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

Some Graphical Aspects of 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...

b b b b b b Frobenius algebras (informal) k a commutative ring A fin. generated projective k -module m : A × A → A algebra s.t. A ∗ = Hom k ( A , k ) dual module with A - A bimodule structure ( afb )( x ) = f ( bxa ) A A ∼ A A ∼ ⇒ = A A ∗ , = A ∗ A ∼ ∼ Example: Img: Ferdinand Georg fin. group algebras C G Frobenius, 1849–1917 Frobenius studied: S n

bc bc bc bc bc bc Finite Hopf algebras (informal) k a commutative ring H fin. generated projective module H is an algebra : product m H is a coalgebra: coproduct ∆ f , g , Id ∈ Hom k ( H , H ) caries conv. product : f ⋆ g := m ; ( f ⊗ g ); ∆ Id H is conv. invertible (antipode S) compatibility axiom: ∼ Example: Img: Heinz Hopf, 1894–1971 fin. group algebras C G

Frobenius algebras (historical) Let A be finitely generated projective over k ∈ cRing i.e ∃ { x i } n i =1 generators for A (e.g. group algebra C S n ) regular representations k f k f k ij ∈ k , mult. table [ f k x i x j = � ij ] ij x k l x i ∼ = [ f i ] k j = [ f k • ij ] left reg. repr. A A l a ∈ End k ( A A ) r x j ∼ = [ f j ] k i = [ f k • ij ] right reg. repr. A A r a ∈ End k ( A A ) k [ f k • � i , j ] a k = [( P ( a ) ) ij ] parastrophic matrix ( a k ∈ k ) Thm. Frobenius: If there exists a k ∈ k such that [( P ( a ) ) ij ] is invertible then A A ∼ = A A . 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 = M n ( k ), k division ring, is Frobenius

Dualities: topological move X object in monoidal category C, rigid ∀ X if ∃ X ∗ , ∗ X such that: ◮ right dual: ev X : X ∗ × X → 1 X cev X : 1 X → X ∗ × X (1 X × cev X ); (ev X × 1 X ) = 1 X (ev X ∗ × 1 X ∗ ); (1 X ∗ × cev X ∗ ) = 1 X ∗ topological Reidemeister 0 move ◮ left dual: X ev : X × ∗ X → 1 X X cev : 1 X → X × ∗ X ( X cev × 1 X ); (1 X × X ev) = 1 X (1 ∗ X ) × ∗ X ev); ( ∗ X cev × 1 ∗ X ) = 1 ∗ X topological Reidemeister 0 move ◮ symmetry (braiding): σ X , Y : X × Y → Y × X ( σ X , Y × 1); (1 × σ X , Z ); ( σ Y , Z × 1) = (1 × σ Y , Z ); ( σ Y , Z × 1); (1 × σ X , Y ) (this is not our yanking move...)

Graphical dualities: topological move, twist ev X ev X ev ev if sym ; ; ; ; ∼ X X ∼ ; ∼ θ ∼ ∼ ∼

Bilinear forms Regular associative bilinear forms Bil r ass ( A , k ) ◮ β : A × A → k ∈ Bil r ass ( A , k ) if β ( ab , c ) = β ( a , bc ) (=ass.) and β non-degenerate ◮ β ′ ∼ = β (homothetic) if ∃ k ∈ k × , ∃ V ∈ Aut k ( A ) such that β ( a , b ) = k β ′ ( Va , Vb ) ◮ β is symmetric if β ( a , b ) = β ( b , a ), ∀ a , b ∈ A (i.g. A � = A op ) ◮ α ∈ Aut k − 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 t a , b ) transposition: ( V t ) t = α V α − 1 , i.g. not identity; α has finite order n then ( · ) t 2 n = ( · ) ◮ λ := β (1 , − ) = β ( − , 1) is called Frobenius homomorphism If λ ( ab ) = λ ( ba ) ( ⇔ α = Id ) ‘trace form’

b bc b b b b b b bc b b bc b b b b Bilinear forms cont. 1 1 ∼ ∼ ∼ sym β β β β λ b V t V ∼ ∼ ∼ α β β β β β

Duality from bilinear forms in Bil r ass ( A , k ) [ A unital algebra, fin. generated projective; generators { x i } ; β ∈ Bil r ass ( A , k ) ] ◮ r : Bil r ass ( A , k ) ∼ → Hom k ( A , A ∗ ) :: β �→ r β , r β ( a ) = β ( a , − ) ass ( A , k ) ∼ ◮ l : Bil r → Hom k ( A , A ∗ ) :: β �→ l β , l β ( a ) = β ( − , a ) → A e = A ⊗ A op ∼ End( A ) ∼ → A ⊗ A ∗ ∼ → A ⊗ A V ∼ x i ⊗ b op ∼ x i ⊗ f i ∼ � � � x i ⊗ y i = = = i i i i ( · ) op maps left to right modules (actions) f i ∈ Hom k ( A , k ) dual elements (indep. of choice) y i ∈ A acts via β (and r β , l β ) (indep. of choice) Frobenius system: A Frobenius system for A is a triple ( β, x i , y i ) such that ∀ a ∈ A : � � x i β ( y i , a ) = a = β ( a , x i ) y i i i this is the ‘yanking move’!. . . but wait a moment. . .

Separability and Frobenius [ k ∈ cRing ; A a k -algebra, A M A an ( A , A )-bimodule, i.e. an A e left module ] ◮ D : A → M s.t. D ( ab ) = D ( a ) b + aD ( b ) derivation Der k ( A , M ) k -module of derivations D m : A → M :: D m ( a ) = am − ma for all m ∈ M inner der. ◮ D m = 0 iff m ∈ M A := { m ∈ M | am = ma , ∀ a ∈ A } 0 → M A → M → Der k ( A , M ) exact, also M A ∼ = Hom A e ( A e , M ), m A : A e → A epi = Hom A e ( A , M ), M ∼ 0 → I ( A ) = Ker( m A ) → A ⊗ A op → A → 0 exact ◮ δ : A → I ( A ) :: a �→ δ ( a ) = a ⊗ 1 − 1 ⊗ a A δ ( A ) = I ( a ) = δ ( A ) A is an ideal Lemma Hom A e ( I ( A ) , M ) ∼ = Der k ( A , M )

Separability and Frobenius, cont. Apply Hom A e ( − , A ) to exact seq., recall A e ( A , M ) ∼ H 1 ( A , M ) = Ext 1 = Der k ( A , M ) / InnDer k ( A , M ) 1st. Hochschild cohomology grp. Thm: For k -algebras A is equivalent: ◮ A is projective as left A e -module ◮ 0 → I ( A ) → A ⊗ A op → A → 0 for A e -modules is split ◮ ∃ e = � e 1 ⊗ e 2 ∈ A ⊗ A s.t. ∀ a ∈ A : ae = ea and � e 1 e 2 = 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 A A ∼ A as right A -modules ] = A ∗ Thm: For an n -dim. algebra A , the following are equivalent: ◮ A is Frobenius ◮ the representations r , l : A → M n ( k ) are equivalent ◮ ∃ a ∈ k n s.t. the parastrophic matrix P a is invertible ◮ ∃ β ∈ Bil r ass and hence a Frobenius homomorphims λ ◮ ∃ hyperplane of A that does not contain any nonzero right ideals of A ◮ ∃ a Frobenius system ( λ, x i , y i ), λ ∈ A ∗ , ( λ = β ; m A ) e = � e 1 ⊗ e 2 = � i x i ⊗ y i ∈ A ⊗ A s.t. � λ ( e 1 ) e 2 = 1 = � e 1 λ ( e 2 ) ae = ea , ( e ⊂ ( A e ) A ) ◮ and many more. . .

Frobenius extensions (needed for Jones idempotents and polynom) i ◮ ring extension A / S , homomorphism S → 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 M S and M A be the categories of right S resp A modules, R : M A → M S restriction functor; Define functors: ◮ adjoint: T : M S → M A :: M S �→ M S ⊗ S A , f �→ f ⊗ Id A ◮ coadjoint: H : M S → M A :: M S �→ Hom S ( A S , M S ), 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 M S → M A . equival. to: = ( A A S ) ∗ and A S fin. proj. 1) S A A ∼ 2) A A S ∼ = ∗ ( S A A ) and S A fin. proj. 3) ∃ λ ∈ Hom S − S ( A , S ), x i , y i ∈ A s.t. ∀ a ∈ A � x i λ ( y i a ) = a = � λ ( ax i ) y i

λ -multiplication End( A S ) ∼ ∗ A implies the multiplication: = A S ⊗ S S a A ; b A = � a i ⊗ f i ; � b j ⊗ g j = � a i f i ( b j ) ⊗ g j Thm: If A / S is a Frobenius extension with system ( λ, x i , y i ), then A ⊗ S A ∼ = End( A S ) as rings, with λ -multiplication on A ⊗ A ( a ⊗ b ); ( c ⊗ d ) := a λ ( bc ) ⊗ d = a ⊗ λ ( bc ) d Cor: If ( λ, x i , y i ) is a Frobenius system for A / S , then e = � x i ⊗ y i ∈ ( A ⊗ S A ) A Thm: Let ( λ, x i , y i ) be a Frobenius system for A / S , all other such systems are in 1-1 correspondence up to equivalence, for d ∈ Cent A ( S ) invertible, by ( λ d , x i , d − 1 y i ).

b b b b b 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’] ev X � x i ⊗ y i � x i ⊗ y i β �→ ∼ ∼ β β

bc b bc bc bc bc b bc bc 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 ) � r ◮ right integral: H ∋ 0 � = µ r : H → k s.t ∀ a ∈ H : µ r a = ǫ ( a ) µ r � r � r � r � r � r H ∼ H is an ideal in H : H H = H = H H , = k � r ◮ right norm: n ∈ H s.t. for λ Frob. hom. and λ n = ǫ , n ∈ H λ ( nax ) = ( λ n )( ax ) = ǫ ( ax ) = ǫ ( a ) ǫ ( x ) = λ ( n ǫ ( a ) x ) b µ l b µ r ǫ ǫ µ l µ r ∼ ∼ ∼ ǫ ǫ [careful: e.g. Clifford algebras don’t have na¨ ıvely such structures. . . ]