On Induced Representations of Lie Algebras, Groups and Coalgebras

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 HomUh(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 n1, n2 and h of g such that g = n1hn2, (vector space direct sum), and [hni] ⊆ ni 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 IωW 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 = n1h, 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 = UtUn2 . n2 . Ut, and this is a left Uh-module, right Ut-module direct sum. Let V: UgUt be the corresponding projection. Define j_hatW: W → HomUt(Ug, W) ⊆ HomUh(Ug, W) by

[j_hatW(w)](s) = w . γ(s) for wW, s ∈ Ug.

j_hatW is a Uh-module monomorphism. Set IωW = [im j_hatW] . Ug. If ψ ∈ HomUh(W1, W2), define Iωψ by (Iωψ)(f) = ψ ° f for f ∈ IωW1 ⊆ HomUh(Ug, W1): it turns out (see [7]) that (Iωψ)(f) ∈ IωW2, and that Iω is a functor.

Wallach shows that his induced modules have various interesting properties, e.g.:

(0) Iω is a functor.

(1) There is an injection of Uh-modules jW: WFIωW, and im jW, generates IωW as Ug-module.

(2) There is a natural injection HomUg(V, IωW) → HomUh(FV, W).

(3) IωW is a submodule of HomUh(Ug, W). If W is a simple Uh-module of finite dimension over k, and HomUh(Ug, W) contains a finite-dimensional simple Ug-module V, then V = IωW, (whereas WUh Ug and HomUh(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 Iω 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: GH, I: HG be functors. We say that I is an injective weak left adjoint to F if for all WH, V ∈ G, there is an injection

  θWV: 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 WH and VG there is an injection

  ηVW: G(V, IW) → H(FV, W) (2)

natural in V and W.

Notation. Given natural injections (1) and (2) and WH, we denote by jW the morphism θW,IW(1IW) ∈ H(W, FIW), and we denote by dW the morphism ηIW,W(1IW) ∈ H(FIW, W). The naturality of (1) and (2) implies that j: 1HFI and d: FI → 1H 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

  WH       dW ° jW = 1W. (3)

1.4 In the terminology defined in Section 1.3,

PROPOSITION. Wallach's functor Iω: 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: GH be a functor. Suppose that F has a left adjoint L: HG and a right adjoint R: HG. Let i: 1HFL denote the unit of the adjoint pair (L, F), and let e: FR → 1H 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 HomUh(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: LR is a natural transformation with the property that

  WH,       eW ° Fφ ° iW = 1W         (4)

then φ determines an injective weak double adjoint to F. Conversely, an injective weak double adjoint to F determines a natural transformation φ: LR satisfying (4).

For the proof of this theorem, we need a Lemma.

LEMMA. Suppose

commutative diagram

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: BC such that the following diagram commutes:

commutative diagram                 (5)

Proof. Suppose α is the coequalizer of f, g as shown in the next diagram:

commutative 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: LR is a natural transformation satisfying condition (4), and let WH. Since G has epimorphic images, φW factorizes as φW = νW ° μW, with μW epi, νW monic:

commutative diagram

Define an object function I: HG by IW = im φW. Suppose W1, W2H and φ ∈ H(W1, W2) and consider the commutative diagram (6); this diagram can be regrouped as in diagram (7).

commutative diagram

In this form, it can be seen that our lemma applies. Thus there exists a unique morphism, which we denote by Iψ: IW1IW2, satisfying

 

μW2 ° Lψ = Iψ ° μW1

νW2 ° Iψ = Rψ ° νW1

(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: LI and ν: IR are natural transformations.

Define j = Fμ ° i: 1HFI and d = e ° Fν: FI → 1H . Then, for WH the following diagram commutes:

commutative diagram

Next, for WH and VG, we define ηVW: G(V, IW) → H(FV, W) by ηVW(α) = dW ° Fα for α ∈ G(V, IW) and θWV: G(IW, V) → H(W, FV) by θWV(β) = Fβ ° jW for β ∈ G(IW, V). We claim that ηVW and θWV are injective, and natural in V and W. In fact, the naturality follows from that of jW and dW, in a trivial way.

Suppose that α1, α2G(V, IW) and that ηVW1) = ηVW2). That is, dW ° Fα1 = dW ° Fα2. Since dW = eW ° FνW, it follows that

eW ° FνW, ° Fα1 = eW ° FνW, ° Fα2
i.e.
eW ° FW, ° α1) = eW ° FW, ° α2)

Since, according to MacLane [3, page 80, theorem 1, part (ii)], the bijection G(V, RW) → H(FV, W) is given by χ → eW ° Fχ the last equation implies that

νW ° α1 = νW ° α2

But, by its definition, νW is monic. Thus α1 = α2. Hence ηVW is injective. A similar argument shows that θWV is injective.

Finally, for WH one calculates that

ηIW,W(1IW) = dW
and
θW,IW(1IW) = jW

Inspection of the commutative diagram (9) shows that

dW ° jW = 1W

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

ηVW: G(V, IW) → H(FV, W)
and
θWV: G(IW, V) → H(W, FV)

for all WH and VG. Set dW = ηIW,W(1IW) and jW = θW,IW(1IW) and suppose, further, that for all WH, dW ° jW = 1W. (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

λWV: H(W, FV) → G(LW, V)
and
ρVW: H(FV, W) → G(V, RW)

respectively.

By MacLane [3, page 80, theorem 1], for α ∈ G(LW, V), and β ∈ G(V, RW),

  λWV-1 = Fα ° iW (10a)
  ρVW-1 = eW ° Fβ (10b)

Now λWV ° θVW: G(IW, V) → H(W, FV) → G(LW, V) is injective. Set μW = λW,IWW,IW(1IW)). Using Yoneda's lemma (MacLane [3, page 61]), one can deduce that μW is natural in W, and that λWV ° θVW: = GW, V). Thus, by a result in MacLane [3, page 89, lemma], μW is epi.

By the definitions of μW and jW, and Eq. (10a),

  jW = θW,IW(1IW) = λW,IW-1W) = FμW ° iW. (11a)

Similarly, setting νW = ρIW,WIW,W(1IW)), we find that νW is monic, natural in W, and satisfies

  dW = ηIW,W(1IW) = ρIW,W-1W) = eW ° FνW. (11b)

Thus φ = ν ° μ is a natural transformation from L to R, and, for all WH,

eW ° FφW ° iW (eW ° FνW) ° (FμW ° iW)
  dW ° jW
  1W.

So φ: LR 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: GH be a functor. Then a junctor L: HG is an injective weak left adjoint to F if and only if there exists a natural transformation j: 1HFI such that

  for all WH, VG, χ ∈ G(IW, V),
im jW ⊆ ker Fχ ⇒ χ = 0
(12)

THEOREM 2.2 DUAL. Let G and H be module categories and let F: GH be a functor. Then a functor L: HG is an injective weak right adjoint to F if and only if there exists a natural transformation d: FI → 1H Such that

  for all WH, VG, χ ∈ G(V, IW),
im Fχ ⊆ ker dW ⇒ χ = 0.
(12')

Remarks. (12) can be restated as: im jW generates FIW as G-object, while (12)' can be restated as: ker dW 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 WH, V G, χ, χ' ∈ G(IW, V),
im jW 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: HG is an injective weak left adjoint to F, so that there is a natural injection θWV: G(IW, V) → H(W, FV) for all WH, VG. In the notation of Section 1.3, jW = θI,IW(1IW) is a natural transformation 1HFI. We must show that jW satisfies condition (12). Let χ ∈ G(IW, V). By naturality of θWV, the following diagram commutes:

commutative diagram

H(W,Fχ)(θW,IW(1IW)) = θWV(G(IW,χ)(1IW)),

or

Fχ ° jW = θWV(χ).

Now, im jW ⊆ ker Fχ ⇒ Fχ ° jW = 0 ⇒ θWV(χ) = 0 ⇒ χ = 0 since θWV is injective. So (12) holds.

Conversely, suppose there is a natural transformation j: 1HFI satisfying condition (12). Define θWV: G(IW, V) → H(W, FV) by θWV(χ) = Fχ ° jW, for WH, VG, χ ∈ G(IW, V). It is routine to check that θWV is natural in W and V. θWV(χ) = 0 ⇔ Fχ ° jW = 0 ⇒ im jW ⊆ 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 Iω described in Section 1.1, is an injective weak double adjoint to the forgetful functor F: Mod-g → Mod-h, where hg are Lie algebras with the decomposition g = n1hn2 (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 = HomUh(Ug, –). To use Theorem 2. 1, we must define a natural transformation φ: LR and show that φ satisfies condition (4), and that the functor I induced by φ is equivalent to Wallach's functor, Iω. Let W ∈ Mod-h = H.

Define a map φ¯W: W ⊗ Ug → HomUh(Ug, W) by φ¯(w, g)(s) = w . γ(gs), where wW, g, sUg, and γ: UgUt are as in Section 1. 1. It is easy to check that φ¯W is bilinear, and, for all hUh, φ¯W(wh, g)(s) = wh . γ(gs) = w . γ(hgs) = φ¯W(w, hg)(s), so φ¯W induces a map phi;W: WUh Ug → HomUh(Ug, W), given by φW(wg)(s) = w . γ(gs). It is easy to check that φW is a Ug-homomorphism, and natural in W. We claim that φ: LR satisfies condition (4) with respect to the double adjoint situation (Mod-h, Mod-g, F, – ⊗Uh, Ug, i, HomUh(Ug, –), e), where i and e are specified below.

The unit iW: WWUh Ug is given by iW(w) = w ⊗ 1Ug, (wW), and the counit eW: HomUh(Ug, W) → W is given by eW(f) = f(1Ug), for f ∈ HomUh(Ug, W).

Thus, for wW, (eW ° FφW ° iW)(w) = eW(FφW(w ⊗ 1)) = [FφW(w ⊗ 1)](1Ug) = w . γ(1Ug) = 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 Iω. IW is defined to be im φw for W ∈ Mod-h. But, by inspection of the definition of φW above, and of j_hatW in Section 1.1, we see that im φW = im j_hatW . Ug. Thus IW = im j_hatW. Ug = IωW, so the object functions of I and Iω 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 ∈ HomUh(W1, W2), then Iψ is the unique Ug-homomorphism making the following diagram commute:

commutative diagram

where μW1, μW2, νW1, νW2 are maps defined as in the proof of Theorem 2.1.

It is routine to check that the diagram above actually does commute with Iωψ in place of Iψ, hence Iωψ = Iψ, and so Iω = I. Thus Iω is an injective weak double adjoint to F: Mod-g → Mod-h.

3.3 It is possible to calculate that jW and dW, the maps arising from φW in the proof of Theorem 2. 1, are given by jW(w)(g) = w . γ(g) and dW(f) = f(1Ug) with wW, gUg, fFIW.

It is also possible to generalize from the situation of universal enveloping algebras Uh, Ug with a projection γ: UgUh, to the case of any ring R with 1 and subring S such that 1 ∈ S, with a ring retraction γ: RS. 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 = UtUt . n2 . Un2 - a left Ut-, right Uh-module direct sum. Let γ': UgUt 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_hatW: WUt UgW by d_hatW(wg) = w . γ'(g). where W ∈ Mod-h, wW, gUg. Consider the Uh-submodule ker d_hatW of WUt Ug. Ker dW contains a unique largest Ug-module, namely the sum of all Ug-modules contained in ker dW. Call this largest Ug-module Y(W), and set JW = (WUt 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 HomUh(Ug, W).

To make J into a functor, we must define its action on Uh-homomorphisms. Let W, ∈ Mod-h, and let ψ ∈ HomUh(W, ). Suppose wW, gUg, so that wg + Y(W) ∈ JW. We set (Jψ)(w ⊗ g + Y(W)) = ψ(w) ⊗ g + Y() and extend this definition to all of JW by linearity.

We must check that Jψ is well-defined. Note firstly that the map wg → ψ(w) ⊗ g; WUt UgUt Ug is well-defined by functoriality of – ⊗Ut Ug. Suppose that w1, …, wnW and g1, …, gnUg, and that ∑i=1n wigiY(W). Then, since Y(W) is a Ug-module contained in ker d_hatW, for all xUg,

 0 =  d_hatW(∑ wigix)
   =  wi . γ'(gix)           by definition of jW.
Hence   
 0 =  ψ(∑ wi . γ'(gix))
   =  ∑(ψ(wi) . γ'(gix))
   =  d_hatW(∑(ψ(wi) ⊗ gix).

That is, for all xUg [∑ ψ(wi) ⊗ gi] . x ∈ ker d_hatW. In other words, [∑ ψ(wi) ⊗ gi] . Ug ⊆ ker d_hat But this forces ∑i=1n ψ(wi) ⊗ giY(), 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 HomUh(Ug, W). Define a map νW: JW → HomUh(Ug, W) by νW(wg + Y(W))(u) = w . γ'(gu) for wW, u, gUg. If hUh, then νW(wg + Y(W))(uh) = w . γ'(guh) = w . γ'(gu) . h = [νW(wg + Y(W))(u)] . h so im νW does consist of Uh-homomorphisms. Also, if x ∈ Ug, then νW(wg + Y(W))x(u) = w . γ'(gxu) = νW(wgx + Y(W))(u), so νW is a Ug-homomorphism. Finally, if ψ: W is a Uh-homomorphism, then HomUh(Ug, ψ) [νW(wg + Y(W))](u) = ψ(w) . γ'(gu) = νW(Jψ(w ⊗ g + Y(W))), so νW is natural in W.

4.4 Proof that J is an injective weak double adjoint. We define μW: WUh UgJW by μW(wg) = wg + Y(W) for wW, gUg. Again, this is easily seen to be a natural transformation. Set φW = νW ° μW.

φ: – ⊗Uh Ug → HomUh(Ug, –) is a natural transformation, and, for wW, eW(FφW(iW(w))) = eW(FφW(w ⊗ 1Ug)) = w . γ'(1) = w, where e, i are as in Section 3.3, so eW ° FφW ° iW = 1W. 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 HG are finite groups, k a field, and M a finite-dimensional right kH-module, then MkH kG ≅ HomkH(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 × GT is a map such that g → (tm(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: sS, gG}, where s × g = {(s . h-1, hg): hH}. We give S ×H G the structure of a right G-set by setting (s × g) . g¯ = s × g.g¯ for gG. Define HomH(G, S) to be the set {f: GS | ∀hHgG f(gh) = f(g) . h}. We give HomH(G, S) the structure of a right G-set by setting f(g) = f(g.g¯) for g¯ ∈ G.

Clearly f ∈ HomH(G, S) is completely determined by its values on a set of left coset representatives for H in G. In fact, |HomH(G, S)| = |S|[H:G]. Again, |s × g| =|H| for sS, gG, so |S ×H G| = |S| . [G:H]. Thus, in general, |S ×H G| ≠ |HomH(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, HomH(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: GH which is split by the inclusion HG, (so γ(h) = h for hH). For each right H-set S, define a map φS: S ×H G → HomH(G, S) by φS(s × g)() = s . γ(gg¯), for sS, g, G. If hH, φS(s x g)(g¯h) = s . γ(gg¯h) = s . γ(gg¯) . γ(h) = s . γ(gg¯) . h = φS(s × g)() . h, and φS(s . h-1 × hg)() = s h-1 . γ(hgg¯) = s h-1 . γ(h) . γ(gg¯) = s h-1 hγ(gg¯) = φS(s × g)() so φS is well-defined. If xG, then φS(s × gx)() = s . γ(gxg¯) = φS(s × g)x() so φS is a G-homomorphism, and it is easy to see that φS is natural in S, cf. Section 4.3.

Let iS: SS ×H G be given by iS(s) = s × 1G and let eS: HomH(G, S) → S be given by eS(f) = f(1G). iS and eS are respectively the unit of the adjoint pair (– ×H G, F), and the counit of the adjoint pair (F, HomH(G, –)). Then for any sS, (eS ° FφS ° iS)(s) = (FφS(s × 1))(1G) = s . γ(1 . 1) = s, so eS ° FφS ° is = 1S. 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_hatS . G, where j_hatS: SF HomH(G, S) is given by j_hatS(s)(g) = s . γ(g), and im j_hatS . G = {fx : f ∈ im j_hatS, xG}.

6. AN APPLICATION TO COREPRESENTATIONS OF COALGEBRAS

6.1 Let D be a coalgebra over k with D0 = Corad(D). Let D = D0I where I is a coideal, and suppose that C is a coalgebra with f: CD a coalgebra surjection which splits, via g: DC, as a left D- and right D0-comodule map; thus f ° g = 1D.

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 φW to denote its structure map, and similarly εC and ΔC will denote the counit and comultiplication of a coalgebra C. If V1, …, Vn are vector spaces, and π ∈ Sn, the symmetric group on n symbols, then

τπ: V1 ⊗ … ⊗ VnV ⊗ ... ⊗ Vnπ

denotes the "twist" map permuting the tensorands. Let W ∈ Comod-D0, let ι: D0D be the inclusion. With Trushin, we define ω: WWD C by ω = (1 ⊗ g)(1 ⊗ ι)ψW.

6.2 We set TW = C* . ω(W), (cf. Trushin [6, Definition 1.6]). If W1, W2 are right D0-comodules and h: W1W2 is a D0-comodule map, then we set Th = (h ⊗ 1)|C*ω(W): TW1TW2; Trushin has shown ([6, Proposition 1.5(ii)(a)]) that im ThTW2, 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 jW: WFTW by jW(w) = ω(w), and dW: FTWW by dW = μW ° (1 ⊗ εC)|FTW, where μW: W × kW is the natural isomorphism. Now dW ° jW = μW(1 × εC)ω, and, as Trushin remarks,

μW(1 × εC)ω = μW(1 ⊗ εCgιW = μW(1 ⊗ εD0W = 1W,

so dW, jW 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 jW generates TW as C*-module, (b) d and j are natural transformations, and (c) ker dW contains no nonzero C*-modules.

(a) im jW generates M by definition of TW.

(b) Naturality of d: if W1, W2 are right D0-comodules and h: W1W2 is a D0-comodule map, then

dW2 ° FTh  = μW2(1 ⊗ εC)|FTW2 (h ⊗ 1)|FTW1
  =  μW2(h ⊗ εC)|FTW1
  =  h ° μW1(1 ⊗ εC)|FTW1 by an easy calculation,
  = h ° dW1, so dW is natural in W.

Naturality of j: jWi = (1 ⊗ gι) ψWi, and FTh = h ⊗ 1, with domains and codomains suitably restricted. So jW2 ° h = (1 ⊗ gι)ψW2h = (1 ⊗ gι)(h ⊗ 1)ψW1 by the comodule morphism property, = (h ⊗ 1)(1 ⊗ gι)ψW1 = FTh ° jW1, so jW is natural in W.

(c) Proof that ker dW contains no nonzero C*-modules: if ∑i ci* . ω(mi) is a nonzero element of a C*-module M contained in ker dW, then for all c* ∈ C*,

0  =  μW-1 dW (c* . ∑i ci* . ω(mi))
   =  (1 ⊗ εC) (∑i c* ci* . ω(mi))
   =  (1 ⊗ εC) (∑i c* ci* . ∑(mi) mi(0)gι(mi(1)))
   =  (1 ⊗ εC) (1 ⊗ μW) (1 ⊗ 1 ⊗ <,>)τ(132)
× (1 ⊗ 1 ⊗ ΔC) (∑i c* ci* ⊗ ∑(mi) mi(0)gι(mi(1)))
   =  (1 ⊗ εC) (1 ⊗ μW) (1 ⊗ 1 ⊗ <,>)τ(132)
× (∑i c* ci* ⊗ ∑(mi) mi(0) ⊗ ΔCgι(mi(1)))
   =  (1 ⊗ εC) (1 ⊗ μW) (1 ⊗ 1 ⊗ <,>)τ(132)
× (∑i c* ci* ⊗ ∑(mi) mi(0) ⊗ (gι ⊗ gι) ΔD0(mi(1)))
   =  (1 ⊗ εC) (1 ⊗ μW) (1 ⊗ 1 ⊗ <,>)τ(132)
× (∑i c* ci* ⊗ ∑(mi) mi(0)gι(mi(1)) ⊗ gι (mi(2)))
    by a property of comodules
   =  (1 ⊗ εC) (1 ⊗ μW) (∑i(mi) mi(0)gι(mi(1)) ⊗ <c*ci*, gι(mi(2))>)
   =  i(mi) mi(0) ⊗ εCgι(mi(1)) <c*ci*, gι(mi(2))>
   =  i(mi) mi(0) ⊗ <c*ci*, gι(mi(1))>
    by properties of comodules and coalgebras
   =  i(mi) mi(0) ⊗ <c* ⊗ ci*, ΔCgι(mi(1))>.
Thus for all c* ∈ C*,
0  =  i(mi) mi(0) ⊗ <c*, gι(mi(1))> <ci*, gι(mi(2))>

as gι is a coalgebra map.

Now a similar, but shorter, calculation shows that

i ci* . ω(mi) = ∑i(mi) mi(0)gι(mi(1))<ci*, gι(mi(2))>.

Since Eq. (13) is true for all c* ∈ C*, it follows that ∑i ci*ω(mi) = 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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