On Induced Representations of Lie Algebras, Groups and Coalgebras

Journal of Algebra, Vol. 58, No. 1, May 1979, pp. 37-50
William H. Wilson

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 n₁, n₂ and h of g such that g = n₁ ⊕ h ⊕ n₂, (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_ωW 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 = n₁ ⊕ h, and make W into a Ut-module by having n₁ act trivially on W. By the Poincare-Birkhoff-Witt theorem (cf. Humphreys [2, page 92]), we can write Ug = Ut ⊕ Un₂ . n₂ . 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_ωW = [im j_hat_W] . Ug. If ψ ∈ Hom_Uh(W₁, W₂), define I_ωψ by (I_ωψ)(f) = ψ ° f for f ∈ I_ωW₁ ⊆ Hom_Uh(Ug, W₁): it turns out (see [7]) that (I_ωψ)(f) ∈ I_ωW₂, 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_ωW, and im j_W, generates I_ωW as Ug-module.

(2) There is a natural injection Hom_Ug(V, I_ωW) → Hom_Uh(FV, W).

(3) I_ωW 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_ωW, (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 W₁, W₂ ∈ H and φ ∈ H(W₁, W₂) 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: IW₁ → IW₂, satisfying

μ_W2 ° Lψ = Iψ ° μ_W1             (8)

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 α₁, α₂ ∈ G(V, IW) and that η_VW(α₁) = η_VW(α₂). That is, d_W ° Fα₁ = d_W ° Fα₂. Since d_W = e_W ° Fν_W, it follows that

e_W ° Fν_W, ° Fα₁ = e_W ° Fν_W, ° Fα₂
i.e.
e_W ° F(ν_W, ° α₁) = e_W ° F(ν_W, ° α₂)

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 ° α₁ = ν_W ° α₂

But, by its definition, ν_W is monic. Thus α₁ = α₂. 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)

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 = n₁ ⊕ h ⊕ n₂ (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_ωW, 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 W₁, W₂, ∈ Mod-h, and ψ ∈ Hom_Uh(W₁, W₂), 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_ωψ in place of Iψ, hence I_ωψ = 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 . n₂ . Un₂ - 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 w₁, …, w_n ∈ W and g₁, …, g_n ∈ Ug, and that ∑_i=1ⁿ w_i ⊗ g_i ∈ Y(W). Then, since Y(W) is a Ug-module contained in ker d_hat_W, for all x ∈ Ug,

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=1ⁿ 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.

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 D₀ = Corad(D). Let D = D₀ ⊕ 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 D₀-comodule map; thus f ° g = 1_D.

Let Comod-D₀, 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-D₀ → 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-D₀.

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 V₁, …, V_n are vector spaces, and π ∈ S_n, the symmetric group on n symbols, then

τ_π: V₁ ⊗ … ⊗ V_n → V_1π ⊗ ... ⊗ V_nπ

denotes the "twist" map permuting the tensorands. Let W ∈ Comod-D₀, let ι: D₀ → 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 W₁, W₂ are right D₀-comodules and h: W₁ → W₂ is a D₀-comodule map, then we set Th = (h ⊗ 1)|_C*ω(W): TW₁ → TW₂; Trushin has shown ([6, Proposition 1.5(ii)(a)]) that im Th ⊆ TW₂, 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_Cg_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 W₁, W₂ are right D₀-comodules and h: W₁ → W₂ is a D₀-comodule map, then

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_W2h = (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*,

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.

ACKNOWLEDGMENTS

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.

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