Modules over regularised algebras

4.1. Regularised Frobenius algebras 107 The definition of bimodules in terms of left and right modules requires an extra con-tinuity assumption, so we spell it out in detail:

Definition 4.1.37. An A-B-bimodule over regularised algebras A and B is an object U ∈ S together with a family of morphisms

U

(a, l, b)

A ρ_a,l,b =

U

B

∈ S(A⊗U ⊗B, U) (4.1.72)

for every a, l, b∈R>0 such that the following conditions hold.

1. For every a=a₁+a₂ =a⁰₁+a⁰₂,b =b₁+b₂ =b⁰₁+b⁰₂ and l=l₁+l₂ ρ_a1,l1,b1 ◦(id_A⊗ρ_a2,l2,b2 ⊗id_B) = ρ_a⁰

1,l,b⁰₁ ◦ µ^A_a0

2 ⊗id_U⊗µ^B_b0 2

, (4.1.73)

the morphisms Q^U_a,l,b :=ρ_a1,l,b1 ◦ η_a^A

2 ⊗id_U⊗η^B_b

2

satisfies that lima,l,b→0Q^U_a,l,b = id_U and

2. the map

(R³>0∪ {0})→ S(U, U)

(a, l, b)7→Q^U_a,l,b (4.1.74) is jointly continuous.

A bimodule is called transmissive, if ρ_a,l,b depends only on a+b, or, in other words, if ρa+u,l,a−u is independent of u. As with the inclusion of the extra parameter l, the notion of transmissivity is motivated by the application to area-dependent quantum field theory, see Section 4.2.3.

Remark 4.1.38. Let U be a left A-module and a right B-module such that the left and right actions ρ^L_a,l and ρ^R_b,m commute. That is for every a, b∈R>0 and l₁+l₂ =m₁+m₂

ρ_a,l,b:=ρ^L_a,l1 ◦ ρ^R_b,l2 ⊗id_B

=ρ^R_a,m1 ◦ id_A⊗ρ^L_b,m2

. (4.1.75)

If ρ_a,l,b is jointly continuous in the parameters, then U is anA-B-bimodule. Note that in contrast to the case of usual bimodules over associative algebras, which are defined to be left and right modules with commuting actions, here we have to impose the extra condition of joint continuity.

Conversely, let U be an A-B-bimodule with action ρa,l,b. If the limit ρ^L_a,l := lim

b→0ρ_a,l,b1 ◦(idA⊗U⊗η_b^B₂) (4.1.76) withb =b₁+b₂ exists for everya, l ∈R>0 and remains jointly continuous in the limit, then U becomes a leftA-module with action ρ^L_a,l. Similarly, if the analogous a→0 limit exists then U becomes a right B-module. In Appendix4.A we give an example which illustrates that these limits do not always exist.

Example 4.1.39. Let AutRFrob(S)(A) denote the invertible morphisms inRFrob(S)(A, A).

Then for α, β ∈AutRFrob(S)(A) we can define a transmissive bimodule structure _αA_β onA by twisting the multiplication from the two sides and letting the l-dependence be trivial.

That is, for every a, b, l∈R>0 we define the action to be

ρa,l,b :=µa◦(id⊗µb)◦(α⊗idA⊗β), (4.1.77) which is jointly continuous in the parameters, since the composition in S is separately continuous, and since we can rewrite ρa,l,b =Pc◦ρa⁰,l,b⁰ witha⁰+b⁰+c=a+b. Note that β⁻¹ :_αA_β →_β⁻¹_◦αA_idA is a bimodule isomorphism, so it is enough to consider twisting on one side.

The proof of the following proposition is similar to that of Proposition 4.1.18.

Proposition 4.1.40. LetF =L

k∈IF_k and G=L

j∈JG_j be RFAs inHilb as in Propos-ition 4.1.18. Let Mkj ∈ Hilb be an Fk-Gj-bimodule with action ρ^M_a,l,b^kj for k ∈I and j ∈J. Then M :=L

k∈I, j∈JM_kj is an F-G-bimodule in Hilb if and only if for every a, l, b∈R>0

sup

k∈I, j∈J

n ρ^M_a,l,b^kj

o

<∞ . (4.1.78)

A morphism U −→^φ V of left modules over a regularised algebra A is a morphism in S which respects the action:

φ◦ρ^U_a,l =ρ^V_a,l◦(id_A⊗φ), (4.1.79) for alla, l ∈R>0. One similarly defines morphisms of right modules and bimodules. Denote with A-Mod(S) the category of left modules over A in S.

Recall from Proposition 4.1.24 that for a regularised algebra A ∈ Vect^fd the pair D(A) = (A, H) consists of the underlying algebra ofAand an elementH in its centre. Let A-Mod^Z Vect^fd

denote the following category. Its objects are pairs (U, H_U), whereU is a left A-module in Vect^fd and H_U ∈End_A(U). Its morphisms are leftA-module morphisms φ:U →V, such thatH_V ◦φ=φ◦H_U.

As in the case of regularised algebras in Vect^fd (cf. Proposition 4.1.17), the semigroup (a, l)7→Q^U_a,l is norm continuous and hence Q^U_a,l =e^a·HA+l·HU for H_A, H_U ∈End_A(U) such that H_A◦H_U =H_U ◦H_A.

Let us define a functor D : A-Mod(Vect^fd) → A-Mod^Z Vect^fd

as follows. The A-module structure on D(U) is given by ρ^U =Q^U_−a,−l◦ρ^U_a,l and H_U is defined as above. On morphisms D is the identity.

Proposition 4.1.41. The functor D :A-Mod(Vect^fd) →A-Mod^Z Vect^fd

is an equival-ence of categories.

Proof. The A-module structure on D⁻¹(U, HU) is given by ρ^U_a,l := e^aHA+lHU ◦ρ^U with H_A=ρ^U(H⊗ −)∈End_A(U), where ρ^U is the action on U.

4.1. Regularised Frobenius algebras 109 Remark 4.1.42. Let A, B ∈ Vect^fd be regularised algebras and U ∈ Vect^fd an A-B -bimodule U ∈ Vect^fd. As before we have Q^U_a,l,b = e^aHA+lHU+bHB, where H_A, H_U, H_B ∈ End_A,B(U) are bimodule homomorphisms. ThenU is transmissive if and only ifH_A =H_B. Let us assume now that S is symmetric. We now introduce a notion of duals for bimodules.

Definition 4.1.43. LetA, B ∈ S be regularised algebras. A dual pair of bimodules is an A-B-bimoduleU ∈ S and aB-A-bimoduleV ∈ S together with families of morphisms for every a, l, b∈R^>0

γ_a,l,b ∈ S(I, V ⊗U), β_a,l,b ∈ S(U ⊗V,I) , (4.1.80) jointly continuous in the parameters, which we denote with

(a, l, b)

β_a,l,b =

U V

(a, l, b)

γ_a,l,b =

V U

, I

I

, (4.1.81)

such that fora₁+a₂ =a,b₁+b₂ =b and l₁+l₂ =l we have

(a2, l2, b2)

Q^U_a,l,b = U V

(a1, l1, b1)

Q^V_a,l,b =

V U

,

(a1, l1, b1) (a2, l2, b2)

I I

I I

, (4.1.82)

and for every a₁+a₂ =a₃+a₄, b₁+b₂ =b₃+b₄ and l₁+l₂ =l₃ +l₄ we have U

(a1, l1, b1)

V

A

(a², l² , b²

)

(a4, l4, b4)

V

A

(a³ ,l³,b³)

U

= and

V

(a1, l1, b1)

U B

(a² ,l²

,b²

) (a4, l4, b4)

U

(a³ ,l³,b³)

V

=

B

B B A A

. (4.1.83)

Let us compare this situation to Lemma 4.1.7. There the continuity of γ_a in the parameter was automatic, but in Definition4.1.43we demanded continuity explicitly. The reason for this is that the argument in the proof of Lemma4.1.7does not apply, as we have not required that id_V ⊗Q^U_a,l,b is continuous in the parameters, see Remark4.1.36. However one can easily check that for every a1+a2 =a3+a4, l1+l2 =l3+l4 and b1+b2 =b3+b4

(id_V ⊗Q^U_a

1,l1,b1)◦γ_a2,l2,b2 = (Q^V_a

3,l3,b3 ⊗id_U)◦γ_a4,l4,b4 . (4.1.84) Furthermore, in Hilb this is equal toγ_{a1+a2,l1+l2,b1+b2}.

Note that (4.1.83) implies that the action onV is determined by the action on U:

V

(a1, l1, b1)

ρ^V_a,l,b =

V (a3, l3, b3)

A

(a2, l2, b2)

B

. (4.1.85)

We similarly define dual pairs of left and right modules and we omit the details here.

Example 4.1.44. Let A ∈ S be a symmetric RFA, α ∈ AutRFrob(S)(A) and _αA_id be the twisted transmissive bimodule from Example 4.1.39. Then (_αA_id,_α⁻¹A_id) is a dual pair of bimodules with duality morphisms

β_a,l,b =ε_a◦µ_b◦(id_A⊗α) and γ_a,l,b = (α⁻¹⊗id_A)◦∆_a◦η_b (4.1.86) for a, l, b∈R>0. Note that these morphisms only depend on a+b.

Remark 4.1.45. If (U, V) is a dual pair of bimodules with duality morphisms γ_a,l,b and βa,l,b, then (V, U) is also a dual pair of bimodules with duality morphisms σV,U ◦γa,l,b and β_a,l,b◦σ_V,U.

Duals of bimodules over associative algebras are unique up to unique isomorphism.

In the following we will see that under some assumptions this will be true for duals of bimodules over regularised algebras too. Let (U, V) and (U, W) be two dual pairs of bimodules and define

U V

(a2, l2, b2)

ϕa,l,b :=

(a1, l1, b1) W

U W ψa,l,b:=

V ,

(a2, l2, b2)

(a1, l1, b1)

, (4.1.87)

which satisfy for a=a₁+a₂, b=b₁+b₂ and l=l₁+l₂ that

ϕ_a1,l1,b1 ◦ψ_a2,l2,b2 =Q^V_a,l,b and ψ_a1,l1,b1 ◦ϕ_a2,l2,b2 =Q^W_a,l,b . (4.1.88) Using separate continuity of the composition and (4.1.88) one can show the following (we omit the details):

Lemma 4.1.46. If the limits

a,l,b→0lim ϕ_a,l,b and lim

a,l,b→0ψ_a,l,b (4.1.89)

exist, then ϕ0,0,0 and ψ0,0,0 are mutually inverse bimodule isomorphisms between V and W.

4.1. Regularised Frobenius algebras 111 In general we do not know if V ∼=W, not even in Hilb.

Remark 4.1.47. A related concept of duals was introduced in [ABP] where duals are parametrised by Hilbert-Schmidt maps. The authors introduced the notion of a nuclear ideal in a symmetric monoidal category, which in Hilb consists of Hilbert-Schmidt maps HSO(H,K) for H,K ∈ Hilb [ABP, Thm. 5.9]. Part of the data is an isomorphism θ : HSO(H,K) −→ B^∼= (C,H ⊗ K), where H now denotes the conjugate Hilbert space. For f ∈HSO(H,K) andg ∈HSO(K,L) in a nuclear ideal the “compactness” relation holds:

(idL⊗θ(f^†)^†)◦(θ(g)⊗idH) = g◦f . (4.1.90) Our definition of duals fits into this framework as follows. Let A, B be a regularised algebras andU anA-B-bimodule in Hilb with dualV. Then one can show that Q^U_a,l,b is a trace class map, and hence Hilbert-Schmidt, cf. Lemma 4.1.13. Using the above notation letH =K=L:=U,

f :=Q^U_a,l,b , g :=Q^U_a⁰_,l⁰_,b⁰ , β_a,l,b^U :=θ(f^†)^† γ_a^U0_,l⁰_,b⁰ :=θ(g) . (4.1.91) Then (4.1.90) is exactly one half of the duality relation (4.1.82) and (U, U) is a dual pair of bimodules in the sense of Definition 4.1.43.

Im Dokument State-sum construction of two-dimensional functorial field theories (Seite 106-111)