Some Graphical Aspects of Frobenius Structures Bertfried Fauser - - PowerPoint PPT Presentation

Some Graphical Aspects

• f Frobenius Structures

Bertfried Fauser

b.fauser@cs.bham.ac.uk The categorical flow of information in quantum physics and linguistics

October 29-31, 2010 @ Oxford

Theme: Why can we yank?

Let’s first see how it works...

Frobenius algebras (informal)

Img: Ferdinand Georg Frobenius, 1849–1917

k a commutative ring A fin. generated projective k-module m : A × A → A algebra s.t. A∗ = Homk(A, k) dual module with A-A bimodule structure (afb)(x) = f (bxa) ⇒

AA ∼

= AA∗, AA ∼ = A∗

A ∼

b b b b

∼

b b

Example:

• fin. group algebras CG

Frobenius studied: Sn

Finite Hopf algebras (informal)

Img: Heinz Hopf, 1894–1971

k a commutative ring H fin. generated projective module H is an algebra : product m H is a coalgebra: coproduct ∆ f , g, Id ∈ Homk(H, H) caries conv. product : f ⋆ g := m; (f ⊗ g); ∆ IdH is conv. invertible (antipode S) compatibility axiom:

bc bc

∼

bc bc bc bc

Example:

• fin. group algebras CG

Frobenius algebras (historical)

Let A be finitely generated projective over k ∈ cRing i.e ∃ {xi}n

i=1 generators for A (e.g. group algebra CSn)

regular representations xixj =

k f k ij xk

f k

ij ∈ k, mult. table [f k ij ]

• lxi ∼

= [fi]k

j = [f k ij ] left reg. repr. AA

la ∈ Endk(AA)

• rxj ∼

= [fj]k

i = [f k ij ] right reg. repr. AA

ra ∈ Endk(AA)

• k[f k

i,j]ak = [(P(a))ij] parastrophic matrix

(ak ∈ k)

• Thm. Frobenius: If there exists ak ∈ k such that [(P(a))ij] is

invertible then AA ∼ = AA. Examples

• A = k[X, Y ]/X 2, Y 2 is Frobenius
• A = k[X, Y ]/X 2, XY 2, Y 3 is not Frobenius
• A = Mn(k), k division ring, is Frobenius

Dualities: topological move

X object in monoidal category C, rigid ∀X if ∃X ∗, ∗X such that:

◮ right dual:

evX : X ∗ × X → 1X cevX : 1X → X ∗ × X (1X × cevX); (evX × 1X) = 1X (evX ∗ × 1X ∗); (1X ∗ × cevX ∗) = 1X ∗ topological Reidemeister 0 move

◮ left dual: Xev : X × ∗X → 1X Xcev : 1X → X × ∗X

(Xcev × 1X); (1X × Xev) = 1X (1∗X) × ∗Xev); (∗Xcev × 1∗X) = 1∗X topological Reidemeister 0 move

◮ symmetry (braiding):

σX,Y : X × Y → Y × X (σX,Y × 1); (1 × σX,Z); (σY ,Z × 1) = (1 × σY ,Z); (σY ,Z × 1); (1 × σX,Y ) (this is not our yanking move...)

Graphical dualities: topological move, twist

; ∼ ; ;

X

X

; ∼ ∼ ∼ θ

∼ ∼ ;

Bilinear forms

Regular associative bilinear forms Bilr

ass(A, k) ◮ β : A × A → k ∈ Bilr ass(A, k)

if β(ab, c) = β(a, bc) (=ass.) and β non-degenerate

◮ β′ ∼

= β (homothetic) if ∃k ∈ k×, ∃V ∈ Autk(A) such that β(a, b) = kβ′(Va, Vb)

◮ β is symmetric if β(a, b) = β(b, a), ∀a, b ∈ A (i.g. A = Aop) ◮ α ∈ Autk−alg(A) s.t. β(a, b) = β(b, α(a)) Nakayama aut.

unique up to inner aut., iff α = Id ⇔ β is symmetric

◮ β(a, Vb) = β(V ta, b) transposition: (V t)t = α V α−1,

i.g. not identity; α has finite order n then (·)t2n = (·)

◮ λ := β(1, −) = β(−, 1) is called Frobenius homomorphism

If λ(ab) = λ(ba) (⇔ α = Id) ‘trace form’

Bilinear forms cont.

b b b b

∼

bc b bc bc b

∼ ∼

b b b b

∼ ∼ α λ 1 1

b

∼

b b b V t

V β β β β β β β β β

Duality from bilinear forms in Bilr

ass(A, k)

[A unital algebra, fin. generated projective; generators {xi}; β ∈ Bilr

ass(A, k) ]

◮ r : Bilr ass(A, k) ∼

→ Homk(A, A∗) :: β → rβ, rβ(a) = β(a, −)

◮ l : Bilr ass(A, k) ∼

→ Homk(A, A∗) :: β → lβ, lβ(a) = β(−, a) End(A) ∼ → Ae = A ⊗ Aop ∼ → A ⊗ A∗ ∼ → A ⊗ A V ∼ =

• i

xi ⊗ bop

i

∼ =

• i

xi ⊗ fi ∼ =

• i

xi ⊗ yi (·)op maps left to right modules (actions) fi ∈ Homk(A, k) dual elements (indep. of choice) yi ∈ A acts via β (and rβ, lβ) (indep. of choice) Frobenius system: A Frobenius system for A is a triple (β, xi, yi) such that ∀a ∈ A:

• i

xi β(yi, a) = a =

• i

β(a, xi)yi

this is the ‘yanking move’!. . . but wait a moment. . .

Separability and Frobenius

[k ∈ cRing; A a k-algebra, AMA an (A, A)-bimodule, i.e. an Ae left module ]

◮ D : A → M s.t. D(ab) = D(a)b + aD(b) derivation

Derk(A, M) k-module of derivations Dm : A → M :: Dm(a) = am − ma for all m ∈ M inner der.

◮ Dm = 0 iff m ∈ MA := {m ∈ M | am = ma, ∀a ∈ A}

0 → MA → M → Derk(A, M) exact, also MA ∼ = HomAe(A, M), M ∼ = HomAe(Ae, M), mA : Ae → A epi 0 → I(A) = Ker(mA) → A ⊗ Aop → A → 0 exact

◮ δ : A → I(A) :: a → δ(a) = a ⊗ 1 − 1 ⊗ a

Aδ(A) = I(a) = δ(A)A is an ideal Lemma HomAe(I(A), M) ∼ = Derk(A, M)

Separability and Frobenius, cont.

Apply HomAe(−, A) to exact seq., recall H1(A, M) = Ext1

Ae(A, M) ∼

= Derk(A, M)/InnDerk(A, M)

• 1st. Hochschild cohomology grp.

Thm: For k-algebras A is equivalent:

◮ A is projective as left Ae-module ◮ 0 → I(A) → A ⊗ Aop → A → 0 for Ae-modules is split ◮ ∃e = e1 ⊗ e2 ∈ A ⊗ A s.t. ∀a ∈ A : ae = ea and

e1e2 = 1 splitting idempotent Thm: Any projective separable A over k ∈ cRing is finitely generated. Thm: A separable algebra A over a field is semisimple.

Frobenius algebra: characterisation

[recall: A fin. dim k-algebra is Frobenius if AA ∼

= A∗

A as right A-modules ]

Thm: For an n-dim. algebra A, the following are equivalent:

◮ A is Frobenius ◮ the representations r, l : A → Mn(k) are equivalent ◮ ∃a ∈ kn s.t. the parastrophic matrix Pa is invertible ◮ ∃β ∈ Bilr ass and hence a Frobenius homomorphims λ ◮ ∃ hyperplane of A that does not contain any nonzero right

ideals of A

◮ ∃ a Frobenius system (λ, xi, yi), λ ∈ A∗, (λ = β; mA)

e = e1 ⊗ e2 =

i xi ⊗ yi ∈ A ⊗ A s.t.

ae = ea, (e ⊂ (Ae)A) λ(e1)e2 = 1 = e1λ(e2)

◮ and many more. . .

Frobenius extensions (needed for Jones idempotents and polynom)

◮ ring extension A/S, homomorphism S i

→ A, Z(A) center

◮ algebra if : S ∈ cRing and i factors as S → Z(A) → A ◮ A/S central if i(S) = Z(A),

proper if i is 1-1 Let MS and MA be the categories of right S resp A modules, R : MA → MS restriction functor; Define functors:

◮ adjoint:

T : MS → MA :: MS → MS ⊗S A, f → f ⊗ IdA

◮ coadjoint: H : MS → MA :: MS → HomS(AS, MS),

ms → (as → mf (a)s)

◮ (T, R) and (R, H) are adjoint pairs of functors

Def: A ring extension A/S is a Frobenius extension iff H, T are naturally adjoint functors from MS → MA.

• equival. to:

1) SAA ∼ = (AAS)∗ and AS fin. proj. 2) AAS ∼ = ∗(SAA) and SA fin. proj. 3) ∃λ ∈ HomS−S(A, S), xi, yi ∈ A s.t. ∀a ∈ A xiλ(yia) = a = λ(axi)yi

λ-multiplication

End(AS) ∼ = AS ⊗S S

∗A implies the multiplication:

aA; bA = ai ⊗ fi; bj ⊗ gj = aifi(bj) ⊗ gj Thm: If A/S is a Frobenius extension with system (λ, xi, yi), then A ⊗S A ∼ = End(AS) as rings, with λ-multiplication on A ⊗ A (a ⊗ b); (c ⊗ d) := aλ(bc) ⊗ d = a ⊗ λ(bc)d Cor: If (λ, xi, yi) is a Frobenius system for A/S, then e = xi ⊗ yi ∈ (A ⊗S A)A Thm: Let (λ, xi, yi) be a Frobenius system for A/S, all other such systems are in 1-1 correspondence up to equivalence, for d ∈ CentA(S) invertible, by (λd, xi, d−1yi).

Frobenius multiplication & ‘yanking’

We are now in the position to produce the ‘yanking move’:

◮ l.h.s: m : A ⊗ A → A in two versions, using dality via the (left)

regular representation l(a) ∈ A ⊗ A∗, and the λ-multiplication from the Frobenius homomorphism λ(−) = β(1, −) = β(−, 1)

◮ r.h.s: duality expressed via Frobenius system

[This is the archetypical move for ‘teleportation’]

→

∼ β

b b b

∼

b b

xi ⊗ yi β β xi ⊗ yi

Frobenius and Hopf

Let k be a ring with trivial Picard group Pic[k] = 0 (e.g. field) H fin. generated projective

◮ augmentation: ǫ : H → k is a homomorphism

ǫ(ab) = ǫ(a)ǫ(b)

◮ right integral:

r

H ∋ 0 = µr : H → k s.t ∀a ∈ H : µra = ǫ(a)µr

r

H is an ideal in H : H

r

H =

r

H =

r

H H,

r

H ∼

= k

◮ right norm: n ∈ H s.t. for λ Frob. hom. and λn = ǫ, n ∈

r

H

λ(nax) = (λn)(ax) = ǫ(ax) = ǫ(a)ǫ(x) = λ(nǫ(a)x)

bc

∼

b µl

∼ ǫ

bc bc bc b bc

µl ǫ ǫ

b µr

∼ ǫ

bc b bc

µr

[careful: e.g. Clifford algebras don’t have na¨ ıvely such structures. . . ]

Frobenius and Hopf, cont

Thm: Larson-Sweedler-Pareigis H is a fin. proj. Hopf algebra over k, Pic[k] = 0 then:

◮ there is a right Hopf module structure on H∗ ◮ there exists a left integral µl ∈

l

H such that Θ : H → H∗

defined by Θ(x)(y) = µl(yS(x)) is a right Hopf module isomorphism

◮ The antipode S is bijective (∃S−1) ◮ H is a Frobenius algebra with Frobenius homomorphism µl

If ∃ e = xi ⊗ yi separable: axi ⊗ yi = xi ⊗ yia, we obtain:

∼

b

∼

b

xi ⊗ yi

b b b

xi ⊗ yi Frobenius coproduct

Frobenius and Hopf: commonalities and differences

∼

b

∼

b b

bc bc bc bc bc bc bc bc

; ↔

bc bc bc bc

;

b b b b

∼

b

• p

• p

↔

b

∼ xiyi = 1 1 ;

b b b b

∼ ∼

b b b b b b b b

∼ ∼

b b b b b b b b

∼

b b

b b

b

∼

b b b

;

aM a

q-Bit: Clifford view

V = R3, β = δ, generators σ1, σ2, σ2, real algebra Cℓ3,0(V , δ)

[actually a comodulue algebra, Grassman alg. deformation, ask Majid]

idempotents: f0 = 1

2(1 + σ3), f1 = 1 2(1 − σ3), f0 + f1 = 1, primitive ◮ regular rep. la 8-dim ⊂ M8(R) ◮ center CR = R ⊕ R, as real algebra 1, σ1σ2σ3 ≡ σ123 = i ◮ spinor rep. by left ideals: Cℓ3,0fi ∼

= Cℓ3,0Si

C ∼

= fi, σ1fi ∼ = |0i, |1i

◮ R → C → Cℓ3,0 factors through center, fi splitting idemp. ◮ matrix rep: [σ1]i =

1 1 i , [σ2]i = −i i i , [σ3]i = 1 1 i

◮ trace: λ(xy) = β(x, y) = ii | xy | i ∈ C ◮ e = i xi ⊗ yi =| i⊗ | i ◮ Frobenius product:

m(| i⊗ | j) = δi,j | j Frobenius coproduct: δ(| i) =| i⊗ | i

. . . as you knew before

Teleportation again

| ψ

b b

What you should know, but weren’t told. . .

◮ Hattori-Stallings ranks, Higman trace, module of trace forms ◮ Azumaya algebras, quadratic algebras, arithmetic Witt groups ◮ Cartan map, Casimir elements ◮ relation K0(A) → G0(A), Grothendieck groups ◮ relation to character theory, regular characters, irreducible characters ◮ relation to non-linear (solvable) differential equations ◮ Frobenius manifolds, Chazy equation, Egoroff metric ◮ topological quantum field theory, Gromov-Witten invariants ◮ character theory of filtered/graded algebras (of polynomial type) ◮ quantum cohomology (Vershik, Olshanski, Okounkov et al.) ◮ symmetric functions and Grothendieck λ-rings, Macdonald polynomials ◮ spherical categories and generalized hyper geometric spherical functions ◮ Frobenius functors, separable functors and entwined modules:

(Doi-Koppinen, Yetter-Drinfeld, Hopf, and Long modules. . . )

Literature: where it was taken, where to go. . . , a partial & subjective view

◮

• L. Kadison, ‘New examples of Frobenius extensions’, Univ. Lect. Ser. Vol 14, AMS, 1999

◮

• S. Caenepeel, G. Militaru, Sh. Zhu, ‘Frobenius and separable functors for generalized module categories

and non-linear equations’, Springer LNM 1787, 2002

◮

• A. Davydov, A. Molev, ‘A categorical approach to classical and quantum Schur-Weyl duality’,

arXiv:1008.3739v2, 37pp.

◮

• W. Murray, ‘Bilinear forms on Frobenius algebras’, J.Alg. 293, 2005:89–101

◮

• M. Lorenz, ‘Some applications of Frobenius algebras to Hopf algebras’, arXiv:1008.4054v1, 22pp
• M. Lorenz, L. Fitzgerald Tokoly, ‘Projective modules over Frobenius algebras and Hopf comodule algebras’,

arXiv:1008.4056v1, 15pp

◮

• N. Hitchin, ‘Frobenius manifolds’, J. Hurtubise, F. Lalonde (eds.) Gauge theory and symplectic geometry,

69–112, 1997 Kluver Acad. Publ.

◮

• J. Kock, ‘Frobenius algebras ad 2D topological quantum field theories’, London Math. Soc. Student Texts

59, CUP 2003

◮

• M. Atiyah, ‘Topological quantum field theories’, IHES, Publ. Math. 68, 1989, 175–186

◮

• M. M¨

uger, ‘From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories’, J. Pure. Appl. Alg. 180, 2003, 81–157

◮

• B. Pareigis, When Hopf algebras are Frobenius algebras’, J. Alg. 18, 1971:588–596

R.G Larson, M.E. Sweedler, ‘An associative orthogonal bilinear form for Hopf algebras’, Amer. J. Math. 91, 1969:75–93

◮

I.G. Macdonald, ‘Symmetric functions and Hall-polynomials‘, Oxford Sci. Publ. 1995

◮

• A. Okounkov, R. Pandharipande, ‘Quantum cohomology of the Hilbert scheme of points in the plane’,
• Invent. Math. 179, 2010:523–557

◮

• A. Okounkov, G. Olshanski, ‘Shifted Schur functions’, Algebra i Analiz 9 (1997), no. 2, 73–146 (Russian);
• eng. in St. Petersburg Math. J. 9 (1998), no. 2, 239-300.

◮

• A. J. Hahn, ‘Quadratic algebras, Clifford algebras, and arithmetic Witt groups’, Springer Verlag, 1993