A Calabi-Yau structure on the representation category of a Frobe-

3.2. Constructing an equivalence between Frobenius algebras and Calabi-Yau categories

is an inverse to ΨP,M. Indeed, (ΨP,M ◦ϕ)(g) = ΨP,M

Xn i=1

f_i⊗g(pi)

!

=x7→

Xn i=1

f_i(x).g(pi) =g. (3.46) The last equality follows by applying gto equation (3.38). On the other hand,

(ϕ◦ΨP,M)(p^∗⊗m) =ϕ(x7→p^∗(x).m) =^Xn

i=1

f_i⊗p^∗(pi).m=^Xn

i=1

f_i.p^∗(pi)⊗m

=^Xn

i=1(x7→f_i(x).p^∗(p_i))⊗m=p^∗⊗m.

(3.47)

In the last equality, we have used that P^∗ is a right A-module. The last equality follows again by applyingp^∗ to equation (3.38). This shows that ΨP,M is an isomorphism.

(4)⇒(3) is trivial, since we may chooseM :=P.

(3)⇒(2): Suppose that ΨP,P :P^∗⊗AP →EndA(P) is an isomorphism. Then, Ψ⁻_P,P¹ (idP) =^Xn

i,j

fi⊗pj (3.48)

is a dual basis. Indeed, Xn i=1

fi(x).pi= ΨP,P

Xn i=1

fi⊗p

!

(x) = idP(x) =x. (3.49)

Corollary 3.20. Let A be a separable algebra over a field K, and let M be a finitely generated A-module. Then, the map ΨM,M :M^∗⊗AM → EndA(M) of lemma 3.19 is an isomorphism of A-modules.

Proof. This follows from the fact that every module over a separableK-algebra is pro-jective, which is proven in corollary 3.16. Hence, by the first part of lemma 3.19, the map ΨM,M :M^∗⊗AM →HomA(M, M) is an isomorphism.

This corollary enables us to define a trace for finitely-generated modules over a sepa-rable symmetric Frobenius algebra, as the next subsection shows.

3.2.2. A Calabi-Yau structure on the representation category of a Frobenius

Definition 3.21. Let (A, λ) be a separable symmetric Frobenius algebra over a field K with Frobenius formλ:A →K. LetM be a finitely-generated left A-module. Denote by

ev :M^∗⊗AM →A

f ⊗m7→f(m) (3.50)

the evaluation.

SinceM is finitely generated, the map ΨM,M : EndA(M)→M^∗⊗AM is an isomor-phism by corollary 3.20. We define a trace tr^λ_M : EndA(M)→K by the composition

tr^λ_M : EndA(M)−−−−→^Ψ−1M,M M^∗⊗AM −→^ev A−→^λ K. (3.51) Remark 3.22. As defined here, the trace tr^λ_M is the composition of the Hattori-Stallings trace with the Frobenius formλ. For more on the Hattori-Stallings trace, see [Hat65], [Sta65] and [Bas76].

Example 3.23. Let (A, λ) be a separable symmetric Frobenius algebra over a fieldK.

Suppose thatF is a free A-module with basis e₁, . . . , en. Then,

tr^λ_F(idF) =nλ(1A). (3.52)

Example 3.24. As a second example, letA:=Mn(K) be the algebra ofn×n-matrices overKwith Frobenius form λgiven by the usual trace of matrices. Then, M :=Kⁿ is a projective (but not free), simpleA-module. We claim:

tr^λ_M(idM) = 1. (3.53)

Indeed, lete₁, . . . , e_n be a vector space basis ofKⁿ. This basis also generatesKⁿ as an A-module. Define for each 1≤i≤n aK-linear mapf_i^∗ :Kⁿ→Mn(K) =A by setting

f_i^∗(ek) :=δ_i,1E_k,1, (3.54) where E_k,1 is the square matrix with (k,1)-entry given by one and zero otherwise. A short calculation confirms that the f_i^∗ are even morphisms of A-modules. Indeed, if M ∈A, then

(M.f_i^∗(ek))p,q =δ_1,iδ_1,qM_p.k= (f_i^∗(M.ek))p,q. (3.55) Next, we claim that

Ψ⁻¹_M,M =^Xn

i=1

f_i^∗⊗e_i ∈M^∗⊗AM. (3.56) Indeed,

ΨM,M

Xn i=1

f_i^∗⊗e_i

!

(ek) =^Xn

i=1

f_i^∗(ek).ei

=^Xn

i=1

δ_i,1E_k,1ei

=E_k,1e₁ =e_k.

(3.57)

3.2. Constructing an equivalence between Frobenius algebras and Calabi-Yau categories Thus,

tr^λ_M(idM) =λ Xn i=1

f_i^∗(ei)

!

=λ Xn i=1

E_i,1δ_1,i

!

=λ(E1,1) = 1. (3.58) Next, we show that tr^λ_M has indeed the properties of a trace. In order to show that the trace is symmetric, we need an additional lemma first.

Lemma 3.25. Let A be an K-algebra, and let M and N be left A-modules. Define a linear map

ξ: (M^∗⊗AN)×(N^∗⊗AM)→M^∗⊗AM

(f⊗n, g⊗m)7→f⊗g(n).m. (3.59) Then, the following diagram commutes:

(M^∗⊗AN)×(N^∗⊗AM) M^∗⊗AM

HomA(M, N)×HomA(N, M) HomA(M, M)

ξ

ΨM,N×ΨN,M ΨM,M

◦

(3.60)

Here, the horizontal map at the bottom is given by composition of morphisms ofA-modules and ΨM,M is defined as in equation (3.40).

Proof. We calculate:

(ΨM,M ◦ξ)(f⊗n, g⊗m) = Ψ(f⊗g(n).m)

=x7→f(x)g(n).m. (3.61)

On the other hand,

ΨM,N(g⊗m)◦ΨN,M(f⊗n) = (x7→g(x).m)◦(x7→f(x).n) =x7→g(f(x).n).m

=x7→f(x)g(n).m.

(3.62) Comparing the right hand-side of equation (3.61) with the right hand-side of equation (3.62) shows that the diagram commutes.

We are now ready to show that the trace is symmetric:

Lemma 3.26. Let (A, λ) be a separable, symmetric Frobenius algebra over a field K. Let M and N be finitely-generated A-modules, and let f :M →N and g :N → M be morphisms of A-modules. Then, the trace is symmetric:

tr^λ_M(g◦f) = tr^λ_N(f ◦g). (3.63) Proof. Write

Ψ⁻¹_M,N(f) =^X

i,j

m^∗_i ⊗n_j ∈M^∗⊗AN and Ψ⁻¹_N,M(g) =^X

k,l

x^∗_k⊗y_l ∈N^∗⊗AM. (3.64)

We calculate:

tr^λ_M(g◦f) = (λ◦ev◦Ψ⁻¹_M,M)(g◦f)

= (λ◦ev)



X

i,j,k,l

m^∗_i ⊗x^∗_k(nj).yl



 (by lemma 3.25)

=λ



X

i,j,k,l

m^∗_i(x^∗_k(nj).yl)





= ^X

i,j,k,l

λ(x^∗_k(nj)·m^∗_i(yl)).

(3.65)

On the other hand,

tr^λ_N(f◦g) = (λ◦ev◦Ψ⁻¹_N,N)(f◦g)

= (λ◦ev)



X

i,j,k,l

x^∗_k⊗m^∗_i(yl).nj



 (by lemma 3.25)

=λ



X

i,j,k,l

x^∗_k(m^∗_i(yl).nj)





= ^X

i,j,k,l

λ(m^∗_i(yl)·x^∗_k(nj)).

(3.66)

Since λ is symmetric, the right hand-sides of equations (3.65) and (3.66) agree. This shows that the trace is symmetric.

Historically, the Hattori-Stallings trace has been defined by using bases. This is also possible for the trace in definition 3.21, as the next remark shows.

Remark 3.27. Let (A, λ) be a separable, symmetric Frobenius algebra over a field K, and letM be a finitely-generated A-module.

IfM is a freeA-module, we may express the trace in a basis ofM: iff ∈EndA(M), choose a basis e₁, . . . , en of M, and let e^∗₁, . . . , e^∗_n be the dual basis. Let (Af)ij :=

e^∗_i(f(ej))∈A. Then, tr^λ_M(f) = (λ◦ev) ^Xn

i=1

e^∗_i ⊗f(ei)

!

=^Xn

i=1

λ(e^∗_i(f(ei))) =^Xn

i=1

λ((Af)ii)∈K. (3.67) Since the trace is symmetric by lemma 3.26, we know that tr^λ_M(g◦f◦g⁻¹) = tr^λ_M(f) for any isomorphismg, and therefore the trace is independent of the basis. IfM is only projective and not necessarily free, there is anA-moduleQ so thatF :=M⊕Q is free.

Lete₁ =m₁⊕q₁, . . . , e_n=m_n⊕q_nbe a basis ofF, and lete^∗₁=m^∗₁⊕q₁^∗, . . . , e^∗_n=m^∗₁⊕q_n^∗ be the dual basis. Using the additivity of the trace as in lemma 3.5, we have

tr^λ_M(f) = tr^λ_M⊕Q(f⊕0) =^Xn

i=1

λ(e^∗_i((f⊕0)(ei))) =^Xn

i=1

λ(m^∗_i(f(mi))). (3.68)

3.2. Constructing an equivalence between Frobenius algebras and Calabi-Yau categories Since the trace is symmetric, this expression is independent of the basis of F. Now, a standard argument (cf. [Sta65, 1.7]) shows that equation (3.68) is also independent of the complement Q. For the sake of completeness, we recall this argument here.

Suppose there is another Q⁰ so that M⊕Q⁰:=F⁰ is free. Letα be the isomorphism α:Q⊕F⁰ =Q⊕(M⊕Q⁰)∼=Q⁰⊕(M⊕Q) =Q⁰⊕F. (3.69) Using that the trace is additive as in lemma 3.5 shows that

tr^λ_M⊕Q(f ⊕0Q) = tr^λ_M⊕Q⊕F0(f⊕0Q⊕0F⁰) and

tr^λ_M⊕Q0(f ⊕0Q⁰) = tr^λ_M⊕Q0⊕F(f⊕0Q⁰⊕0F). (3.70) Since

(idM ⊕α)⁻¹◦(f ⊕0Q⁰ ⊕0F)◦(idM⊕α) =f⊕0Q⁰⊕0F⁰, (3.71) using the cyclic invariance of the traces shows that

tr^λ_M⊕Q(f ⊕0Q) = tr^λ_M⊕Q0(f ⊕0Q⁰), (3.72) as required. Thus, equation (3.68) is independent of the complementQ.

Next, we show that the trace is non-degenerate.

Lemma 3.28. Let (A, λ) be a separable, symmetric Frobenius algebra over an alge-braically closed field K, and let M and N be finitely-generated A-modules. Then, the bilinear pairing of vector spaces induced by the trace in definition 3.21

h−,−i: HomA(M, N)×HomA(N, M)→K

(f, g)7→tr^λ_M(g◦f) (3.73) is non-degenerate.

Proof. By Artin-Wedderburn’s theorem, the algebra Ais isomorphic to a direct product of matrix algebras over K:

A∼=^Yr

i=1

M_ni(K). (3.74)

Since the sum of the usual trace of matrices gives each Athe structure of a symmetric Frobenius algebra, lemma 2.9 shows that the Frobenius formλof Ais given by

λ=^Xr

i=1

λitri, (3.75)

where tri:M_ni(K)→Kis the usual trace of matrices and λ_i ∈K^∗ are non-zero scalars.

Recall that a module over a finite-dimensional algebra is finite-dimensional (as a vector space) if and only if it is finitely generated as a module, cf. [SY11, Proposition 2.5]. A classical theorem in representation theory (cf. theorem 3.3.1 in [EGH⁺11]) asserts that

the only finite-dimensional simple modules of A are given by V₁ :=Kⁿ¹, . . . V_r := K^nr. Since the category (A-Mod)^fg is semisimple, we may decompose the finitely-generated A-modulesM andN as the direct sum of simple modules:

M ∼=





l1

M

i1=1Kⁿ¹



⊕





l2

M

i2=1Kⁿ²



⊕ · · · ⊕





lr

M

ir=1K^nr





N ∼=





l⁰₁

M

i1=1Kⁿ¹



⊕





l⁰₂

M

i2=1Kⁿ²



⊕ · · · ⊕





l⁰_r

M

ir=1K^nr



.

(3.76)

By Schur’s lemma, anyf ∈HomA(M, N) is given byf =f₁⊕f₂⊕. . .⊕f_r wheref_i is a l_i⁰×li-matrix. Similarly, anyg∈HomA(N, M) is given by g=g₁⊕g₂⊕. . .⊕gr where eachg_i is al_i×l⁰_i matrix. Thus,

tr^λ_M(g◦f) = tr^λ_M((g₁f₁)⊕(g₂f₂)⊕. . .⊕g_rf_r)

=^Xr

i=1

tr^λ_(Kni)li(gif_i) (by additivity)

=^Xr

i=1 li

X

j=1

tr^λ_Kni((gif_i)j,j) (by lemma 3.6)

=^Xr

i=1 li

X

j=1 l⁰_i

X

k=1

tr^λ_Kni((gi)j,k◦(fi)k,j).

(3.77)

Sincef was assumed to be non-zero, at least one (fi)j,kis non-zero. Suppose that (f˜i)˜j,k˜ ∈ EndA(Kⁿⁱ) is not the zero morphism. By Schur’s lemma, (f˜i)_˜j,˜k is an isomorphism. Now defineg∈HomA(N, M) as

(gi)j,k := (λ˜i)⁻¹δ_i,˜iδ_j,˜jδ_k,˜k(f˜i)⁻_˜k,¹_˜j. (3.78) Then, by example 3.24,

tr^λ_M(g◦f) = (λ˜i)⁻¹tr^λ_Kⁿ˜i(id_Kⁿ˜i) = 1_K 6= 0. (3.79)

We summarize the situation with the following proposition:

Proposition 3.29. Let (A, λ) be a separable symmetric Frobenius algebra over an alge-braically closed field K. Then, the category of finitely-generated A-modules (A-Mod)^fg has got the structure of a Calabi-Yau category with tracetr^λ_M : EndA(M)→Kas defined in equation (3.51).

Proof. Since A a separable K-algebra, A is finite-dimensional by corollary 3.18. By lemma B.5, all finitely generatedA-modules are necessarily finite-dimensional. It is well-known that the category of finite-dimensional modules over a finite-dimensional algebra

3.2. Constructing an equivalence between Frobenius algebras and Calabi-Yau categories is a finite, linear category, cf. [DSPS14]. SinceA is a separableK-algebra, allA-modules are projective by corollary 3.16. Hence, (A-Mod)^fg is semisimple.

If M is a finitely-generated A-module, the trace tr^λ(M) : End(M) → K as defined in equation (3.51) is symmetric by lemma 3.26, while the induced bilinear form is non-degenerate by lemma 3.28. This shows that (A-Mod)^fg is a Calabi-Yau category.

The following example shows that the assumption that the algebra Ais separable is a necessary condition.

Example 3.30 (Counter-example). LetKbe a field of characteristic two, and consider the group algebraA:=K[Z2]. Then,A∼=K[x]/(x²−1)∼=K[x]/(x−1)². This is in fact a Frobenius algebra with Frobenius formλ(g) =δg,e, which is not separable. Let S be the trivial representation, and consider a projective two-dimensional representation of A which we shall call P. Here, the non-trivial generator g ofA acts on P by the matrix

g= 1 1 0 1

!

. (3.80)

One easily computes that Hom(P, S)∼=ⁿ0 b|b∈K

o, and Hom(S, P)∼= ( a

0

!

|a∈K )

. (3.81) We claim that there is no trace on the representation category of A. Indeed, let trS : End(S)→K be any linear map. Then, the pairing

Hom(S, P)⊗Hom(P, S)→K (3.82)

a0

!

⊗0 b7→trS

0 b a 0

!!

= 0 (3.83)

is always degenerate. Therefore, a non-degenerate pairing does not exist.

Im Dokument Group Actions on Bicategories and Topological Quantum Field Theories (Seite 77-83)