g¯) for g¯ ∈ G.
Clearly f ∈ Hom_{H}(G, S) is completely determined by its values on a set of left coset representatives for H in G. In fact, |Hom_{H}(G, S)| = |S|^{[H:G]}. Again, |s × g| =|H| for s ∈ S, g ∈ G, so |S ×_{H} G| = |S| . [G:H]. Thus, in general, |S ×_{H} G| ≠ |Hom_{H}(G, S)|, so these G-sets cannot be isomorphic. Let F: Set-G → Set-H be the usual forgetful functor. One can check that there are natural bijections
Set-G(S ×_{H} G, T) →
Set-H(S, FT)
and
Set-G(T, Hom_{H}(G, S)) →
Set-H(FT, S),
so we have a double adjoint situation as in Section 2.1.
5.3 Weak Induction from Set-H to Set-G. Suppose that there exists a group epimorphism. γ: G → H which is split by the inclusion H → G, (so γ(h) = h for h ∈ H). For each right H-set S, define a map φ_{S}: S ×_{H} G → Hom_{H}(G, S) by φ_{S}(s × g)(g¯) = s . γ(gg¯), for s ∈ S, g, g¯ ∈ G. If h ∈ H, φ_{S}(s x g)(g¯h) = s . γ(gg¯h) = s . γ(gg¯) . γ(h) = s . γ(gg¯) . h = φ_{S}(s × g)(g¯) . h, and φ_{S}(s . h^{-1} × hg)(g¯) = s h^{-1} . γ(hgg¯) = s h^{-1} . γ(h) . γ(gg¯) = s h^{-1} hγ(gg¯) = φ_{S}(s × g)(g¯) so φ_{S} is well-defined. If x ∈ G, then φ_{S}(s × gx)(g¯) = s . γ(gxg¯) = φ_{S}(s × g)^{x}(g¯) so φ_{S} is a G-homomorphism, and it is easy to see that φ_{S} is natural in S, cf. Section 4.3.
Let i_{S}: S → S ×_{H} G be given by i_{S}(s) = s × 1_{G} and let e_{S}: Hom_{H}(G, S) → S be given by e_{S}(f) = f(1_{G}). i_{S} and e_{S} are respectively the unit of the adjoint pair (– ×_{H} G, F), and the counit of the adjoint pair (F, Hom_{H}(G, –)). Then for any s ∈ S, (e_{S} ° Fφ_{S} ° i_{S})(s) = (Fφ_{S}(s × 1))(1_{G}) = s . γ(1 . 1) = s, so e_{S} ° Fφ_{S} ° is = 1_{S}. Thus, by Theorem 2. 1, we may conclude that φ induces an injective weak double adjoint functor I: Set-H → Set-G, and calculations show SIF = im j_hat_{S} . G, where j_hat_{S}: S → F Hom_{H}(G, S) is given by j_hat_{S}(s)(g) = s . γ(g), and im j_hat_{S} . G = {f^{x} : f ∈ im j_hat_{S}, x ∈ G}.
6. AN APPLICATION TO COREPRESENTATIONS OF COALGEBRAS
6.1 Let D be a coalgebra over k with D_{0} = Corad(D). Let D = D_{0} ⊕ I where I is a coideal, and suppose that C is a coalgebra with f: C → D a coalgebra surjection which splits, via g: D → C, as a left D- and right D_{0}-comodule map; thus f ° g = 1_{D}.
Let Comod-D_{0}, Comod-D and Comod-C denote the obvious categories of comodules. Trushin, in [6], has, inter alia, constructed a type of induced comodule functor Comod-D_{0} → Comod-C, along similar lines to Wallach's functor for Lie algebras (see [7], [8], and sects. 1, 3). In this section, we shall show that Trushin's functor is an injective weak double adjoint to the obvious forgetful functor F: Comod-C → Comod-D_{0}.
From now we shall assume some familiarity with chapters 1, 2 of Sweedler [5], and with the notation and results of Section 1 of Trushin's paper [6]. If W is a comodule we shall use φ_{W} to denote its structure map, and similarly ε_{C} and Δ_{C} will denote the counit and comultiplication of a coalgebra C. If V_{1}, …, V_{n} are vector spaces, and π ∈ S_{n}, the symmetric group on n symbols, then
τ_{π}: V_{1} ⊗ … ⊗ V_{n} → V_{1π} ⊗ ... ⊗ V_{nπ}
denotes the "twist" map permuting the tensorands. Let W ∈ Comod-D_{0}, let ι: D_{0} → D be the inclusion. With Trushin, we define ω: W → W ⊗^{D} C by ω = (1 ⊗ g)(1 ⊗ ι)ψ_{W}.
6.2 We set TW = C* . ω(W), (cf. Trushin [6, Definition 1.6]). If W_{1}, W_{2} are right D_{0}-comodules and h: W_{1} → W_{2} is a D_{0}-comodule map, then we set Th = (h ⊗ 1)|_{C*ω(W)}: TW_{1} → TW_{2}; Trushin has shown ([6, Proposition 1.5(ii)(a)]) that im Th ⊆ TW_{2}, it follows from this that T has the multiplicative property of a functor. We shall show that T is an injective weak double adjoint to F.
Define j_{W}: W → FTW by j_{W}(w) = ω(w), and d_{W}: FTW → W by d_{W} = μ_{W} ° (1 ⊗ ε_{C})|_{FTW}, where μ_{W}: W × k → W is the natural isomorphism. Now d_{W} ° j_{W} = μ_{W}(1 × ε_{C})ω, and, as Trushin remarks,
μ_{W}(1 × ε_{C})ω = μ_{W}(1 ⊗ ε_{C}g_{ι})ψ_{W} = μ_{W}(1 ⊗ ε_{D0})ψ_{W} = 1_{W},
so d_{W}, j_{W} satisfy condition (3) of Section 1.3. In order to prove that T is an injective weak double adjoint to F, it remains to prove (according to Theorems 2.2, 2.2 dual, and the remarks following them) that (a) im j_{W} generates TW as C*-module, (b) d and j are natural transformations, and (c) ker d_{W} contains no nonzero C*-modules.
(a) im j_{W} generates M by definition of TW.
(b) Naturality of d: if W_{1}, W_{2} are right D_{0}-comodules and h: W_{1} → W_{2} is a D_{0}-comodule map, then
d_{W2} ° FTh | = | μ_{W2}(1 ⊗ ε_{C})|_{FTW2} (h ⊗ 1)|_{FTW1} |
= | μ_{W2}(h ⊗ ε_{C})|_{FTW1} | |
= | h ° μ_{W1}(1 ⊗ ε_{C})|_{FTW1} by an easy calculation, | |
= | h ° d_{W1}, so d_{W} is natural in W. |
Naturality of j: j_{Wi} = (1 ⊗ gι) ψ_{Wi}, and FTh = h ⊗ 1, with domains and codomains suitably restricted. So j_{W2} ° h = (1 ⊗ gι)ψ_{W2}h = (1 ⊗ gι)(h ⊗ 1)ψ_{W1} by the comodule morphism property, = (h ⊗ 1)(1 ⊗ gι)ψ_{W1} = FTh ° j_{W1}, so j_{W} is natural in W.
(c) Proof that ker d_{W} contains no nonzero C*-modules: if ∑_{i} c_{i}^{*} . ω(m_{i}) is a nonzero element of a C*-module M contained in ker d_{W}, then for all c* ∈ C*,
0 | = | μ_{W}^{-1} d_{W} (c* . ∑_{i} c_{i}* . ω(m_{i})) |
= | (1 ⊗ ε_{C}) (∑_{i} c* c_{i}* . ω(m_{i})) | |
= | (1 ⊗ ε_{C}) (∑_{i} c* c_{i}* . ∑_{(mi)} m_{i(0)} ⊗ gι(m_{i(1)})) | |
= | (1 ⊗ ε_{C})
(1 ⊗ μ_{W})
(1 ⊗ 1 ⊗ <,>)τ_{(132)} × (1 ⊗ 1 ⊗ Δ_{C}) (∑_{i} c* c_{i}* ⊗ ∑_{(mi)} m_{i(0)} ⊗ gι(m_{i(1)})) | |
= | (1 ⊗ ε_{C})
(1 ⊗ μ_{W})
(1 ⊗ 1 ⊗ <,>)τ_{(132)} × (∑_{i} c* c_{i}* ⊗ ∑_{(mi)} m_{i(0)} ⊗ Δ_{C}gι(m_{i(1)})) | |
= | (1 ⊗ ε_{C})
(1 ⊗ μ_{W})
(1 ⊗ 1 ⊗ <,>)τ_{(132)} × (∑_{i} c* c_{i}* ⊗ ∑_{(mi)} m_{i(0)} ⊗ (gι ⊗ gι) Δ_{D0}(m_{i(1)})) | |
= | (1 ⊗ ε_{C})
(1 ⊗ μ_{W})
(1 ⊗ 1 ⊗ <,>)τ_{(132)} × (∑_{i} c* c_{i}* ⊗ ∑_{(mi)} m_{i(0)} ⊗ gι(m_{i(1)}) ⊗ gι (m_{i(2)})) | |
by a property of comodules | ||
= | (1 ⊗ ε_{C}) (1 ⊗ μ_{W}) (∑_{i} ∑_{(mi)} m_{i(0)} ⊗ gι(m_{i(1)}) ⊗ <c*c_{i}*, gι(m_{i(2)})>) | |
= | ∑_{i} ∑_{(mi)} m_{i(0)} ⊗ ε_{C}gι(m_{i(1)}) <c*c_{i}*, gι(m_{i(2)})> | |
= | ∑_{i} ∑_{(mi)} m_{i(0)} ⊗ <c*c_{i}*, gι(m_{i(1)})> | |
by properties of comodules and coalgebras | ||
= | ∑_{i} ∑_{(mi)} m_{i(0)} ⊗ <c* ⊗ c_{i}*, Δ_{C}gι(m_{i(1)})>. | |
Thus for all c* ∈ C*, | ||
0 | = | ∑_{i} ∑_{(mi)} m_{i(0)} ⊗ <c*, gι(m_{i(1)})> <c_{i}*, gι(m_{i(2)})> |
as gι is a coalgebra map.
Now a similar, but shorter, calculation shows that
∑_{i} c_{i}* . ω(m_{i}) = ∑_{i} ∑_{(mi)} m_{i(0)} ⊗ gι(m_{i(1)})<c_{i}*, gι(m_{i(2)})>.
Since Eq. (13) is true for all c* ∈ C*, it follows that ∑_{i} c_{i}*ω(m_{i}) = 0. Thus M = {0} and condition (c) is established.
So T is an injective weak double adjoint to F, as claimed.
Sections 1-4 formed part of the author's Ph.D. thesis at the University of Sydney. The author would like to express thanks to his supervisor, Dr D.W. Barnes, to Dr J.N. Ward for helpful advice and criticism of the material in these sections, and to the referee, for a suggestion which shortened the calculations in Section 6.2.