Examples of regularised algebras and RFAs in Vect fd and Hilb

4.1. Regularised Frobenius algebras 95 We give an estimate for the last term in (4.1.34). Recall that each T_n was chosen in the algebraic tensor product of H₁ and H₂. Thus T_n is a finite sum of elementary tensors,

T_n =

tn

X

j=1

x^j_n⊗y_n^j (4.1.39)

for tn ∈Z^≥1, x^j_n ∈ H⁽¹⁾ and y_n^j ∈ H⁽²⁾. Using this, we get:

(S_a⁽¹⁾−S_a⁽¹⁾₀ )⊗S_b⁽²⁾T_n+S_a⁽¹⁾₀ ⊗(S_b⁽²⁾−S_b⁽²⁾

0 )T_n

≤

tn

X

j=1

(S_a⁽¹⁾−S_a⁽¹⁾

0 )x^j_n ·

S_b⁽²⁾

·

y_n^j +

(S_a⁽¹⁾

0

·

x^j_n ·

(S_b⁽²⁾−S_b⁽²⁾

0 )y_n^j

.

(4.1.40)

By strong continuity of a7→Sa⁽ⁱ⁾ we can chose δ₂ >0 such that for everya, b∈X with

|a−a₀|+|b−b₀|< δ₂ we have (S_a⁽¹⁾−S_a⁽¹⁾

0 )x^j_n

< ε 4t_nκ

y^jn

and

(S_b⁽²⁾−S_b⁽²⁾

0 )y^j_n

< ε 4t_n

Sa⁽¹⁾0

·

x^jn

, (4.1.41)

for every j = 1, . . . , t_n, since these are only finitely many conditions to satisfy. Let δ :=

min{δ₁, δ₂}. Then for everya, b∈X with |a−a₀|+|b−b₀|< δ we have that

(S_a⁽¹⁾⊗S_b⁽²⁾−S_a⁽¹⁾

0 ⊗S_b⁽²⁾

0 )T_n ≤ ε

2 . (4.1.42)

Finally, using (4.1.38) and (4.1.42) we have that

(S_a⁽¹⁾⊗S_b⁽²⁾−S_a⁽¹⁾₀ ⊗S_b⁽²⁾₀ )T

< ε . (4.1.43)

By iterating the previous lemma we see that the definition of a regularised algebra simplifies in Hilb. Namely it is enough to check that a 7→ P_a is continuous, rather than having to consider multiple tensor products.

Corollary 4.1.16. The continuity condition (4.1.4) in Hilb is automatically satisfied for any n ≥2if it holds for n= 1.

4.1.3 Examples of regularised algebras and RFAs in V ect

^fd

and

1. Let A be an algebra in Vect^fd with multiplicationµ and unit η and setµ_a :=µ·e^aσ, η_a := η·e^aσ for some σ ∈ C. Then A is a regularised algebra. One can similarly obtain an RFA from a Frobenius algebra.

A Frobenius algebra in Vect^fd is always finite-dimensional. In Example 1 we equipped them with an RFA structure for which all a →0 limits exist. The converse also holds in the following sense.

Proposition 4.1.17. LetA ∈ RFrob(Hilb). The following are equivalent.

1. A is finite-dimensional.

2. All of the following limits exist:

lima→0η_a , lima→0µ_a , lima→0ε_a , lima→0∆_a .

Proof. (1⇒2): IfAis finite-dimensional, then the map a7→P_a is norm continuous, hence P_a = e^aH for some H ∈ B(A). Then η₀ := e^−aHη_a is independent of a and η_a = P_a◦η₀, hence lima→0ηa =η0 exists. One similarly proves that the other limits exist as well.

(1⇐2): The morphisms given by these limits define a Frobenius algebra structure on A, hence A is finite-dimensional.

2. Consider the polynomial algebraC[x] and complete it with the Hilbert space structure given by hxⁿ, x^mi = δ_n,mf(m) for some monotonously decreasing function f : N → (0,1] ⊂ R and denote by C[x] its Hilbert space completion. Let P_a(xⁿ) := e^aσxxⁿ for σ ∈ R (note the x in the exponent). We now show that this defines a bounded operator. Let y∈C[x] with y=P

n∈Ny_nxⁿ. Then kP_a(y)k² = X

n,m∈N

a^m m!

2

f(n+m)|y_n|² ≤ X

n,m∈N

a^2m

(2m)!f(n)|y_n|² ≤e^akyk² . where we used that f is monotonously decreasing.

Let us assume that

sup

k∈N

( _k X

l=0

f(k)f(l)f(k−l) )

<∞ (4.1.44)

holds, e.g. f(m) = (1 +m)⁻² or f(m) =e^−m. Then the operator M :C[x]→C[x]⊗C[x]

x^k 7→

k

X

l=0

f(k)x^k−l⊗x^l (4.1.45)

4.1. Regularised Frobenius algebras 97 is bounded. The adjoint of M is the standard multiplication

µ:C[x]⊗C[x]→C[x]

x^k⊗x^l7→x^k+l , (4.1.46)

which is therefore also a bounded operator. Then defining µa :=Pa◦µand ηa(1) :=

P_a(1) gives a regularised algebra inHilb. Note, however, that this regularised algebra cannot be turned into a regularised Frobenius algebra because P_a is not trace class, cf. Lemma4.1.13.

3. Consider the Frobenius algebra A := C[x]/hx^di in Vect^fd with ε(x^k) = δk,d−1. Let h ∈ A and define Pa(f) := e^ahf, εa :=ε◦Pa, ηa :=Pa◦η and µa := Pa◦µ. Then C[x]/hx^diis an RFA, denoted A_h. Unless d= 1, this RFA is not separable.

Proposition 4.1.18. LetI be a countable (possibly infinite) set. For k∈I letF_k ∈ Hilb be a (possibly infinite-dimensional) RFA. Then L

k∈IF_k (the completed direct sum of Hilbert spaces) is an RFA in Hilb if and only if, for every a∈R>0,

sup

k∈I

µ^k_a

<∞ and sup

k∈I

∆^k_a

<∞ , (4.1.47)

X

k∈I

ε^k_a

2 <∞ and X

k∈I

η_a^k

2 <∞ , (4.1.48)

where µ^k_a, ∆^k_a, ε^k_a and η_a^k denote the structure maps of F_k. Proof. LetF :=L

k∈IF_k and fix the value of a.

(⇒): Let us writex_k for the k’th component of x∈F =L

k∈IF_k. Then for every k ∈I ∆^k_a

= sup

xk∈F_k kx_kk=1

∆^k_a(x_k)

= sup

xk∈F_k kx_kk=1

k∆_a(x_k)k ≤ sup

xk∈F_k kx_kk=1

k∆_ak · kx_kk=k∆_ak<∞ ,

so in particular sup_k

∆^k_a

< ∞. A similar proof applies to the case of µ_a. We calculate the norm of ηa:

kη_ak² =kη_a(1)k² =X

k∈I

η_a^k(1)

2 =X

k∈I

η_a^k

2 ,

which is finite if and only if η_a is a bounded operator. If ε_a is bounded, then by the Riesz Lemma there exists a unique v ∈ F such that εa(x) = hv, xi and kεak = kvk. Then hv_k, x_ki=hv, x_ki=ε_a(x_k) =ε^k_a(x_k). So again by the Riesz Lemma

ε^k_a

=kv_kk. We have that

kε_ak² =kvk² =X

k∈I

kv_kk² =X

k∈I

ε^k_a

2 .

(⇐): The operators η_a and ε_a are bounded by the previous discussion. For ∆_a one has that

k∆_ak² = sup

x∈F kxk=1

k∆_a(x)k² = sup

x∈F kxk=1

X

k∈I

∆_a(x_k)

2

= sup

x∈F kxk=1

X

k∈I

∆^k_a(x_k)

2

≤ sup

x∈F kxk=1

X

k∈I

∆^k_a

2kxkk² ≤

sup

l

∆^l_a

2

· sup

x∈F kxk=1

X

k∈I

kxkk² = sup

l

∆^l_a

2 <∞ ,

so ∆a is bounded. Forµa the proof is similar.

Then one needs to check thata 7→P_a:=P

k∈IP_a^k is continuous. Letε∈R>0,a₀ ∈R^≥0 and f ∈ L

k∈IF_k with components f_k be fixed. Let a⁰ > a₀ and 0 < E < ε be arbitrary.

SincePa−Pa0 is a bounded operator, one can findJa⁰ ⊂I finite, such that for everya < a⁰ X

j∈I\J_a0

(P_a^j−P_a^j

0)f_j

2 < E .

Then let δ⁰ >0 be such that for every |a−a0|< δ⁰ X

j∈J_a0

(P_a^j −P_a^j₀)fj

2 < ε−E ,

which can be chosen since the sum is finite and each P_a^j is continuous by assumption.

Finally let δ := min{δ⁰, a⁰−a₀}. By construction we have that for every |a−a₀|< δ, k(P_a−P_a0)fk² =X

j∈I

(P_a^j−P_a^j₀)f_j

2 < ε .

All examples of RFAs known to us are of the above form. For Hermitian RFAs, which we will introduce in Section 4.1.5, we can show that they are necessarily of the above form.

Note that the same RFA F_k cannot appear infinitely many times, as otherwise the bounds (4.1.48) would be violated.

Now we continue our list of examples with some special cases.

4. Let (_k, σ_k)_k∈I be a countable family of pairs of complex numbers such that for all a >0

sup

k∈I

ke^−aσk

<∞ and X

k∈I

e^−aσk _k

2

<∞ . (4.1.49) Then A,σ := L

k∈ICfk, the Hilbert space generated by orthonormal vectors fk, becomes an RFA by Proposition4.1.18 via

µ_a(f_k⊗f_j) :=δ_k,jkf_ke^−aσk , η_a(1) :=X

k∈I

f_k

_ke^−aσk , (4.1.50)

∆_a(f_k) := f_k⊗f_k k

e^−aσk , ε_a(f_k) :=_ke^−aσk . (4.1.51)

4.1. Regularised Frobenius algebras 99 This RFA is strongly separable (with τ_a=η_a) and commutative.

5. Let I := Z^>0 and consider the one-dimensional Hilbert spaces Cf_k and Cg_k with kf_kk² =k² and kg_kk² =k⁻¹. Let F :=L∞

k=1Cf_k and G:=L∞

k=1Cg_k be the Hilbert space direct sums, so that

hf_k, f_ji_F =δ_k,jk² and hg_k, g_ji_G=δ_k,jk⁻¹ . (4.1.52) Define the maps

µ^F_a(f_k⊗f_j) :=δ_k,je^−ak2f_k , η^F_a(1) :=

∞

X

k=1

e^−ak2f_k ,

∆^F_a(fk) :=e^−ak2fk⊗fk , ε^F_a(fk) :=e^−ak2 ,

(4.1.53)

and similarly for G by changing f_k to g_k. These formulas define strongly separable (with τ_a =η_a) commutative RFAs by the previous example with (_k, σ_k) = (k⁻¹, k²) for F and with (k, σk) = (k, k²) for G. Note that lima→0µ^F_a exists and has norm 1, but lima→0µ^G_a does not: the set µ^G₀(g_k⊗g_k)

/kg_k⊗g_kk=k

k∈Z>0 is not bounded.

Define the morphism of RFAs ψ :F →Gas

ψ(f_k) = g_k for k= 1,2, . . . .

It is an operator with kψk= 1 and is mono and epi, but it does not have a bounded inverse, as the set { kψ⁻¹(g_k)k/kg_kk=k² |k ∈Z>0} is not bounded. This is an ex-ample illustrating that the categoryHilbis not abelian: a morphism can be mono and epi without being invertible. The example also shows that RFA morphisms which are mono and epi need not preserve the existence of zero-area limits. Isomorphisms, on the other hand, being continuous with continuous inverse, do preserve the existence of limits.

6. Consider L²(G), the Hilbert space of square integrable functions on a compact semisimple Lie group G with the following morphisms:

η_a(1) := X

V∈Gˆ

e^−aσVdim(V)χ_V , µ(F)(x) :=

Z

G

F(y, y⁻¹x)dy , P_a(f) := µ(η_a(1)⊗f) , µ_a:=P_a◦µ ,

εa(f) :=

Z

G

ηa(1)(x)f(x⁻¹)dx , ∆(f)(x, y) :=f(xy), ∆a:= ∆◦Pa ,

(4.1.54)

where f ∈L²(G), F ∈L²(G×G)∼=L²(G)⊗L²(G), ˆG is a set of representatives of isomorphism classes of finite-dimensional simple unitary G-modules,σ_V is the value of the Casimir operator of the Lie algebra of G in the simple module V, χ_V is the character of V, and R

G denotes the Haar integral on G. These formulas define a strongly separable RFA in Hilb (with τ_a=η_a), for details see Section 4.4.1.

7. The centre of the previous RFA isCl²(G), the Hilbert space of square integrable class functions onG, with multiplication, unit and counit given by the same formulas, but with the following coproduct:

∆_a(f) = X

V∈Gˆ

e^−aσV (dim(V))⁻¹χ_V ⊗χ_Vf_V , (4.1.55) where f = P

V∈Gˆf_Vχ_V ∈ Cl²(G). This is a strongly separable RFA in Hilb (with τ_a(1) = P

V∈Gˆe^−aσV (dim(V))⁻¹χ_V and τ_a⁻¹(1) = P

V∈Gˆe^−aσV (dim(V))³χ_V). For more details see Section4.4.1.

Im Dokument State-sum construction of two-dimensional functorial field theories (Seite 95-100)