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arXiv:1701.00020v2 [math.OA] 19 Apr 2017
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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RALF MEYER AND SUTANU ROY
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Abstract. We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product multiplicative unitary have the same Hilbert space representations. We also show that the multiplicative unitaries associated to two regular objects for the same tensor category are equivalent and hence generate isomorphic C-quantum groups. In particular, a C-quantum group is determined uniquely by its tensor category of representations on Hilbert space, and any functor between representation categories that does not change the underlying Hilbert spaces comes from a morphism of C-quantum groups.
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1. Introduction
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The TannakaKrein Theorem by Woronowicz [11] recovers a compact quantum group from its tensor category of finite-dimensional representations, together with the forgetful functor to the tensor category of Hilbert spaces. We shall prove an analogue of this result for C-quantum groups, that is, quantum groups generated by manageable multiplicative unitaries. Our result asserts that an isomorphism between the tensor categories of Hilbert space representations that does not change the underlying Hilbert spaces lifts to an isomorphism of the underlying Hopf -algebras. More generally, we shall explain how to extract multiplicative unitaries from representation categories and how to lift tensor functors between representation categories to morphisms of multiplicative unitaries.
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This article grew out of a suggestion by David Bücher to clarify the construction of a semidirect product multiplicative unitary from a braided multiplicative unitary in [6,9]. A braided multiplicative unitary is supposed to describe a braided C-quantum group, which should be a YetterDrinfeld algebra over some other C-quantum group, equipped with a comultiplication B B B into its YetterDrinfeld twisted tensor square. The semidirect product is constructed in [6, 9] by writing down a unitary and checking that it is multiplicative. The data of a braided multiplicative unitary consists of four unitaries, subject to seven conditions. All four unitaries must appear in the explicit formula, and all seven conditions must be used in the proof that the semidirect product is multiplicative. Thus the direct verification in [6] is rather complicated. Here we offer a conceptual explanation for this construction.
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The main idea behind this is the theory of regular objects in tensor categories by Pinzari and Roberts [8]. We prefer to call them natural right absorbers because the adjective "regular" is already used for too many other purposes. A natural right absorber in C gives rise to a multiplicative unitary W and a tensor functor from C
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2000 Mathematics Subject Classification. 46L89 (81R50 18D10). Key words and phrases. quantum group, braided quantum group, multiplicative unitary, braided multiplicative unitary, tensor category, quantum group representation, quantum group morphism, TannakaKrein Theorem. Dedicated to Professor Shôichirô Sakai. The second author was partially supported by an INSPIRE faculty award given by D.S.T., Government of India.
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1
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2
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RALF MEYER AND SUTANU ROY
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to the tensor category of Hilbert space representations of W. Representations of the semidirect product multiplicative unitary should be equivalent to representations of the braided multiplicative unitary. This idea already appears in a special case in [1]. Here we extend this result to the general case. Starting with a braided multiplicative unitary, we define its representation category and describe a natural right absorber in it by combining two rather obvious pieces. The corresponding multiplicative unitary turns out to be the semidirect product. We also show that the functor from representations of the braided multiplicative unitary to representations of the semidirect product is an isomorphism of categories. The most difficult point here is to prove that any representation of the semidirect product comes from a representation of the braided multiplicative unitary.
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The semidirect product comes with a projection, which is another multiplicative unitary linked to it by pentagon-like equations. We interpret this projection through a projection on the representation category. More generally, we show that any tensor functor between representation categories that does not change the underlying Hilbert spaces lifts to a morphism between the associated multiplicative unitaries as defined in [3,7]. This also implies the weak TannakaKrein Theorem for C-quantum groups mentioned above. And it gives yet another equivalent description of quantum groups with projection.
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2. Natural right absorbers in Hilbert space tensor categories
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We are going to recall the notion of a (right) regular object of a tensor category from [8]. We call such an object a natural right absorber, avoiding the overused adjective "regular". Going beyond [8], we show that different natural right absorbers give isomorphic multiplicative unitaries with respect to the morphisms of C-quantum groups defined in [3,7]. We also add a further equivalent description of such quantum group morphisms through functors between representation categories, and we show that isomorphic multiplicative unitaries generate isomorphic C-quantum groups.
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Notation 2.1. Let Hilb denote the W-category of Hilbert spaces. This is a symmetric monoidal category for the usual tensor product of Hilbert spaces, with the obvious associator (H1 H2) H3 = H1 (H2 H3), the obvious unit transformations C H = H = H C, and the obvious symmetric braiding
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: H1 H2 H2 H1, x1 x2 x2 x1.
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Let C be a W-category with a faithful forgetful functor For : C Hilb. Faithfulness allows us to assume that C(x1, x2) B(For(x1), For(x2)) for all objects x1, x2 C (we write for objects of categories, for arrows). We say that a B(For(x1), For(x2)) comes from C if it belongs to C(x1, x2). We think of objects in C as Hilbert spaces with some extra structure, such as a representation of a (braided) multiplicative unitary; the morphisms are those bounded linear maps that preserve this extra structure. Motivated by this interpretation, we assume the following throughout this article:
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Assumption 2.2. If For(x) = For(x ) and the identity map on this Hilbert space comes from an arrow x x , then x = x .
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We also want a functor : Hilb C with For = idHilb. Thus acts as the identity on arrows, and the arrows (H1) (H2) in C are exactly all bounded linear operators H1 H2. We abbreviate x := For(x) for x C. We interpret as the functor that equips a Hilbert space H with the "trivial" extra structure to get an object in C. The existence of is a very weak assumption, which follows, for instance, if C is monoidal and has direct sums.
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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3
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We assume that C is also a monoidal category, but not necessarily braided, such that both For and are strict monoidal functors. This means, first, that For(x1 x2) = For(x1)For(x2) for all x1, x2 C and (H1 H2) = (H1) (H2) for all H1, H2 Hilb. Secondly, that the tensor unit in C is C, which For maps back to the tensor unit in Hilb. Thirdly, For and map associators and unit transformations in C to the obvious associators and unit transformations in Hilb. Finally, we require the following assumption, which is trivial to check in all cases we shall consider:
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Assumption 2.3. Let x1, x2, y C and let a : For(x1) For(x2) be such that a id comes from an arrow x1 y x2 y in C or id a comes from an arrow y x1 y x2 in C. Then a itself comes from an arrow x1 x2 in C.
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Definition 2.4. A Hilbert space tensor category is a monoidal W-category C with a faithful, strict monoidal functor For : C Hilb and a strict monoidal functor : Hilb C satisfying For = idHilb and Assumptions 2.2 and 2.3.
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Example 2.5. Let W U(H H) be a multiplicative unitary. Let Rep(W) be the W-category of its (right) Hilbert space representations, with intertwiners as arrows. That is, the objects are pairs (K, U) where K is a Hilbert space and U U(K H) satisfies W23U12 = U12U13W23 in U (KHH). The arrows (K1, U1) (K2, U2) are operators a B(K1, K2) with U2a1 = a1U1, where a1 := aidH in the leg numbering notation. The forgetful functor Rep(W) Hilb forgets the representation, and (K) := (K, 1). The tensor product of two representations Ui U(Ki H), i = 1, 2, is U1 U2 := U113U223 U (K1 K2 H). Quick computations show that this is again a representation, that is associative, and that (C) is a tensor unit, with the usual associator and unit transformations from Hilb. Since an operator of the form a1 B(K1 K2) for a B(K1) commutes with U223, it is an intertwiner for U113U223 if and only if a is one for U1. Hence Assumption 2.3 holds. Assumption 2.2 holds because our objects are indeed Hilbert spaces with extra structure.
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Lemma 2.6. Let x1, x2 C, H Hilb. Then an operator a : For(x1) H For(x2) comes from an arrow a^ C(x1 H, x2) if and only if the operators a : For(x1) For(x2), a( ), come from arrows in C(x1, x2) for all H. Analogous statements hold for operators H For(x1) For(x2), For(x1) For(x2) H, For(x1) H For(x2).
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Proof. An arrow a^ C(x1 H, x2), gives arrows a^ in C(x1, x2) with For(a^) = a by taking a^ := a^ idx1 (| ) , where | : C H, , and where we implicitly identify x1 = x1 C.
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For the converse, let (Pi)iI be a net of finite-rank projections converging weakly to the identity operator on H. Then the endomorphisms idx1 (Pi) of x1 H still converge weakly to the identity operator. Since C(x1 H, x2) is weakly closed, it suffices to lift a (idx1 (Pi)) to an arrow in C for all i and then take a limit. Writing the finite-rank projection Pi as a sum of rank-1 projections, we further reduce to the lifting of an operator of the form a | | for a unit vector H. This is lifted by the following composite in C:
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x1 (H) -i-d--(--|) x1 (C) = x1 -a^ x2.
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Remark 2.7. The functor is unique if it exists. Let H be a Hilbert space. Then any bounded linear operator C H comes from an arrow (C) (H) in C. Conversely, let x be an object of C with For(x) = H such that any bounded linear map C H comes from an arrow C x. Hence the identity map (H) x comes from an arrow in C by Lemma 2.6. Then (H) = x by Assumption 2.2.
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4
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RALF MEYER AND SUTANU ROY
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Given objects x1, x2, x3, y1, y2, y3 C, there are canonical maps
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C(x1 x2, y1 y2) C(x1 x2 x3, y1 y2 x3), C(x2 x3, y2 y3) C(x1 x2 x3, x1 y2 y3),
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T T12 = T idx3 , T T23 = idx1 T.
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An arrow T13, however, cannot always be defined because this would require a braiding on C. Nevertheless, the operator T13 may be defined if the object in the middle is of the form H. Lemma 2.6 implies that the flip operator
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: For(x) H H For(x), ,
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comes from an arrow in C(x H, H x) for all x C, H Hilb. We use these arrows in C to define
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C(x1 x2, y1 y2) C(x1 H x2, y1 H y2), T T13 := 23T1223 = 12T2312.
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Definition 2.8. Let C be a Hilbert space tensor category as above. A natural right absorber in C is an object C together with unitaries
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U x : x (x) for all x C
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with the following properties: (2.8.1) the unitaries U x are natural in the sense that the following diagram com-
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mutes for any arrow a C(x1, x2), x1, x2 C:
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x1
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U x1 =
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(x1)
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aid
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aid= (a)id
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x2
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U x2 =
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(x2)
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(2.8.2) for all x1, x2 C, the following diagram of unitaries commutes:
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x1 x2 Ux1x2 (x1 x2)
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U2x32
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x1 (x2) U1x31 (x1) (x2)
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Lemma 2.9. If and (U x)xC are a natural right absorber for C, then U (H) = id(H) for any Hilbert space H.
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Proof. Assumption (2.8.2) for x1 = x2 = C = (C) implies U (C) = idC. Any vector H gives an arrow | : (C) (H). The naturality assumption (2.8.1) applied to these arrows gives U (H)( ) = for all H, For(). Thus U (H) = id.
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Example 2.10. Let W be a multiplicative unitary and let C = Rep(W) as in Example 2.5. The pentagon equation says that the unitary W is also a representation of itself. A unitary U U(K H) is a representation if and only if it is an intertwiner
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(K H, U13W23) = (K H, U W) (K H, idK W) = (K H, W23).
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We claim that W with the family of arrows U : (K, U) (H, W) (K, idK) (H, W) is a natural right absorber in Rep(W). First, the arrows in Rep(W) are exactly those operators for which the arrows U above are natural. Secondly, the tensor product of two representations is defined exactly so as to verify (2.8.2).
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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5
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Proposition 2.11 ([8, Theorem 2.1]). Let (C, For, , ) be a Hilbert space tensor category and let and (U x)xC be a natural right absorber for C. For x C, let Hx := For(x), and let us also write U x for For(U x) U (Hx H). Then U is a multiplicative unitary, and U x for x C is a right representation of U . This
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construction gives a fully faithful, strict tensor functor from C to the tensor category Rep(U ) of representations of the multiplicative unitary U , which intertwines the forgetful functors from C and Rep(U ) to Hilb.
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Proof. The condition (2.8.2) and Lemma 2.9 give U x(H) = U1x3 : x (H) (x) (H) .
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Let x C. Then U x C(x , x ) is an intertwiner. So we may apply naturality to it. This and condition (2.8.2) give the commuting diagram of unitaries
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(x) U1x2 x U23 x ()
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U23=U x
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(x) () U1x2
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U x
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(x )
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U1x3
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(x) ()
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That is, U1x2U1x3U23 = U23U1x2. When we take x = , this is the pentagon equation for U . For general x, it says that U x is a right representation of U .
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The naturality of U x says that arrows x1 x2 in C are intertwiners U x1 U x2 . To prove that we have a fully faithful functor, we must show the converse. So let a : Hx1 Hx2 be an intertwiner U x1 U x2 . Then we get an arrow
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x1
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-U-x1
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(x1)
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--(-a-)-i-d
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(x2)
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(U x2 )-1
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-----
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x2
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Since a is an intertwiner, the forgetful functor maps this composite arrow to a idH . Since this operator comes from C, Assumption 2.3 ensures that a also comes from C.
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Thus any intertwiner comes from an arrow in C. This finishes the proof that the functor from C to the category of right representations of U is fully faithful. By
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construction, our functor intertwines the forgetful functors to Hilb. The condition (2.8.2) says exactly that U x1x2 is the tensor product representation
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U x1 U x2 . Since we assumed For to map the associator and unit transformations in C to the usual ones in Hilb, the functor x U x from C to the representation category of U is a strict tensor functor.
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We have not found a "nice" characterisation when the functor C Rep(U ) is essentially surjective, that is, when every representation of U comes from an
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object of C. An artificial example where this is not the case is the subcategory of Rep(U ) consisting of all representations that are either trivial or a direct sum of
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subrepresentations of . This has all the structure that we require. And it is also closed under direct sums and subrepresentations. If Rep(U ) is, say, the category
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of representations of the group Z of integers, then the representations given by non-trivial characters on Z are missing in this subcategory.
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Example 2.12. Let W U(H H) be a multiplicative unitary. A left representation of W on a Hilbert space K is a unitary V^ U(H K) satisfying
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V^ 23W12 = W12V^ 13V^ 23 U (H H K). The tensor product of two left representations V^ i U (H Ki), i = 1, 2, is the left representation on K1 K2 defined by
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V^ 1 V^ 2 := V^ 213V^ 112 U (H K1 K2).
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Left representations of W also form a Hilbert space tensor category with the obvious forgetful functor and (H) = (H, 1). Actually, this tensor category is isomorphic to
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6
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RALF MEYER AND SUTANU ROY
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the category of right representations of the dual multiplicative unitary W = W: the isomorphism takes a left representation V^ U(K H) to the right representation V^ U(HK) of W. Since W is a natural right absorber for right representations
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of W by Example 2.10, the unitary W, viewed as a left representation, is a natural right absorber in the tensor category of left representations of W. The natural intertwiner is
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V^ : (K H, V^ W) (K H, 1K W).
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Next we want to prove that the multiplicative unitaries for two natural right absorbers of C are isomorphic in the category of multiplicative unitaries introduced in [7] and further studied in [3].
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Proposition 2.13. Let (, (U x)xC) and (, (U x)xC) be two natural right absorbers for (C, For). Let H := H, H := H, U := U U (H H), U := U
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U(H H) be the corresponding multiplicative unitaries. The unitaries
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V := U U (H H), W := U U (H H)
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satisfy the following pentagon-like equations:
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U23V12 = V12V13U23, V23U12 = U12V13V23, V23W12 = W12U13V23,
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U23W12 = W12W13U23, W23U12 = U12W13W23, W23V12 = V12U13W23.
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If the multiplicative unitaries U and U are manageable, then V and W give morphisms between the corresponding C-quantum groups that are inverse to each other in the category of C-quantum groups defined in [3].
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Proof. Our assumptions are symmetric in (, U ) and (, U ). When we exchange
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them, the equations in the first column become the corresponding ones in the second
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column. So it suffices to prove those in the first column. We already know that V = U is a right representation of U , which gives the first equation. The other
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equations are proved similarly. For the second equation, we use the naturality of U for the intertwiner U : () and rewrite U = U13U23 = V13V23 and U () = U23 = V23. For the third equation, we use the naturality of U for the intertwiner W : () and rewrite U = U13U23 = U13V23 and U () = U23 = V23.
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Morphisms of quantum groups are described in [3, Lemma 3.2]. The equations U23V12 = V12V13U23 and V23U12 = U12V13V23 say that V is a morphism from U to U . The equations U23W12 = W12W13U23 and W23U12 = U12W13W23 say that W is a morphism from U to U . The product of two morphisms is defined in [3, Definition 3.5]
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as the solution of a certain operator equation. The equation V23W12 = W12U13V23 says that the product of V and W is U . The equation W23V12 = V12U13W23 says that the product of W and V is U . Manageability is needed in [3] to ensure that
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the equation in [3, Definition 3.5] always has a solution. So manageability is needed
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to talk about a category of morphisms between multiplicative unitaries.
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Example 2.14. Let (, U ) be a natural right absorber for C and let y C. Then = y with U x := U x idy for all x C is a natural right absorber as well. The corresponding multiplicative unitary is
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(2.1)
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U y = (U y)123 = U13U2y3 U (H Hy H Hy).
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Proposition 2.13 shows that U and U y are isomorphic multiplicative unitaries
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when they are both manageable, compare [6, Theorem 3.7].
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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7
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We now extend the analysis above to describe functors between representation categories. Let C1 and C2 be Hilbert space tensor categories with natural right absorbers (1, U1) and (2, U2), respectively. Let : C1 C2 be a strict tensor functor with For2 = For1. If C1 and C2 are representation categories, then this means that turns a representation of one sort into one of the other on the same Hilbert space in a natural way and preserving tensor products. Such a functor also satisfies 1 = 2 by the argument in Remark 2.7.
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Proposition 2.15. Let : C1 C2 be a strict tensor functor with For2 = For1. The unitary V := U2(1) U (H1 H2 ) satisfies
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(U22 )23V12 = V12V13(U22 )23,
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V23(U11 )12 = (U11 )12V13V23,
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that is, V is a bicharacter from U11 to U22 . Moreover, for any x C1,
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(2.2)
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V23(U1x)12 = (U1x)12(U2(x))13V23 U (Hx H1 H2 ).
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If the multiplicative unitary U11 is manageable and C2 = Rep(U22 ), then the map from functors : C1 C2 as above to unitary bicharacters V U (H1 H2 ) from U11 to U22 is bijective. If the multiplicative unitaries U11 and U22 are both manageable, then V is a morphism between the corresponding C-quantum groups in the category defined in [3].
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Proof. The first two equations in the proposition say that V is a morphism of C-quantum groups as in [3, Lemma 3.2] provided the multiplicative unitaries U11 and U22 are manageable, so that they generate C-quantum groups. We already know that V is a right representation of U22 , which is the first equation. The second equation is the special case x = 1 of the third one. The third equation says that the functor on representation categories induced by V is , as expected. We
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prove this third equation by identifying
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(x 1) = (x) (1), ( (x) 1) = ((x)) (1),
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U2(x1) = (U2(x))13V23, U2( (x)1) = (U2(1))23 = V23,
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and using the naturality of U2 for the intertwiner (U1x) : (x 1) ( (x) 1). This gives (2.2). This is equivalent to (U2(x))13 = (U1x)12V23(U1x)12(V23), which determines the object (x) of C2 by Proposition 2.11. This describes how acts on
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objects. Then its action on arrows is determined by the faithful forgetful functor to Hilbert spaces. So V determines the functor .
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Now assume that U11 is manageable. Let V U (H1 H2 ) be a bicharacter. Any bicharacter induces a functor between the representation categories by
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[3, Proposition 6.5]. The proof of this proposition does not describe this functor
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explicitly. An explicit formula for is similar to the formula for the composition
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of bicharacters, which is a special case. Namely, let x C1. As in the proof of [3, Lemma 3.6], manageability shows that there is a unitary operator U2(x) that verifies (2.2); moreover, U2(x) is a representation of U22 , and there is a unique functor : C1 Rep(U22 ) with For = For that sends x C to this representation and that acts by the identity map on arrows, viewed as Hilbert space operators. This
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functor is a strict tensor functor. Any functor is of this form for the corresponding bicharacter V . This gives the desired bijection.
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Proposition 2.15 gives yet another equivalent characterisation of the quantum
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group morphisms of [3]: they are equivalent to strict tensor functors between the
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representation categories with For = For. This result is similar in spirit to [3, Theorem 6.1], which uses coactions on C-algebras instead of representations.
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8
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RALF MEYER AND SUTANU ROY
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2.1. Left and right absorbers. A natural left absorber in C is defined like a natural right absorber, but on the other side:
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Definition 2.16. A natural left absorber in C is an object C with unitaries
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Ux : x (x) for all x C
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with the following properties:
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(2.16.1) the unitaries Ux are natural in the sense that the following diagram commutes for any arrow a : x1 x2:
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x1
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Ux1 =
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(x1)
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id a
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id a
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x2
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Ux2 =
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(x2)
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(2.16.2) for all x1, x2 C, the following diagram commutes:
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x1 x2 Ux1x2 (x1 x2)
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(Ux1 )12
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(x1) x2 (Ux2 )13 (x1) (x2)
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The analogue of Lemma 2.9 holds for natural left absorbers as well, that is, U(H) = id(H) for any Hilbert space H.
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Let W be a multiplicative unitary. Then the categories of left and of right representations of W have a canonical natural right absorber by Examples 2.10 and 2.12. It is unclear, in general, whether they have a natural left absorber as well. The only construction of left absorbers that we know uses the contragradient operation to turn a right into a left absorber. For contragradients to exist, we assume W to be manageable. We work with right representations of W. The contragradient of a representation U on a Hilbert space H is a representation U on the complex-conjugate Hilbert space H. The contragradient construction becomes a covariant functor Rep(W) Rep(W) when we map an intertwiner a : H1 H2 to a : H1 H2. This is not quite a W-functor because it is conjugate-linear, not linear. The contragradient of a trivial representation remains trivial. The
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contragradient operation is involutive, that is, U = U for representations and a = a for intertwiners. It reverses the order of tensor factors: the flip operator : H1 H2 H2 H1 = H2 H1 intertwines U1 U2 with the contragradient of U1 U2, see [10, Section 3].
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Let (, (U x)xRep(W)) be a natural right absorber for Rep(W). For instance, we may take the canonical one described in Example 2.10. Let := be the contragradient of , so = . Let Ux : x () x for x Rep(W) be the composite unitary intertwiner
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x = x - x~ ~ -U-x~ (x~) = (x) - (x) = (x).
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Routine computations show that (, (Ux)) is a natural left absorber if (, (U x)) is a natural right absorber. This proves the following:
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Proposition 2.17. Let W be a multiplicative unitary. If W is manageable, then its tensor category of representations Rep(W) contains both a natural right absorber and a natural left absorber.
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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9
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If C has both a right absorber and a left absorber , then
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() = = ().
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That is, the direct sums of infinitely many copies of and are isomorphic. This common direct sum is both a left and a right absorber, and its isomorphism class does not depend on the choice of or . These observations go back already to [8], and they need only absorption, without any naturality. We are going to use the uniqueness of two-sided absorbers to prove that any isomorphism between the representation categories of two C-quantum groups comes from an isomorphism of Hopf -algebras. First we need a preparatory result, which would really belong into [3], but was not proved there.
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Theorem 2.18. The isomorphisms in the category of C-quantum groups defined in [3] are the same as the Hopf -isomorphisms of the underlying C-bialgebras.
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Proof. It is trivial that a Hopf -isomorphism induces an isomorphism in the category of [3]. Conversely, an isomorphism between two C-quantum groups (Ci, Ci ), i = 1, 2, in the category of [3] only gives a Hopf -isomorphism between their universal dual quantum groups C^1u = C^2u (or C1u = C2u, but we shall use the dual isomorphism below). For locally compact quantum groups with Haar weights, an isomorphism C1u = C2u implies a Hopf -isomorphism between (C1, C1 ) and (C2, C2 ) because the invariant weights on C1u = C2u are unique, see [2, p. 873]. We shall generalise this to C-quantum groups generated by manageable multiplicative unitaries. The Hopf -isomorphism C^1u = C^2u induces an isomorphism between the representation categories of (C1, C1 ) and (C2, C2 ).
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Let Wi U (Hi Hi), i = 1, 2, be manageable multiplicative unitaries that generate (Ci, Ci ). We view Wi as a right representation of (Ci, Ci ) on Hi. The representation of C^iu associated to Wi descends to a faithful representation of C^i: this is the standard construction of C^i B(Hi) from a multiplicative unitary in [10]. Thus we have to prove that the representations of C^1u = C^2u associated to W1 and W2 have the same kernel. Since our multiplicative unitaries are manageable, the representation category
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C := Rep(W1) = Rep((C1, C1 )) = Rep((C2, C2 )) = Rep(W2)
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contains both a natural left and a natural right absorber by Proposition 2.17. Both W1 and W2 are natural right absorbers. By the remarks above, the direct sums (W1) and (W2) of infinitely many copies of W1 and W2 are isomorphic objects of C because they are both isomorphic to the direct sum of infinitely many copies of a natural left absorber. Therefore, the representations of C^1u associated to (W1) and (W2) have the same kernel. Then the representations of C^1u associated to W1 and W2 also have the same kernel. Thus our Hopf -isomorphism C^1u = C^2u descends to a Hopf -isomorphism C^1 = C^2. This implies a Hopf -isomorphism C1 = C2.
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Corollary 2.19. A C-quantum group (C, C) is determined uniquely by its tensor category Rep(C, C) of representations with the forgetful functor to Hilb.
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Proof. Assume to begin with that there is an equivalence of tensor categories F0 from Rep(C, C ) to Rep(D, D) such that the forgetful functors For F0 and For to Hilb are naturally isomorphic. This natural isomorphism consists of natural unitaries (H,V ) : For(F0(H, V )) - H for all Hilbert spaces H with a representation V of (C, C ). We use (H,V ) on the first leg to transfer the representation F0(H, V ) of (D, D) to the Hilbert space H. This gives another equivalence of tensor categories F from Rep(C, C) to Rep(D, D) such that the tensor functors For F
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10
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RALF MEYER AND SUTANU ROY
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and For are equal. Thus F turns a representation of (C, C) on a Hilbert space H into a representation of (D, D) on the same Hilbert space and maps an intertwiner for (C, C) to the same operator, now as an intertwiner for (D, D). Since the forgetful functor to Hilbert spaces is faithful and strict, the functor F is a strict
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tensor functor as well. We may improve the inverse equivalence to a strict tensor
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functor acting identically on objects as well. Thus F is an isomorphism of tensor
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categories. Let WC and WD be manageable multiplicative unitaries that generate (C, C )
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and (D, D). A representation of (C, C ) is equivalent to one of WC on the same Hilbert space. So Rep(C, C ) = Rep(WC ). Similarly, Rep(D, D) = Rep(WD). So WC and WD are natural right absorbers in Rep(C, C ) = Rep(D, D) by Example 2.10. By Proposition 2.13, the multiplicative unitaries WC and WD are
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isomorphic in the category of [3]. Theorem 2.18 shows that this isomorphism is a Hopf -isomorphism.
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Proposition 2.11 has a variant for natural left absorbers. Let and (U x)xC be a natural left absorber for C. For x C, let Hx := For(x), and write U x for For(U x) U (H Hx). Then U is an "antimultiplicative" unitary:
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U12U23 = U23U13U12.
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Moreover, U x for x C is a left representation of U :
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U2x3U1x3U12 = U12U2x3.
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We define a tensor product for representations of U by
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U V := V13U12.
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The map x U x gives a fully faithful, strict tensor functor from C to Rep(U ), which intertwines the forgetful functors from C and Rep(U ) to Hilb.
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Similarly, there is an analogue of Proposition 2.13, saying that the antimultiplicative unitaries H := H, H := H, U := U , U := U associated to two natural left absorbers (, (U x)xC) and (, (U x)xC) are "isomorphic" in a suitable sense. Namely, the unitaries
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V := U U (H H), W := U U (H H)
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satisfy the following pentagon-like equations:
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U12V23 = V23V13U12, V12U23 = U23V13V12, V12W23 = W23U13V12,
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U12W23 = W23W13U12, W12U23 = U23W13W12, W12V23 = V23U13W12.
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It is also interesting to apply the same technique to a tensor category with a natural right absorber (, (U x)xC) and a natural left absorber (, (U x)xC). Let H := H, H := H, U := U , U := U be the associated multiplicative and antimultiplicative
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unitaries. Define
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V := U U (H H), W := U U (H H).
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These unitaries satisfy the following pentagon-like equations:
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U12V23 = V23V13U12, U23V12 = V12V13U23, V13W12 = W12V13U23,
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U12W23 = W23W13U12, U23W12 = W12W13U23, W13V23 = V23W13U13.
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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11
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The proofs are similar to those in Proposition 2.13. In addition, let x be any object of C. Naturality of U with respect to the intertwiner U x : x (x) gives
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(2.3)
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U1x2 = (U13)(U2x3)U13U2x3 = W13(U2x3)W13U2x3.
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Naturality of U with respect to the intertwiner U x : x (x) gives
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(2.4)
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U2x3 = (U23)(U1x2)U23U1x2 = V23(U1x2)V23U1x2.
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Here U x and U x are the representations of U and U associated to x, respectively. So
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these determine each other. If C = Rep(U ) for a manageable multiplicative unitary U and U comes from its contragradient as above, then also C = Rep(U ). So for a given representation U x of U , there is a unique representation U x of U satisfying (2.3). And for a given representation U x of U , there is a unique representation U x of U
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satisfying (2.4).
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Multiplicative and antimultiplicative unitaries are closely related to the Heisenberg
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and anti-Heisenberg pairs studied in [4]. By definition, a Heisenberg pair for a C-quantum group (C, C) is a pair of representations (, ^) of (C, C^) such that (^ )W for the reduced bicharacter W U(C^ C) is a multiplicative unitary. And an anti-Heisenberg pair is a pair of representations (, ^) of (C, C^) such that
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(^ )W is an antimultiplicative unitary.
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3. Representations of braided multiplicative unitaries
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Let H and L be Hilbert spaces and let W U(H H) be a multiplicative unitary. Let
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U U(L H), V^ U(H L), F U(L L),
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be a braided multiplicative unitary over W (see [6]). We are first going to define a tensor category Rep(W, U, V^ , F) of right representations.
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Definition 3.1. A (right) representation of (W, U, V^ , F) is a triple (K, S, T), where K is a Hilbert space, S U(K H) is a right representation of W on K, that is,
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(3.1)
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W23S12 = S12S13W23 in U (K H H),
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and T U(K L) is equivariant with respect to the tensor product representation S U of W,
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(3.2)
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S13U23T12 = T12S13U23 in U (K L H),
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and satisfies the (top-braided) representation condition
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(3.3)
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F23T12 = T12(L L)23T12(L L)23F23 in U (K L L).
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We recall how the braiding operators L K : L K K L are defined, where K carries a representation S U(K H) of W. Namely, L K := Z for the unique Z U(K L) with
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(3.4)
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Z13 = V^ 23(S12)V^ 23S12
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in U(K H L).
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The braiding in (3.3) is the same as in the top-braided pentagon equation for F.
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Hence (L, U, F) is an example of such a right representation. A morphism (K1, S1, T1) (K2, S2, T2) is a bounded operator a : K1 K2 that
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intertwines both representations, that is, a1 S1 = S2 a1 and a1 T1 = T2 a1. This turns the representations of (W, U, V^ , F) into a W-category Rep(W, U, V^ , F).
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Forgetting both representations S and T gives the forgetful functor to Hilbert spaces.
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The functor maps K (K, 1, 1). If the identity map on K is an intertwiner (K, S1, T1) (K, S2, T2), then S1 = S2 and T1 = T2. So Assumption 2.2 is satisfied.
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12
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RALF MEYER AND SUTANU ROY
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We define a tensor product operation on Rep(W, U, V^ , F) by (K1, S1, T1) (K2, S2, T2) := (K1 K2, S1 S2, T1 T2)
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with
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S1 S2 = S113S223 U (K1 K2 H), T1 T2 = (L K2)23T112(K2 L)23T223 U (K1 K2 L).
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The braiding operators L
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K2 23
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and
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K2
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L use only the representations S on K2 and V^
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on L and therefore make sense. In contrast, K2 L and L K2 would be defined if we
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had a left representation of W on K2 instead of a right one.
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Lemma 3.2. The above definitions turn Rep(W, U, V^ , F) into a Hilbert space tensor category.
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Proof. First, we ought to check that the tensor product above is well-defined, that
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is, gives representations again. We check associativity of the tensor product first
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because we want to use it to prove that the tensor product is again a representation. Let Si U (Ki H) and Ti U (Ki L) for i = 1, 2, 3 be corepresentations of W, U, V^ , F. The definition of T T makes sense for any W-equivariant unitary operators T, T . Thus both (T1 T2) T3 and T1 (T2 T3) are defined even if we do not yet know that T1 T2 and T2 T3 give representations again. We claim that both (T1 T2) T3 and T1 (T2 T3) are equal to the W-equivariant unitary
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(3.5)
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(L K2K3)234T112(K2K3 L)234(L K3)34T223(K3 L)34T334
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in U (K1 K2 K3 L). The operators L Ki : L Ki Ki L are defined by L Ki := Zi, where Zi U (Ki L) satisfies
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(3.6)
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Z1i3 = V^ 23(Si12)V^ 23Si12
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in U (Ki H L)
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for i = 1, 2, 3. And L K1K2 = Z122312, where Z12 U (K1 K2 L) satisfies
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(3.7)
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Z11224 = V^ 34(S1 S2)123V^ 34(S1 S2)123
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in U (K1 K2 H L).
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This equation gives Z12 = Z223Z113 when we plug in the definition of and eliminate S1, S2 and V^ using (3.6). Therefore,
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L
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K1K2 = Z122312 = Z223Z1132312 = Z22323Z11212 = L
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K2 L 23
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K112.
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Similarly, L
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K2K3 = L
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K3 L 23
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K212. Now both T1
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(T2
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T3) and (T1
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T2)
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T3
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and the expression in (3.5) are equal because they all simplify to
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L
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K3 L 34
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K2 23 T112 K2
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L 23
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T223
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K3
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L 34
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T334.
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Next, we check that the tensor product of two representations is again a representation. The proof will also help to construct a natural right absorber later. We claim that an operator T U(K L) together with (K, S) Rep(W) gives a representation if and only if T is an intertwiner
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(K L, S U, T F) (K L, S U, 1 F).
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Indeed, being such an intertwiner means being equivariant with respect to S U and intertwining T F = (L K2)23T12(K2 L)23F23 with 1 F = F23. The latter is exactly our representation condition. Assume that T1 U (K1 L) and T2 U (K2 L) are braided representations. Since T223 is equivariant, when we conjugate it with the braiding operator (L K2L)234 on K1 K2 L L, then we merely transfer it to T234, which commutes with T112. Thus (3.5) shows that T223 is also an intertwiner
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(K1 K2 L, S1 S2 U, T1 T2 F) (K1 K2 L, S1 S2 U, T1 1 F).
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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13
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Similarly, the braiding operator K2 L gives an intertwiner
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K2 L
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(3.8) (K1K2L, S1 S2 U, T1 1 F) -----23 (K1LK2, S1 U S2, T1 F 1).
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Now the operator T112 is an intertwiner (K1 L K2, S1 U S2, T1 F 1) (K1 L K2, S1 U S2, 1 F 1).
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The unitary L K2 gives an intertwiner from the last representation back to
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(K1 K2 L, S1 S2 U, 1 1 F).
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Hence T1 T2 has the expected intertwining property to be a representation. Now we check Assumption 2.3. Let a B(K1) be such that a1 U (K1 K2) is
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an intertwiner for the tensor product representation. Since a commutes with S223, the equivariance with respect to S1 S2 gives that a is S1-equivariant. Since a1 commutes with T1 T2, (L K2K3)234, and T223, it follows that a is an intertwiner for T112 as well.
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Finally, the functors For and are strict tensor functors by definition, and (C) with the canonical unit transformations is indeed a tensor unit.
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Proposition 3.3. The representation
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= (H L, W U, 1 F) is a natural right absorber for the tensor category Rep(W, U, V^ , F).
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Proof. We must construct an intertwiner Ax : x (x) for any representation x = (K, S, T) of (W, U, V^ , F). We claim that the composite operator
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(K H L, S W U, T 1H F) -T--1-H (K H L, S W U, 1K 1H F)
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-S-12 (K H L, 1K W U, 1K 1H F)
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has the properties required in Definition 2.8. The triple (K, S, 1) is a representation of (W, U, V^ , F) for any right representation S of W, and a map between representations
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of this form is an intertwiner if and only if it is an intertwiner for the representations
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of W. In particular, the second map S12 above is an intertwiner, see Example 2.10. Moreover, since there are representations (H, W, 1) and x (H, W, 1) = (K H, S
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W, F 1), the map T 1H above is an intertwiner as well, see the proof of Lemma 3.2. Thus the composite map is an intertwiner x (x) as needed. These two
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operators and their composite are natural by construction, that is, (2.8.1) holds.
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We check condition (2.8.2). Let (Ki, Si, Ti) be representations of (W, U, V^ , F). We
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shall use the diagram in Figure 1. This diagram uses short-hand notation for
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representations.
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For instance,
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S1 S2 U W T1 T2 F 1
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denotes the representation
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(K1 K2 L H, S1 S2 U W, T1 T2 F 1).
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All braiding operators in this diagram exist because the Hilbert space L is on the top strand. They are intertwiners of braided representations, compare (3.8). The remaining arrows are also intertwiners of braided representations by the proof of Lemma 3.2. Before we show that the diagram in Figure 1 commutes, we deduce the condition (2.8.2) from it. The arrow from the (2, 1)-entry to the (2, 5)-entry in Figure 1 along the top boundary is the intertwiner
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T1 T2 1H : (K1 K2 H L, S1 S2 W U, T1 T2 1 F) (K1 K2 H L, S1 S2 W U, 1 1 1 F ),
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compare the proof in Lemma 3.2 that the tensor product is associative. And the arrow going downward from there is (S1 S2)123. So the composite arrow is the
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14
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RALF MEYER AND SUTANU ROY
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S1 S2 U W
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T223
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S1 S2 U W K2 L S1 U S2 W
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T112
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S1 U S2 W L K2 S1 S2 U W
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T1 T2 F 1
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T1 1 F 1
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T1 F 1 1
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1F1 1
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1 1F1
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HL
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K2H L
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LH
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S234
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L K2H
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S234
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LH
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S1 S2 W U T1 T2 1 F
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A2234
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S1 S2 W U T1 1 1 F
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K2H L S223
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S1 U 1 W T1 F 1 1
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K2 L
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S1 1 W U T1 1 1 F
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HL
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S1 1 U W T1 1 F 1
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T112
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S1 U 1 W
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1 F1 1
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K2 L
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T113
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S1 1 U W
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1 1F 1
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S1 S2 W U 1 1 1F
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L K2H S223
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LH
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S1 1 W U 111F
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A1134
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S113
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11W U 11 1 F
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Figure 1. Commuting diagram in Rep(W, U, V^ , F) that proves the condition (2.8.2).
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absorbing intertwiner for the tensor product (K1 K2, S1 S2, T1 T2). Similarly, the arrows labeled A2 and A1 are the absorbing intertwiners for (K2, S2, T2) and (K1, S1, T1), respectively. Hence the commutativity of the boundary of the diagram in Figure 1 is exactly (2.8.2).
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Now we check that the diagram in Figure 1 commutes. The four triangles of braiding operators commute because the braiding operators have enough of the properties of a braided monoidal category, compare the proof in Lemma 3.2. The two pentagons with A1 and A2 as one of the faces commute by definition of our absorbing intertwiners. The two parallelograms with S2 and braiding operators commute because the braiding operators are natural with respect to intertwiners of W-representations. The square with T112 and S234 commutes because we operate on different legs. Finally, we consider the square involving T1 and the braiding operator K2 L. Here K2 carries the trivial representation of W, so that the braiding is just the tensor flip 23. Thus the square commutes, and now we have seen that the entire diagram commutes.
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Theorem 3.4. The operator
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WC := W13U23V^ 34F24V^ 34 U (H L H L)
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is a multiplicative unitary such that there is a fully faithful, strict tensor functor : Rep(W, U, V^ , F) Rep(WC) with For = For. The functor maps a representation (K, S, T) of (W, U, V^ , F) to the following representation of WC:
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S12(T 1H) = S12V^ 23T13V^ 23 U (K H L).
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The functor is an isomorphism of categories if WC and W are manageable.
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The manageability of WC is expressed in [6] in terms of the braided multiplicative unitary (W, U, V^ , F).
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Proof. We have found a natural right absorber (, A) in Proposition 3.3. Proposition 2.11 shows that A is a multiplicative unitary and that x Ax is a fully faithful, strict tensor functor Rep(W, U, V^ , F) Rep(A). By definition, Ax = S12(T 1H) = S12(L H)23T12(H L)23 and, in particular,
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A = (W U)123(1H F 1H) = W13U23(L H)34F23(H L)34.
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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15
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The braiding unitary L H U (L H, H L) is equal to Z for the unique unitary Z U(H L) that satisfies
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Z13 = V^ 23W12V^ 23W12
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in U(H H L),
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compare (3.4) and [5, (6.10)]. We have V^ 23W12V^ 23 = W12V^ 13 because V^ is a left representation of W. Hence Z13 = V^ 13. Since 34F2334 = F24, we get the asserted formulas for Ax and A. We still have to prove that every representation of WC comes from one of (W, U, V^ , F). This will take a while and require some further
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results. This proof will be completed at the end of this article.
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Proposition 3.5. The operators W13U23 U (H L H) and W12 U (H H L) are bicharacters from the multiplicative unitary WC to W U(H H) and back, whose composite from WC to itself is equal to P := W13U23 U (H L H L). Equivalently, the following pentagon-like equations hold:
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(3.9) P23WC12 = WC12P13P23,
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WC23P12 = P12P13WC23,
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P23P12 = P12P13P23.
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Proof. There are two obvious strict tensor functors between the Hilbert space tensor categories Rep(W, U, V^ , F) and Rep(W), namely, the forgetful functor
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Rep(W, U, V^ , F) Rep(W), (K, S, T) (K, S),
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and the functor Rep(W) Rep(W, U, V^ , F),
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(K, S) (K, S, 1K).
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The definitions imply immediately that these are strict tensor functors that are
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compatible with the forgetful functors to Hilb. Both tensor categories involved have natural right absorbers, and the associated multiplicative unitaries are WC and W, respectively. Proposition 2.15 produces bicharacters from strict tensor functors like the ones above. Furthermore, the composite functor on Rep(W) is the identity. Correspondingly, the composite bicharacter from W to itself is the bicharacter that describes the identity functor, which is W itself. And the composite bicharacter from WC to itself is idempotent, which means that it satisfies the pentagon equation. It remains to compute the bicharacters that we get from the
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formulas in Proposition 2.15. The bicharacter describing the functor Rep(W, U, V^ , F) Rep(W) is the canoni-
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cal unitary intertwiner
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W U = W13U23 : (H L, W U) (H, W) (H L, 1) (H, W),
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that is, we get W13U23 U (H L H). The bicharacter describing the functor Rep(W) Rep(W, U, V^ , F) is the natural
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isomorphism
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(H, W, 1) (H L, W U, 1 F) (H, 1, 1) (H L, W U, 1 F)
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described during the proof of Proposition 3.3. Since the representation of F is 1
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here, this simplifies to the unitary W12 U (H H L). By the definition of the composition of bicharacters in [3, Definition 3.5], the
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composite bicharacter from WC to itself is W13U23 if and only if the following equation holds in U(H L H H L):
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W34(W13U23) = (W13U23)(W14U24)W34 Indeed, the representation property of U and the pentagon equation for W give
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W13U23W14U24W34 = W13W14U23U24W34 = W13W14W34U23 = W34W13U23
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as desired. The general theory says that the unitaries in (3.9) are bicharacters and that the bicharacter P is idempotent, that is, satisfies the pentagon equation.
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16
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RALF MEYER AND SUTANU ROY
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It remains to prove that every representation of WC comes from a representation of the braided multiplicative unitary if WC is manageable. That is, we want it to be of the form S12V^ 23T13V^ 23 for some representation (K, S, T) of (W, U, V^ , F). So we start with a representation (K, A) of WC. The Hilbert space must remain K. We have described the functor
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Rep(W, U, V^ , F) Rep(W), (K, S, T) (K, S),
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through the bicharacter W13U23 from WC to W in Proposition 3.5. The proof of Proposition 2.15 shows that there is a unique unitary S U(K H) with
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(3.10)
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(W24U34)A123 = A123S14(W24U34) U (K H L H)
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because the multiplicative unitary W is manageable: this is the functor on representation categories induced by the bicharacter W13U23. Now T should satisfy A123 = S12V^ 23T13V^ 23, that is,
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T13 = V^ 23S12A123V^ 23
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in U(K H L).
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It remains to prove, first, that the right hand side has trivial second leg, so that it
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comes from a unitary T U(KL); and, secondly, that (K, S, T) is a representation of (W, U, V^ , F). Since these computations are quite unpleasant, we proceed indirectly.
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During this proof, we say that a representation of WC comes from a braided representation if it belongs to the image of the functor Rep(W, U, V^ , F) Rep(WC).
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Lemma 3.6. Let (K1, A1) and (K2, A2) be representations of WC . If (K1, A1) and (K1 K2, A1 A2) come from braided representations, then so does (K2, A2).
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Proof. We define Si and Ti for i = 1, 2 as above. We know that (K1, S1, T1) is a braided representation. But at first, we only know T2 U (K2 L H). We may, nevertheless, recycle the diagram in Figure 1, treating it as a diagram in Rep(W) only, and replacing the top left arrow T223 by T2234. The two pentagons still commute by definition of Si, Ti. The four triangles of braiding operators in Figure 1 commute as before. So do the parallelograms containing S223 and braiding operators, and the two squares in the middle: this only needs (S1, T1) to be a braided representation, which we have assumed. Hence the entire diagram commutes. The composite arrow from the (2, 1)-entry to the (5, 4)-entry is the tensor product representation A1 A2. We have assumed that this comes from a braided representation. This must be of the form (S1 S2, T) for some T U (K1 K2 L). Hence
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(K2 L)23T112(L K2)23T2234 = T123.
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Therefore, T2234 acts trivially on the fourth leg. So A2 = S212V^ 23T213V^ 23 for some T2 U (K2 L). In the proof of Lemma 3.2, we have shown that a unitary T in U(K L) together with a representation (K, S) of W is a braided representation if and only if T is an intertwiner from T F to 1K F. Therefore, T1 T2 = (L K2)23T112(K2 L)23T223 and T1 are intertwiners of braided representations. So are the braiding operators, compare (3.8). Hence T223 U (K1 K2 L) is an intertwiner of braided representations. Then so is T2 itself. This means that (S2, T2) is a braided representation.
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Since WC is manageable, Proposition 2.17 shows that Rep(WC ) contains a (natural) left absorber A1. Even more, the proof shows that we may choose A1 to be isomorphic to a direct sum of copies of WC . By definition, WC comes
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from the braided representation (H L, W U, 1H F). Hence the direct sum of countably many summands of WC also comes from a braided representation. Since A1 A2 = A1 1K2 = A1 for any representation (K2, A2), A1 A2 also comes from a braided representation. Now Lemma 3.6 shows that any representation (K2, A2) of WC comes from a braided representation. This finishes the proof of Theorem 3.4.
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BRAIDED MULTIPLICATIVE UNITARIES AS REGULAR OBJECTS
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17
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References
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[1] Pawel Kasprzak, Ralf Meyer, Sutanu Roy, and Stanislaw Lech Woronowicz, Braided quantum SU(2) groups, J. Noncommut. Geom. 10 (2016), no. 4, 16111625, doi: 10.4171/JNCG/268.
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[2] Johan Kustermans and Stefaan Vaes, Locally compact quantum groups, Ann. Sci. École Norm.
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Sup. (4) 33 (2000), no. 6, 837934, doi: 10.1016/S0012-9593(00)01055-7. MR 1832993
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groups, Münster J. Math. 5 (2012), 124, available at http://nbn-resolving.de/urn:nbn:
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Math. Phys. 351 (2017), no. 1, 249282, doi: 10.1007/s00220-016-2727-3.
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[8] Claudia Pinzari and John E. Roberts, Regular objects, multiplicative unitaries and conjugation,
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Internat. J. Math. 13 (2002), no. 6, 625665, doi: 10.1142/S0129167X02001423. MR 1915523 [9] Sutanu Roy, C-Quantum groups with projection, Ph.D. Thesis, Georg-August Universität
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MR 2357350
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SU(N ) groups, Invent. Math. 93 (1988), no. 1, 3576, doi: 10.1007/BF01393687. MR 943923
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E-mail address: rmeyer2@uni-goettingen.de
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Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 35, 37073 Göttingen, Germany
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E-mail address: sutanu@niser.ac.in
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School of Mathematical Sciences, National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni, 752050, India
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