aa r X i v : . [ m a t h . QA ] J a n QUANTUM GALOIS GROUP OF SUBFACTORS
SUVRAJIT BHATTACHARJEE AND DEBASHISH GOSWAMI
Dedicated to the memory of Prof. V.F.R. Jones
Abstract.
Given a II subfactor N ⊂ M with finite index, we prove theexistence of a universal Hopf ∗ -algebra (or, a discrete quantum group in theanalytic language) which acts on M such that the action preserves the canon-ical trace of M and N is in the fixed point subalgebra for the action. We callthis Hopf ∗ -algebra the quantum Galois group for the subfactor and computeit in some examples of interest, notably an arbitrary irreducible finite indexdepth two subfactor of type II . Introduction
The theory of subfactors is one of the cornerstones of the modern theory ofoperator algebras. Since Jones’ beautiful and path-breaking discovery of a deepconnection between subfactors and knot theory, there have been numerous applica-tions of subfactors to diverse fields of mathematics and beyond (e.g. physics). It isnatural to try to understand symmetry objects of subfactors. In fact, since the earlydays ([Ocn88]), it was well recognized that one must go beyond group symmetryto capture symmetries of a general subfactor. Link with Hopf algebras/quantumgroups was conspicuous as several classes of subfactors were constructed from Hopfalgebra (more generally, weak Hopf algebra) (co) actions. In this context, one mayrecall the characterizations of depth two finite index subfactors in terms of weakHopf algebra actions (see for instance, [NV00]).The similarity between a finite field extension and a finite index inclusion of II factors suggested the existence of a Galois-type object. After the pioneering work[Ocn88] of Ocneanu, an intense activity along this line followed which culminatedin, for example, Hayashi’s face algebras and his version of Galois theory [Hay99],Izumi-Longo-Popa’s compact group version [ILP98], Nikshych-Vainerman’s weakHopf algebra version [NV00b]. In the purely algebraic context, Chase-Harrison-Sweedler’s and Chase-Sweedler’s study of Galois theory for commutative rings ledto the notion of a Hopf-Galois extension which in modern form is equivalent to thenotion of a Jones triple, as shown by [NSW98]. In fact, Sweedler’s study of “groupsof algebras” and that of Takeuchi’s in the noncommutative setting gave birth tothe notion of a Hopf algebroid ([Sch00]) and finally to [KS03]. The Galois featureof the crossed product construction which is one of our main examples is in facta special case of a more general characterization by Doi-Takeuchi of Hopf-Galoisextensions with “normal basis property.”Moreover, beginning with [Ocn88] and explored in detail in the series of works[Kaw99, EK95, Kaw95] (see also the beautiful book [EK98]), the four bimodulecategories Bimod M − M , Bimod M − N , Bimod N − M and Bimod N − N associated to a subfactor N ⊂ M became the centre of study as some sort of symmetry objects.The strength of this viewpoint, already exemplified in the papers mentioned justabove, may also be observed from the fact that such categories are intimately relatedto Hopf algebroids ([Sch00]) via the Tannaka reconstruction theorem ([Hai08]). Forexample, the bimodule category Bimod N − N with the obvious fibre functor shouldcorrespond to a Hopf algebroid over N which in some sense is a universal symmetryobject of the subfactor. This viewpoint is in spirit closer to Grothendieck’s versionof Galois theory which takes as its motivation the Galois theory of covering spacesand fundamental groups.However, Hopf algebroids and their actions/fixed point subalgebras are a bittechnical and not as standard or canonical as Hopf algebras and their represen-tations/actions. For this reason, we feel that it is perhaps useful to formulate anotion of symmetry of subfactors in terms of Hopf algebra actions. This is whatwe achieve in this paper and describe such symmetry by universal properties, as pi-oneered by Wang ([Wan98]). We prove the existence of a universal Hopf ∗ -algebraQGal( N ⊂ M ) for a type II subfactor N ⊂ M of finite index, which acts on M such that the action preserves the canonical trace of M and N is in the fixedpoint subalgebra for the action. In more precise terms (see the next section for thenotation and unexplained terminologies), Theorem.
Let N ⊂ M be a pair of finite factors with [ M : N ] < ∞ . Then QGal τ ( N ⊂ M ) exists. Explicitly, QGal τ ( N ⊂ M ) consists of those elements h ∈ Q ∗ aut (End( N M N ) , τ ) such that h · ( xy ) = ( h · x )( h · y ) for all x, y ∈ M . This universal Hopf ∗ -algebra is computed for a general irreducible depth twosubfactor of type II and finite index and as expected, in this case this is a completeinvariant: Theorem.
Let M be a finite factor admitting an outer-action by a finite dimen-sional Hopf C ∗ -algebra H . Then QGal( M ⊂ M ⋊ H ) = H ∗ . Let us end this introduction by briefly describing the plan of this short article. InSection 2, we recall some terminology from subfactor theory, state and prove Wang’sresults in the action language and finally give the definition of the quantum Galoisgroup and prove its existence. Section 3 computes the quantum Galois groups ofthe inclusions coming from the crossed product construction. In the final Section 4,we describe some of the directions which we are looking at presently and we hopeto include these in future versions of this preliminary draft.
Acknowledgments.
Both authors are grateful to Prof. Kawahigashi for pointingout the reference [NV02] to us.2.
Quantum Galois group - definition and existence
In this section, we define what we call the quantum Galois group of an inclusionof II -factors. Examples are given in the next section. UANTUM GALOIS GROUP OF SUBFACTORS 3
Preliminaries.
For the convenience of the reader, we recall some of the ter-minology and results from subfactor theory. Some references are [EK98, GdlHJ89,PP86, Tak03, Jon83].Let M be a finite von Neumann algebra with a fixed normal faithful trace τ , τ (1) = 1. The Hilbert norm on M given by τ is denoted as k x k = τ ( x ∗ x ) / andlet L ( M, τ ) be the completion of M in this norm. Thus L ( M, τ ) is the Hilbertspace of the GNS representation of M , given by τ , and M acts on L ( M, τ ) byleft multiplication. This representation is called the standard representation. Thecanonical conjugation on L ( M, τ ) is denoted by J . It acts on the dense subspace M ⊂ L ( M, τ ) by Jx = x ∗ . Then J satisfies JM J = M ′ and in fact JxJ is theoperator of multiplication on the right with x ∗ : JxJ ( y ) = yx ∗ , y ∈ M ⊂ L ( M, τ ).If N ⊂ M is a von Neumann subalgebra with the same unit, then E N denotesthe unique τ -preserving conditional expectation of M onto N . E N is in fact therestriction to M of the orthogonal projection of L ( M, τ ) onto L ( N, τ ), L ( N, τ )being the closure of N in L ( M, τ ). This orthogonal projection is denoted by e N .We list some properties of e N .(1) e N xe N = E N ( x ), for x ∈ M .(2) If x ∈ M then x ∈ N if and only if e N x = xe N .(3) N ′ = ( M ∪ { e N } ) ′′ .(4) J commutes with e N .Let M denote the von Neumann algebra on L ( M, τ ) generated by M and e N .It follows that M = JN ′ J . M is called the basic construction for N ⊂ M . Werecord two properties of M .(5) M is a factor if and only if N is.(6) M is finite if and only if N ′ is.If M is a finite factor and acts on the Hilbert space H , the Murray-von Neumanncoupling constant dim M H is defined as τ ([ M ′ ξ ]) /τ ′ ([ M ξ ]), where 0 = ξ ∈ H andfor A ⊂ B ( H ) a von Neumann algebra [ Aξ ] is the orthogonal projection onto Aξ .The definition is independent of ξ = 0. For a pair of finite factors N ⊂ M , the Jonesindex of N inside M , denoted [ M : N ], is defined to be the number dim N L ( M, τ ).Some properties of the Jones index are listed below.(7) [ M : M ] = 1. If N ⊂ P ⊂ M then [ M : P ][ P : N ] = [ M : N ].(8) If [ M : N ] < ∞ then N ′ ∩ M is finite dimensional.(9) If [ M : N ] < ∞ then M is a finite factor and the canonical trace on M ,say τ , has the Markov property: τ ( e N x ) = 1[ M : N ] τ M ( x ) ∀ x ∈ M. (10) [ M : M ] = [ M : N ].We now recall a theorem due to Pimsner and Popa. Theorem 2.1.1.
Let N ⊂ M be a pair of finite factors with [ M : N ] < ∞ .Then(1) As a right module over N , the algebra M is projective of finite type.(2) M = { P nj =1 a j e N b j | n ≥ , a j , b j ∈ M } . SUVRAJIT BHATTACHARJEE AND DEBASHISH GOSWAMI (3) If α : M → M is a right N -module map, then α extends uniquely to anelement of M on L ( M, τ ) .(4) If x ∈ M then x ( M ) ⊂ M , where M is viewed as a dense subspace of L ( M, τ ) . As a corollary, one has
Corollary 2.1.2.
Let N ⊂ M be a pair of finite index factors with [ M : N ] < ∞ .Then(1) End( M N ) ∼ = M as C -algebras, and(2) M ⊗ N M ∼ = M as ( N, N ) -bimodules. We have the following proposition.
Proposition 2.1.3.
Let N ⊂ M be a pair of finite index II factors. Then End( N M N ) is finite dimensional.Proof. By the above Corollary, End( N M N ) ∼ = N ′ ∩ M . The result follows from (7),(8) and (10) above. (cid:3) Quantum symmetry of finite spaces.
In this subsection, we restate Wang’s[Wan98] results on quantum symmetry groups of finite spaces in the action picture,which will be used below for the existence theorem. However, the statements of thetheorems are interesting in their own right.2.2.1.
The finite space X n . Let X n = { x , · · · , x n } be the finite space which isidentified with the C ∗ -algebra B = C ( X n ) of continuous functions on X n . Thealgebra has the presentation, B = h e i | e i = e ∗ i = e i , n X j =1 e j = 1 , i = 1 · · · n i . Theorem 2.2.1.
Let H be a Hopf ∗ -algebra such that B = C ( X n ) is an H -module ∗ -algebra. Moreover suppose that H preserves the canonical “integration” τ : B → C . Then there exists a unique pairing h , i : H ⊗ Q aut ( X n ) → C , such that for all h ∈ H , h · e j = X i e i h h, a ij i , j = 1 , . . . , n. Proof.
For h ∈ H , let h · e j = P i h ij e i , h ij ∈ C . Now we draw a series of conclusionsfrom the various conditions on B . First, h · ( e i e j ) = h · ( δ ij e j ) = X k δ ij h kj e k . UANTUM GALOIS GROUP OF SUBFACTORS 5
On the other hand, h · ( e i e j ) =( h · e i )( h · e j )= ( X k ( h ) ki e k )( X l ( h ) lj e l )= X k,l ( h ) ki ( h ) lj e k e l = X k ( h ) ki ( h ) kj e k . Comparing the coefficients of e k from both expressions, we get( h ) ij ( h ) ik = δ jk h ij . (2.2.1)In particular, ( h ) ij ( h ) ij = h ij . (2.2.2)Now, h · ε ( h ) = ⇒ h · ( X j e j ) = ε ( h ) X j e j = ⇒ X j X i h ij e i = X i ε ( h ) e i = ⇒ X i ( X j h ij ) e i = X i ε ( h ) e i = ⇒ X j h ij = ε ( h ) . Thus we obtain, X j h ij = ε ( h ) . (2.2.3)Since B is an H -module ∗ -algebra, we have for all h ∈ H and b ∈ B ,( h · b ) ∗ = S ( h ) ∗ · b ∗ . But ( h · e j ) ∗ = ( X i h ij e i ) ∗ = X i h ij e i . This uses e ∗ i = e i . Also, S ( h ) ∗ · e j = X i ( S ( h ) ∗ ) ij e i . These together imply ( S ( h ) ∗ ) ij = h ij . (2.2.4)Finally, τ ( h · e j ) = ε ( h ) τ ( e j ) = ε ( h )implies τ ( X i h ij e i ) = ε ( h )which gives X i h ij = ε ( h ) . (2.2.5) SUVRAJIT BHATTACHARJEE AND DEBASHISH GOSWAMI
Before going onto defining the pairing, let us observe that, on one hand, X i,k ( h ) ij ( h ) ik S ( h ) kl = X k δ jk ε ( h ) S ( h ) kl = ε ( h ) S ( h ) jl = S ( h ) jl . This uses (2.2.1) summed over i and the fact P i h ij = ε ( h ). On the other hand, X i,k ( h ) ij ( h ) ik S ( h ) kl = X i ( h ) ij ε ( h ) δ il = h lj . Thus we obtain S ( h ) jl = h lj . (2.2.6)Now to define a pairing, as in [VD93], it is sufficient to define the pairing forgenerators only and then check the algebra relations. In our case, define h , i : H ⊗ Q aut ( X n ) → C , h h, a ij i = h ij . We refer to Wang’s paper [Wan98] for the relations of quantum permutation group,(3.i) means equation (3.i) on page 199 of Wang’s paper. We thus have(1) (3.1) holds: h h, a ij i = h h , a ij ih h , a ij i = ( h ) ij ( h ) ij and by (2.2.2) thisequals h ij = h h, a ij i .Also observe that h · e j = P i e i h h, a ij i . Then h · e ∗ j = P i e ∗ i h h, a ∗ ij i which implies ( h · e ∗ j ) ∗ = P i e i h h, a ∗ ij i . On the other hand, S ( h ) ∗ · e j = P i e i h S ( h ) ∗ , a ij i . Since, ( h · e ∗ j ) ∗ = S ( h ) ∗ · e j , we have h h, a ∗ ij i = h S ( h ) ∗ , a ij i .Now, using (2.2.4), h h, a ∗ ij i = h S ( h ) ∗ , a ij i = ( S ( h ) ∗ ) ij = h ij = h h, a ij i . (2) (3.2) holds: h h, P j a ij i = P j h ij = ε ( h ) = h h, i , where we use (2.2.3).(3) (3.3) holds: h h, P i a ij i = P i h ij = ε ( h ) = h h, i , where we use (2.2.5). (cid:3) The finite space M n ( C ) . Let u = ( a klij ) ni,j,k,l =1 and v = ( b klij ) ni,j,k,l =1 be twomatrices. Define ( uv ) klij = X r,s a klrs b rsij , i, j, k, l = 1 , . . . , n. Let ψ = T r be the trace functional on M n . The algebra M n has the followingpresentation: B = h e ij | e ij e kl = δ jk e il , e ∗ ij = e ji , n X r =1 e rr = 1 , i, j, k, l = 1 , . . . , n. i Theorem 2.2.2.
Let H be a Hopf ∗ -algebra such that B = M n ( C ) is an H -module ∗ -algebra. Moreover suppose that H preserves the canonical trace ψ = T r . Thenthere exists a unique pairing h , i : H ⊗ Q aut ( M n ( C )) → C , such that for all h ∈ H , h · e ij = X k,l e kl h h, a klij i , i, j = 1 , . . . , n. UANTUM GALOIS GROUP OF SUBFACTORS 7
Proof.
For h ∈ H , let h · e ij = P k,l e kl h klij . We now draw a series of conclusionsfrom the relations on B . Again, first, h · ( e ij e kl ) = ( h · e ij )( h · e kl )= ( X α,β e αβ ( h ) αβij )( X γ,δ e γδ ( h ) γδkl )= X α,β,γ,δ e αβ e γδ ( h ) αβij ( h ) γδkl = X α,β,γ e αγ ( h ) αβij ( h ) βγkl . On the other hand, h · ( e ij e kl ) = h · ( δ jk e il ) = δ jk X α,γ e αγ h αγil . Comparing coefficients of e αγ from both expressions, we get X β ( h ) αβij ( h ) βγkl = δ jk h αγil . (2.2.7)Now, h · ε ( h ) = ⇒ h · ( X r e rr ) = ε ( h ) X r e rr = ⇒ X r,k,l e kl h klrr = ε ( h ) X kl δ kl e kl = ⇒ X r h klrr = ε ( h ) δ kl . Thus we obtain, X r h klrr = ε ( h ) δ kl . (2.2.8)The preservation of the trace implies ψ ( h · e ij ) = ε ( h ) ψ ( e ij ) ε ( h ) d ij . On the other hand, ψ ( h · e ij ) = ψ ( X k,l e kl h klij ) = X k,l δ kl h klij . We get X r h rrij = ε ( h ) δ ij . (2.2.9)Again, preservation of ∗ implies,( h · e ij ) ∗ = S ( h ) ∗ · e ∗ ij . ( h · e ij ) ∗ = ( X k,l h klij ) ∗ = X k,l e lk h klij . Also, S ( h ) ∗ · e ∗ ij = S ( h ) ∗ · e ji = X k,l e lk ( S ( h ) ∗ ) lkji . These together imply ( S ( h ) ∗ ) lkji = h klij (2.2.10) SUVRAJIT BHATTACHARJEE AND DEBASHISH GOSWAMI
Before going onto getting information about the antipode, let us observe that theassociativity of the H -action follows the multiplication rule stated at the beginning.For h, h ′ ∈ H , h · ( h ′ · e ij ) = h · ( X αβ e αβ h ′ αβij ) = X α,β,k,l e kl h klαβ h ′ αβij . This implies ( hh ′ ) klij = X α,β h klαβ h ′ αβij . Now the antipode should behave like S ( h ) klij = h jilk . Consider X α,β,k,l ( h ) lkxy ( h ) klαβ S ( h ) αβij = X α,β,l δ yα ( h ) llxβ S ( h ) αβij = X α,β δ yα ε ( h ) δ xβ S ( h ) αβij = S ( h ) yxij . The first equality uses (2.2.7) and the second one uses (2.2.9). On the other hand, X α,β,k,l ( h ) lkxy ( h ) klαβ S ( h ) αβij = X k,l ( h ) lkxy ε ( h ) δ klij = h jixy . These together imply, S ( h ) yxij = h jixy , (2.2.11)which was to be obtained. Applying (2.2.11) to (2.2.7), we get, X β ( h ) lkγβ ( h ) jiβα = δ jk h liγα . (2.2.12)Define the pairing by, h , i : H ⊗ Q aut ( M n ( C )) → C , h h, a klij i = h klij . We refer to page 202 of [Wan98] for the relations; (4.i) below means equation (4.i)on page 202 of Wang’s paper. We thus have(1) (4.1) is satisfied because of (2.2.7).(2) (4.2) is satisfied because of (2.2.12).(3) (4.3) is satisfied because of (2.2.10).(4) (4.4) is satisfied because of (2.2.8).(5) (4.5) is satisfied because of (2.2.9). (cid:3)
UANTUM GALOIS GROUP OF SUBFACTORS 9
The finite space ⊕ mk =1 M n k ( C ) . Let u = ( a klrs,xy ) and v = ( b klrs,xy ) be twomatrices with entries from a ∗ -algebra, where k, l = 1 , . . . , n x r, s = 1 , . . . , n y , x, y = 1 , . . . , m. Define uv to be the matrix whose entries are given by( uv ) klrs,xy = m X p =1 n p X i,j =1 a klij,xp b ijrs,py . The algebra B = ⊕ mk =1 M n k has the following presentation: B = h e kl,i | e kl,i e rs,j = δ ij δ lr e ks , j, e ∗ kl,i = e lk,i , m X q =1 n q X p =1 e pp,q = 1 ,k, l = 1 , . . . , n i , r, s = 1 , . . . , n j , i, j = 1 , . . . , m. i Let ψ be the positive functional on B defined by ψ ( e kl,i ) = T r ( e kl,i ) = δ kl , k, l = 1 , . . . , n i , i = 1 , . . . , m. Theorem 2.2.3.
Let H be Hopf ∗ -algebra such that B = ⊕ mk =1 M n k ( C ) is an H -module ∗ -algebra. Moreover, suppose that H preserves the functional ψ . Then thereexists unique pairing h , i : H ⊗ Q aut ( B ) → C , such that for all h ∈ H , h · e rs,j = m X i =1 n i X k,l =1 e kl,i h h, a klrs,ij i , r, s = 1 , . . . , n j , j = 1 , . . . , m. Proof.
The proof follows exactly the same route of the proof of the Theorem 2.2.2.See [Wan98] for the relations among a klrs,ij . (cid:3) The existence theorem.
We now define the quantum Galois group of a pairof finite factors N ⊂ M . Definition 2.3.1.
Let N ⊂ M be a pair of finite factors. Let C( N ⊂ M ) be thecategory whose • objects are Hopf ∗ -algebras Q admitting an action on M making it a module ∗ -algebra such that N ⊂ M Q , where M Q is the invariant subalgebra; • morphisms between two objects, say Q and Q ′ , are Hopf ∗ -algebra mor-phisms φ : Q → Q ′ such that the following diagram commutes: Q ⊗ M Q ′ ⊗ MM φ ⊗ id (2.3.1) where the unadorned arrows are the respective actions. We define the quantumGalois group of the inclusion N ⊂ M denoted QGal( N ⊂ M ) to be a terminalobject of the category C( N ⊂ M ) . By definition, QGal( N ⊂ M ) is unique up to unique isomorphism. There is noreason for it to exists, however, as we shall prove below, under some assumption, italways does. It is useful, in fact necessary, to introduce the following. Definition 2.3.2.
Let C τ ( N ⊂ M ) be the full subcategory of C( N ⊂ M ) consistingof Hopf ∗ -algebras admitting a τ -preserving action on M . A terminal object in thiscategory is denoted as QGal τ ( N ⊂ M ) . Proposition 2.3.3.
Let N ⊂ M be a pair of finite factors with [ M : N ] < ∞ and let H be a Hopf ∗ -algebra belonging to the category C( N ⊂ M ) . If the uniquenormalized trace τ on M is preserved under the H -action, i.e., τ ( h · x ) = ε ( h ) τ ( x ) ,for all x ∈ M , then so is τ . Here, the H -action on M is obtained from Corollary2.1.2.Proof. First observe that τ ( X j a j e N b j ) = X j τ ( e N b j a j ) = X j M : N ] τ M ( b j a j ) . The first equality follows from the traciality of τ . The second from the Markovproperty (9)above. So it suffices to check the claim for elements of the form e N x ,where x ∈ M . Now, τ ( h · ( e n x )) = τ ( e N ( h · x ))= 1[ M : N ] τ M ( h · x )= 1[ M : N ] ε ( h ) τ M ( x )= ε ( h ) τ ( e N x ) . The first equality follows from the fact that H acts trivially on e N which can be seenas follows. Each h ∈ H , viewed as a right N -linear endomorphism of M extendsuniquely to a bounded linear map on L ( M, τ ) by (3) of Theorem 2.1.1. Moreover, L ( N, τ ) is an invariant subspace under such a map. The H -action is ∗ -preserving,so by the standard trick, invariant subspace becomes reducing. (cid:3) Theorem 2.2.3 together with Proposition 2.1.3 imply
Theorem 2.3.4.
Let N ⊂ M be a pair of finite factors with [ M : N ] < ∞ andlet H be a Hopf ∗ -algebra belonging to the category C( N ⊂ M ) . Moreover, assumethat the unique normalized trace τ on M is preserved under the H -action. Thenthe H -action factors through the dual action of a Hopf ∗ -subalgebra of the dual Q ∗ aut (End( N M N ) , τ ) of Q aut (End( N M N ) , τ ) ; Theorem 2.3.4 at once implies
Theorem 2.3.5.
Let N ⊂ M be a pair of finite factors with [ M : N ] < ∞ . Then QGal τ ( N ⊂ M ) exists. Explicitly, QGal τ ( N ⊂ M ) consists of those elements h ∈ Q ∗ aut (End( N M N ) , τ ) such that h · ( xy ) = ( h · x )( h · y ) for all x, y ∈ M . Examples of quantum Galois groups
In this section, we compute the quantum Galois group of some of the genericexamples of finite factor inclusions.
UANTUM GALOIS GROUP OF SUBFACTORS 11
Inclusions arising from crossed products.
One of the standard ways ofproducing a finite-index pair of finite factors from a given one, say M is by takingcrossed product with an outer-action of a finite dimensional Hopf C ∗ -algebra H .It is also well-known that such a pair M ⊂ M ⋊ H is a generic example of anirreducible “depth 2” inclusion. More precisely, let N ⊂ M be a pair of finitefactors with [ M : N ] < ∞ and N ′ ∩ M = C M . Then, by [Szy94], there existsa finite dimensional Hopf C ∗ -algebra H which acts on N such that M = N ⋊ H .We compute the quantum Galois group of such an inclusion in this subsection. Wework in a bit more general situation as it needs no extra effort.Let H be a (not necessarily finite dimensional) Hopf algebra and A be an H -module algebra. Denote by A ⋊ H the smash product of A by H . We recall thefollowing lemma. Lemma 3.1.1.
Let V ∈ Hom C ( H, A ⋊ H ) be the map V ( h ) = 1 ⋊ h. (3.1.1) Then V is convolution invertible and “innerifies” the H -action, i.e., h · x ⋊ V ( h )( x ⋊ V − ( h ) , (3.1.2) where h ∈ H , x ∈ A , ∆ h = h ⊗ h . Proof.
This is standard. But we nevertheless check the details. Let V − : H → A ⋊ H be defined as V − ( h ) = 1 ⋊ S ( h ). Then we compute V V − ( h ) = V ( h ) V − ( h )= (1 ⋊ h )(1 ⋊ S ( h ))= h · ⋊ h S ( h )= ε ( h ) ⋊ ε ( h )= 1 ⋊ ε ( h )1 . The other equality can be proven similarly. Now, for x ∈ A , V ( h )( x ⋊ V − ( h ) = (1 ⋊ h )( x ⋊ ⋊ S ( h ))= ( h · x ⋊ h )(1 ⋊ S ( h ))= ( h · x )( h · ⋊ h S ( h )= h · x ⋊ h S ( h )= h · x ⋊ ε ( h )= h · x ⋊ . (cid:3) Let Q be a Hopf algebra such that A ⋊ H is Q -module algebra and A ⊂ ( A ⋊ H ) Q ,where ( A ⋊ H ) Q is the invariant subalgebra. Such a Hopf algebra exists; for example,let H ∗ be a Hopf algebra dual to H . By this we mean, H ∗ is a Hopf algebra andthere is a nondegenerate pairing h , i : H ∗ ⊗ H → C satisfying the usual compatibilityconditions. For u ∈ H ∗ , x ∈ A and h ∈ H , define u · ( x ⋊ h ) = x ⋊ ( u ⇀ h ) , (3.1.3)where u ⇀ h = h h u, h i . Then it is clear that the H ∗ -action is one such exam-ple. What we show below is that this example is the universal example, under certain conditions. By universality we mean that there should exist a Hopf algebramorphism φ : Q → H ∗ such that the following diagram commutes: Q ⊗ ( A ⋊ H ) H ∗ ⊗ ( A ⋊ H ) A ⋊ H φ ⊗ (3.1.4)Observe that, a necessary condition for this to happen is that for q ∈ Q , h ∈ H , q · (1 ⋊ h ) = φ ( q ) · (1 ⋊ h ) = 1 ⋊ h h φ ( q ) , h i . (3.1.5)That is Q takes H into H in a very special way. We first achieve this. Proposition 3.1.2.
Let q ∈ Q , thought of as a map from H → A ⋊ H , h q · (1 ⋊ h ) .Then for each h ∈ H , V − q ( h ) ∈ A ′ ∩ ( A ⋊ H ) , (3.1.6) where V − q is the convolution product, A ′ ∩ ( A ⋊ H ) is the centralizer of A in A ⋊ H .Proof. Let x ∈ A and h ∈ H . We compute( x ⋊ V − ( h ) q ( h ) = V − ( h ) V ( h )( x ⋊ V − ( h ) q ( h )= V − ( h )( h · x ⋊ q ( h )= V − ( h ) q · (( h · x ⋊ ⋊ h ))= V − ( h ) q · ((1 ⋊ h )( x ⋊ V − ( h ) q ( h )( x ⋊ . The second equality follows from Lemma 1.1, the third and the fifth follow fromthe fact that Q acts trivially on A . Therefore, we are done. (cid:3) Corollary 3.1.3.
Let the extension A → A ⋊ H be irreducible, i.e., A ′ ∩ ( A ⋊ H ) = C .Then for each q ∈ Q , there exists unique λ q ∈ Hom C ( H, C ) such that q · (1 ⋊ h ) = 1 ⋊ h λ q ( h ) . (3.1.7) Therefore, Q actually takes H inside H . Before proving the Corollary, we remark that the condition A ′ ∩ ( A ⋊ H ) = C isalso expressed by saying that the action is outer. Proof of Corollary 3.1.3.
By the previous Proposition, for each q ∈ Q and h ∈ H there exists λ q ( h ) ∈ C such that V − q ( h ) = λ q ( h )(1 ⋊ q ∈ Hom C ( H, A ⋊ H )be defined as Λ q ( h ) = 1 ⋊ λ q ( h )1 . (3.1.8)Then V − q = Λ q which implies q = V Λ q . So for each h ∈ H , q · (1 ⋊ h ) = V ( h )Λ q ( h ) = (1 ⋊ h )(1 ⋊ λ q ( h )1) = 1 ⋊ h λ q ( h ) , which was to be obtained. Uniqueness follows from applying ε . (cid:3) Now using this λ q , we define a dual pairing between Q and H , from whichuniversality follows automatically. Define h , i : Q ⊗ H → C , h q, h i = λ q ( h ) = (1 ⋊ ε )( q · (1 ⋊ h )) . (3.1.9)We show that this defines a dual pairing. UANTUM GALOIS GROUP OF SUBFACTORS 13
Step 1 : h qq ′ , h i = h q ⊗ q, ∆ h i = h q, h ih q ′ , h i holds: For, by associativity, qq ′ · (1 ⋊ h ) = q · (1 ⋊ h λ q ′ ( h )) = 1 ⋊ h λ q ( h ) λ q ′ ( h ) . Therefore h qq ′ , h i = ε ( h ) λ q ( h ) λ q ′ ( h ) = λ q ( h ) λ q ′ ( h ) = h q, h ih q ′ , h i . Step 2 : h q, hh ′ i = h q , h ih q , h ′ i holds: Since A ⋊ H is a Q -module algebra, wehave q · (1 ⋊ hh ′ ) = q · (1 ⋊ h ) q · (1 ⋊ h ′ ) . Now q · (1 ⋊ hh ′ ) = 1 ⋊ h h ′ λ q ( h h ′ )and q · (1 ⋊ h ) q · (1 ⋊ h ′ ) = h λ q ( h ) h ′ λ q ( h ′ ) = h h ′ λ q ( h ) λ q ( h ′ ) . Applying ε yields the result. Step 3 : h , h i = ε ( h ) obviously holds. Step 4 : h q, i = ε ( q ) too is obvious. Step 5 : h q, S ( h ) i = h S ( q ) , h i holds: Obviously, the pairing defines a bialgebramorphism from Q → H ∗ and since a bialgebra morphism is in fact a Hopf algebramorphism, we get the result. Nevertheless, we present an argument which uses onlythe pairing and which is itself pretty! For that, we first recall that the map H ⊗ H → H ⊗ H, x ⊗ y x ⊗ x y, (3.1.10)is a bijection. Now, using this, for h ∈ H , find x i , y i ∈ H such that P i x i ⊗ x i y i = h ⊗
1. We compute, h S ( q ) , h i = h S ( q ) ε ( q ) , h i = h S ( q ) , h ih q , i = X i h S ( q ) , x i ih q , x i y i i = X i h S ( q ) , x i ih q , x i ih q , y i i = X i h S ( q ) q , x i ih q , y i i = X i h ε ( q ) , x i ih q , y i i = X i ε ( q ) ε ( x i ) h q , y i i = X i h q, ε ( x i ) y i i = X i h q, S ( x i ) x i y i i = h q, S ( h ) i . Summarizing all these, we get
Theorem 3.1.4.
Let Q be a Hopf algebra such that A ⋊ H is a Q -module algebraand A ⊂ ( A ⋊ H ) Q , where ( A ⋊ H ) Q is the invariant subalgebra, that is Q ∈ C( A ⊂ A ⋊ H ) . Moreover, suppose that the extension A → A ⋊ H is irreducible, i.e., A ′ ∩ ( A ⋊ H ) = C . Then there exists a unique pairing h , i : Q ⊗ H → C , q ⊗ h
7→ h q, h i , (3.1.11) such that q · ( a ⋊ h ) = a ⋊ h h q, h i , (3.1.12) for all a ∈ A , h ∈ H and q ∈ Q . We note that we do not right away write QGal( A ⊂ A ⋊ H ) = H ∗ . As H isnot assumed to be finite dimensional, we have stated everything in the language ofHopf-pairing. The actual identification of QGal( A ⊂ A ⋊ H ) requires a bit morework which we now aim to describe.We begin by observing that the proof of Proposition 3.1.2 only used the fact that q ∈ Q acts on A ⋊ H as an ( A, A )-bimodule morphism. The question arises if thatis true for all (
A, A )-bimodule morphisms and the answer is yes.
Lemma 3.1.5.
Assume the H -action is outer and let T ∈ End( A A ⋊ H A ) be an ( A, A ) -bimodule morphism of A ⋊ H . Then there exists a unique ˆ T ∈ Hom C ( H, C ) such that T (1 ⋊ h ) = 1 ⋊ h ˆ T ( h ) . (3.1.13) Proof.
The proof is exactly the same as the proofs of Proposition 3.1.2 and theCorollary 3.1.3. (cid:3)
Let us write ˆ H for the vector space Hom C ( H, C ) which is an algebra under theconvolution product. For λ ∈ ˆ H , defineˇ λ : A ⋊ H → A ⋊ H, a ⋊ h a ⋊ h λ ( h ) . (3.1.14)The map ˇ λ is manifestly left A -linear. Let us check that is also right A -linear. For x ∈ A , ˇ λ (( a ⋊ h )( x ⋊ λ ( a ( h · x ) ⋊ h )= a ( h · x ) ⋊ h λ ( h ) , ˇ λ ( a ⋊ h )( x ⋊
1) = ( a ⋊ h λ ( h ))( x ⋊ a ( h · x ) ⋊ h λ ( h ) . Thus ˇ λ is also right A -linear, i.e., ˇ λ ∈ End( A A ⋊ H A ). DefineΦ : End( A A ⋊ H A ) → ˆ H, T ˆ T , (3.1.15)and Ψ : ˆ H → End( A A ⋊ H A ) , λ ˇ λ. (3.1.16) Proposition 3.1.6. Φ is an algebra isomorphism with inverse Ψ .Proof. That Φ is an algebra homomorphism follows from similar computation as in
Step 1 above. The uniqueness in Lemma 3.1.5 together withˇ λ (1 ⋊ h ) = 1 ⋊ h λ ( h )yields that ΦΨ( λ ) = λ . ΨΦ( T ) = T again follows from Lemma 3.1.5. (cid:3) Let Q be a Hopf algebra such that A ⋊ H is a Q -module algebra and A ⊂ ( A ⋊ H ) Q ,where ( A ⋊ H ) Q is the invariant subalgebra. Let φ be the composite Q → ˆ H obtained using the isomorphism from Proposition 3.1.6. Then the computation in Step 2 above implies that φ ( q )( hh ′ ) = φ ( q )( h ) φ ( q )( h ′ ) , i.e., m ∗ φ ( q ) ∈ ˆ H ⊗ ˆ H, (3.1.17)where m is the multiplication of H , m ∗ is the transpose. This says φ ( q ) ∈ H ◦ , thefinite or Sweedler dual of H . Rephrasing Theorem 3.1.4 in this way, we get UANTUM GALOIS GROUP OF SUBFACTORS 15
Theorem 3.1.7.
Let Q be a Hopf algebra such that A ⋊ H is a Q -module algebraand A ⊂ ( A ⋊ H ) Q , where ( A ⋊ H ) Q is the invariant subalgebra. Moreover, supposethat the extension A → A ⋊ H is irreducible, i.e., A ′ ∩ ( A ⋊ H ) = C . Then there existsa a unique Hopf algebra morphism φ : Q → H ◦ such that the following diagram Q ⊗ ( A ⋊ H ) H ◦ ⊗ ( A ⋊ H ) A ⋊ H φ ⊗ (3.1.18) is commutative. Here, H ◦ denotes the Sweedler dual to H , and the unadornedarrows denote respective actions. Thus QGal( A ⊂ A ⋊ H ) = H ◦ . We state the above theorem in the von Neumann algebra setting,
Theorem 3.1.8.
Let M be a finite factor admitting an outer-action by a finitedimensional Hopf C ∗ -algebra H . Then QGal( M ⊂ M ⋊ H ) = H ∗ . We end this subsection with a remark.
Remark 3.1.9.
We note that our computation did not require the preservationof the canonical trace. Since H ∗ automatically preserves the canonical trace, thetrace-preserving quantum Galois group matches is also identified with H ∗ . The invariant subalgebra.
Let M be a finite factor admitting a saturated([SP94]) and outer action of a finite dimensional Hopf C ∗ algebra H . Thus N ≡ M H ⊂ M ⊂ M ⋊ H is a Jones triple, i.e., M ⊂ M ⋊ H is the basic constructionof M H ⊂ M . By [KT09], M can be identified with N ⋊ H ∗ . Then, using Theorem3.1.8, without any extra computation, we can conclude Theorem 3.2.1.
The quantum Galois group
QGal( M H ⊂ M ) of the pair M H ⊂ M is H . Here also, we remark that no trace-preserving assumption was needed. Thisleads us to the following conjecture.
Conjecture 3.2.2.
Let N ⊂ M be a pair of finite factors with [ M : N ] < ∞ and N ′ ∩ M = C M . Then QGal( N ⊂ M ) exists without any trace-preservingassumption. Epilogue
In this section, we describe some of the things we are looking at presently whichwill be included in future versions or sequel of this article.
Depth 2, reducible inclusions.
We have treated examples coming from depth2, irreducible inclusions and as we have already mentioned, these are nothing butcrossed products by finite dimensional Hopf C ∗ -algebras, see [Szy94]. However, inthe reducible situation, Hopf algebras don’t suffice and [NV00] have shown that oneneeds weak Hopf algebras. For details on weak Hopf algebras and their relation tosubfactor theory, see also [BNS99, Nik01, NSW98, NV00b, NV02]. We are trying toimitate the computation in Subsection 3.1 when the inclusion arises from a weakHopf crossed product. Intermediate factors and Galois correspondence.
Any theory regarding thenotion of a “Galois group” should also answer the Galois correspondence, i.e., a (pos-sibly bijective) correspondence between intermediate subalgebras and “subgroups”of the “Galois group”. Indeed, for Kac algebras, a similar correspondence wasobtained in [ILP98] and for weak Hopf algebras, it was shown in [NV00b]. Ourexamples show that such a correspondence hold in our version too. We are lookingat the general situation presently. What is more is that [NV00b] has shown thatany finite depth subfactor may be realized as an intermediate subfactor of a depth2 subfactor and using the correspondence, as a crossed product by a coideal subal-gebra of a weak Hopf algebra. Thus understanding the weak Hopf crossed productis essential and it may lead to a more direct proof of our existence result whichrelies on [Wan98].
Some remarks on a categorical construction.
We recall that for a pair offinite factors N ⊂ M with [ M : N ] < ∞ , there exists a canonical tensor categoryBimod N − N ( N ⊂ M ) ([Ocn88,NV00b]) which is generated by the simple subobjectsof the bimodules M n , n ≥ − M − = N , M = M , i.e., the objects are finite directsums of simple objects of M ⊗ n . The forgetful functor for : Bimod N − N ( N ⊂ M ) → Bimod N − N is then a fibre functor and the pair seems to satisfy the requirements of[Hai08] for a reconstruction, yielding a Hopf algebroid over N , in the sense [Sch00],see also [BS04]. However, this Hopf algebroid does not keep N fixed for the followingsimple reason, as observed in [KS03], page 83: the fixed point subalgebra commuteswith N . Nevertheless, it would be interesting to see what the reconstruction of[Hai08] actually yields. Remark.
We also remark that the weak Hopf algebra, say H , obtained in [NV00b] moreover satisfies Rep( H ∗ ) ∼ = Bimod N − N ( N ⊂ M ) as tensor categories. Infinite index inclusion.
In [HO89], the authors considered infinite index inclu-sions and a corresponding Galois theory. The finiteness of index was crucial to ourexistence result. A deeper look is needed for our version of the Galois group in theinfinite-index situation.
Quantum Galois theory in algebra.
The reader familiar with cleft extensions([Mon93]) might have observed that our computation in Subsection 3.1 uses thecleaving map for the Galois extension A → A ⋊ H . It would be interesting to seeif the method adapts to more general cleft extensions. We would also like to pointout that cleft extensions are special types of Galois extensions, namely a Galoisextension with a normal basis. A purely algebraic investigation in this directionwould be very welcome. Quantum Galois theory in the weak context.
The developments in [NV00b,NV00, NV02, NSW98] already indicate a need to undertake an investigation of uni-versal symmetries in the weak context. The recent preprint [HWWW20] initiatessuch a program in the algebraic world. We hope a similar undertaking in theanalytic setting would be beneficial in understanding quantum symmetries.
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