Deformation cohomology of Schur-Weyl categories. Free symmetric categories
aa r X i v : . [ m a t h . QA ] A ug Deformation cohomology of Schur-Weyl categories.Free symmetric categories
Alexei Davydov a ) and Mohamed Elbehiry b )August 27, 2019 a ) Department of Mathematics, Ohio University, Athens, OH 45701, USA b ) Department of Mathematics, Northeastern University, Boston, MA 02115, USA Abstract
The deformation cohomology of a tensor category controls deformations of its monoidalstructure. Here we describe the deformation cohomology of tensor categories generated byone object (the so-called Schur-Weyl categories). Using this description we compute thedeformation cohomology of free symmetric tensor categories generated by one object with analgebra of endomorphism free of zero-divisors. We compare the answers with the exteriorinvariants of the general linear Lie algebra.
Contents R ep ( g ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Deformation cohomology of R ep ( gl ( V )) . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Deformation cohomology of R ep ( gl ( V ) ⊗ A ) . . . . . . . . . . . . . . . . . . . . . . 22 Possible monoidal structures on a given tensor category naturally form an object of an algebro-geometric nature (the moduli space). The tangent space to the moduli space of tensor structuresis computed by the third cohomology of a certain complex, the deformation complex of the tensorcategory [2, 11] . the deformation cohomology of a tensor category is sometimes called its Davydov-Yetter cohomology. S be the free symmetric ten-sor category generated by one object whose endomorphisms are scalar. We show (assuming thatthe ground field is of odd characteristic) that the deformation cohomology H ∗ ( S ) is the exterioralgebra Λ( e , e , e , ... ) on odd degree generators deg ( e i − ) = 2 i − S ( A ) bethe free symmetric tensor category generated by one object whose endomorphism algebra A is adomain. Then (assuming that A is a commutative algebra, which is more than one dimensional)the deformation cohomology of the free symmetric category S ( A ) is the exterior algebra of thefirst cohomology H ( S ( A )) = A , i.e. H ∗ ( S ( A )) ≃ Λ ∗ ( A ) (theorem 4.14). We also use ourmethods to compute the deformation cohomology of the degenerate affine Hecke category L , i.e.the Schur-Weyl category corresponding to the collection of the degenerate affine Hecke algebras.We show that H ∗ ( L ) is the exterior algebra Λ ∗ ( k [ x ]) of the algebra of polynomials (theorem 4.18).The free symmetric tensor category S ( A ) can be thought of as the limiting case of the repre-sentation category R ep ( gl ( V ) ⊗ A ) of the general linear Lie algebra gl ( V ) ⊗ A , when the dimensionof the vector space V goes to infinity. The deformation cohomology of the representation category R ep ( g ) can be identified with the adjoint g -invariants of the exterior algebra Λ ∗ ( g ) g (theorem5.3, see also [2]). Here we assume that the characteristic of the ground field is zero. Classicalinvariant theory says that Λ ∗ ( gl ( V )) gl ( V ) is he exterior algebra Λ( x , x , ..., x d − ) with generatorsof degree deg ( x i − ) = 2 i − d = dim ( V ) (see e.g. [9, 7]). We use the Schur-Weylduality functor SW : S → R ep ( gl ( V )), sending the generator to the vector representation V , torelate the deformation cohomology of S and of R ep ( gl ( V )). Note that the functoriality propertyof the deformation cohomology is not straightforward and is similar to the functoriality of thecentre, or the Hochschild cohomology (see [4]): a tensor functor F : C → D gives rise to a cospanof homomorphisms of graded algebras H ∗ ( F ) H ∗ ( C ) : : ✉✉✉✉✉✉✉✉✉ H ∗ ( D ) d d ■■■■■■■■■ where H ∗ ( F ) is the deformation cohomology of tensor functor F . We show that the deformationcohomology of the Schur-Weyl functor SW is the exterior algebra H ∗ ( SW ) = Λ( e , e , ..., e d − ),the homomorphism H ∗ ( S ) → H ∗ ( SW ) is the quotioning by the ideal generated by e s , s > d −
1, and the homomorphism H ∗ ( R ep ( gl ( V ))) → H ∗ ( SW ) is an isomorphism sending x m to (( m − − e m (theorem 5.7). This in particular give a precise formulation to the intriguingconnection between the combinatorics of partitions (giving the answer for H ∗ ( S )) and the exteriorinvariants of gl ( V ) observed by Kostant in [9]. We also relate the deformation cohomology of S ( A )and of R ep ( gl ( V ⊗ A )) under the assumption that A be a commutative algebra with dim ( A ) > A ⊗ n has no zero-divisors for any n . Then the deformation cohomology of the Schur-Weyl functor SW : S ( A ) → R ep ( gl ( V ⊗ A )) (sending the generator to the vector representation V ⊗ A ) is the exterior algebra H ∗ ( SW ) = Λ ∗ ( A ) and the homomorphisms H ∗ ( S ( A )) → H ∗ ( SW ) ← H ∗ ( R ep ( gl ( V ⊗ A )))are isomorphisms (theorem 5.11). 2he one-dimensional cohomology H ( S ) suggests that the moduli space of tensor structuresof S is (locally) one-dimensional. This was shown to be true in [2]. The detailed analysis of thismoduli space and its relation to the one-parameter family of Hecke categories from [5] will be thesubject of a subsequent publication.Throughout k will be the ground field. We denote by V ect the category of vector spaces over k . By a tensor category over k we mean a monoidal category enriched over V ect . A tensor functorbetween tensor categories is a V ect -enriched monoidal functor. Acknowledgment
This paper had a rather difficult early life. Started in 2016, when the second author was amaster student at Ohio University it was largely complete by the time the second author movedto Northeastern University. Mostly due to the ineffectiveness of the first author in managing hisever increasing load it took three years to do the final polishing. The first author would like tothank Max Planck Institute for Mathematics (Bonn, Germany) for hospitality during the Summerof 2019, crucial for the completion of this paper.
Let C , D be tensor categories, which we assume to be strict for simplicity of exposition. For atensor functor F : C → D define its n -th power by F ⊗ n : C × ... × C → D , F ⊗ n ( X , ..., X n ) = F ( X ⊗ ... ⊗ X n )For n = 0 denote F ⊗ : V ect → D , F ⊗ ( V ) = V ⊗ I where I ∈ D is the unit object. Denote by E ∗ ( F ) = End ( F ⊗∗ ) the collection of endomorphisms algebras of tensor powers ofa monoidal functor F . An element of E n ( F ) is a collection of endomorphisms a X ,...,X n ∈ End D ( F ( X ⊗ ... ⊗ X n )) for X , ..., X n ∈ C natural in the following sense: the diagram F ( X ⊗ ... ⊗ X n ) a X ,...,Xn / / F ( f ⊗ ... ⊗ f n ) (cid:15) (cid:15) F ( X ⊗ ... ⊗ X n ) F ( f ⊗ ... ⊗ f n ) (cid:15) (cid:15) F ( Y ⊗ ... ⊗ Y n ) a Y ,...,Yn / / F ( Y ⊗ ... ⊗ Y n ) (1)commutes for any f i ∈ C ( X i , Y i ).Here we recall (following [2]) how to equip the collection of algebras E ∗ ( F ) with the structure ofcosimplicial algebra.More precisely the image of the coface map ∂ i : End ( F ⊗ n ) → End ( F ⊗ n +1 ) i = 0 , ..., n + 1of an endomorphism a ∈ End ( F ⊗ n ) has the following specialisation on objects X , ..., X n +1 ∈ C : ∂ i ( a ) X ,...,X n +1 = φ (1 F ( X ) ⊗ a X ,...,X n +1 ) φ − , i = 0 a X ,...,X i ⊗ X i +1 ,...,X n +1 , ≤ i ≤ nψ ( a X ,...,X n ⊗ F ( X n +1 ) ) ψ − , i = n + 13ere φ is the tensor structure constraint F ( X ) ⊗ F ( X ⊗ ... ⊗ X n +1 ) → F ( X ⊗ ... ⊗ X n +1 ) , and ψ is the tensor structure constraints F ( X ⊗ ... ⊗ X n +1 ) → F ( X ⊗ ... ⊗ X n ) ⊗ F ( X n +1 ) . The specialisation of the image of the codegeneration map σ i : End ( F ⊗ n ) → End ( F ⊗ n +1 ) i = 0 , ..., n − σ i ( a ) X ,...,X n − = a X ,...,X i ,I,X i +1 ,...,X n − . The zero component of this complex is the endomorphism algebra
End D ( I ) of the unit object I of the category D , which can be regarded as the endomorphism algebra of the functor F ⊗ .The coface maps ∂ i : End D ( I ) → End ( F ) , i = 0 , ∂ ( a ) X = ρ F ( X ) ( a ⊗ F ( X ) ) ρ − F ( X ) , ∂ ( a ) X = λ F ( X ) (1 F ( X ) ⊗ a ) λ F ( X ) − ;here ρ F ( X ) : I ⊗ F ( X ) → F ( X ) and λ F ( X ) : F ( X ) ⊗ I → F ( X ) are the structural isomorphisms ofthe unit object I .It is straightforward to verify the following. Proposition 2.1.
The maps σ i and ∂ j make E ∗ ( F ) a cosimplicial complex. The cohomology of the corresponding cochain differential ∂ : E n ( F ) → E n +1 ( F ) , ∂ = n +1 X i =0 ( − i ∂ in +1 (2)was called the tangent cohomology of F in [2]. Here we call it the deformation cohomology of F . Example . The space of 1-cocycles Z ( F ) coincides with the space Der ( F ) = { a ∈ End ( F ) | F − X,Y a X ⊗ Y F X,Y = 1 F ( X ) ⊗ a Y + a X ⊗ F ( Y ) , X, Y ∈ C} of derivations (or primitive endomorphisms) of F .The subspace of 1-coboundaries B ( F ) ⊂ Z ( F ) corresponds to the subspace Der inn ( F ) of innerderivations of F . The first cohomology H ( F ) is the space OutDer ( F ) = Der ( F ) /Der inn ( F ) of outer derivations of F .The deformation complex E ∗ ( F ) is equipped with the ∪ -product ∪ : E m ( F ) ⊗ E n ( F ) → E m + n ( F )( a ∪ b ) X ,...,X m ,X m +1 ,...,X m + n = φ ( a X ,...,X m ⊗ b X m +1 ,...,X m + n ) φ − , a ∈ E m ( F ) , b ∈ E n ( F ) . Here φ = F X ⊗ ... ⊗ X m ,X m +1 ⊗ ... ⊗ X m + n is the coherence isomorphism F ( X ⊗ ... ⊗ X m ) ⊗ F ( X m +1 ⊗ ... ⊗ X m + n ) → F ( X ⊗ ... ⊗ X m + n ) . The ∪ -product induces an associative multiplication on the cohomology ∪ : H m ( F ) ⊗ H n ( F ) → H m + n ( F ) . (3)Methods similar to those from [8] (see also [1]) show that the cup-product is graded commutative b ∪ a = ( − | a || b | a ∪ b a ∈ H m ( F ) , b ∈ H n ( F ) . Indeed, the commutativity homotopy for the ∪ -product can be chosen as a ∗ b = m X i =1 ( − ( n − i a ∗ i b , a ∗ i b ) X ,...,X m + n − == (1 F ( X ⊗ ... ⊗ X i − ) ⊗ b X i ,...,X i + m − ⊗ F ( X i + m ⊗ ... ⊗ X m + n ) ) a X ,...,X i − ,X i ⊗ ... ⊗ X i + m − ,X i + m ,...,X m + n − . Let now F : C → D and G : D → E be tensor functors. We have two collections of algebrahomomorphisms E n ( G ) → E n ( G ◦ F ) , E n ( F ) → E n ( G ◦ F ) . (4)The first sends a ∈ E n ( G ) into G ( F ( X ⊗ ... ⊗ X n )) G ( ψ ) − (cid:15) (cid:15) G ( F ( X ⊗ ... ⊗ X n )) G ( F ( X ) ⊗ ... ⊗ F ( X n )) a F ( X ,...,F ( Xn ) / / G ( F ( X ) ⊗ ... ⊗ F ( X n )) G ( ψ ) O O where ψ : F ( X ) ⊗ ... ⊗ F ( X n ) → F ( X ⊗ ... ⊗ X n ) is the tensor constraint of F , while the secondsends b ∈ E n ( F ) into G ( F ( X ⊗ ... ⊗ X n )) G ( b X ,...,Xn ) / / G ( F ( X ⊗ ... ⊗ X n )) . It is straightforward to see that the homomorphisms (4) are cosimplicial maps and give rise tohomomorphisms of deformation cohomology H n ( G ) → H n ( G ◦ F ) , H n ( F ) → H n ( G ◦ F ) . (5)We are mostly interested in the case F = Id C . We denote E ∗ ( C ) = E ∗ ( Id C ) , H ∗ ( C ) = H ∗ ( Id C )and call them the deformation complex and the deformation cohomology of the tensor category C .Using the construction (5) a tensor functor F : C → D gives rise to a pair of homomorphisms H n ( C ) → H n ( F ) , H n ( D ) → H n ( F ) . (6) Lemma 2.3.
Let F : C → D be a fully faithful tensor functor. Then the homomorphism H ∗ ( C ) → H ∗ ( F ) is an isomorphism.Let F : C → D be a full tensor functor essentially surjective on objects. Then the homomorphism H ∗ ( D ) → H ∗ ( F ) is an isomorphism.Proof. Let F : C → D be a fully faithful tensor functor. To show bijectivity of E n ( C ) → E n ( F )note that for any X , ..., X n ∈ C the effect on morphisms End C ( X ⊗ ... ⊗ X n ) → End D ( F ( X ⊗ ... ⊗ X n ))is an isomorphism. Moreover the naturality conditions (1) for Id C and F are the same.Let now F : C → D be a full tensor functor essentially surjective on objects. To show bijectivity of E n ( D ) → E n ( F ) note that for any Y , ..., Y n ∈ C there are X , ..., X n ∈ C such that F ( X i ) ≃ Y i .Thus we have isomorphisms End D ( Y ⊗ ... ⊗ Y n ) → End D ( F ( X ) ⊗ ... ⊗ F ( X n )) → End D ( F ( X ⊗ ... ⊗ X n )) . Using the surjections C ( X i , X ′ i ) → D ( F ( X i ) , F ( X ′ i )) ≃ D ( Y i , Y ′ i ) we can see that the naturalityconditions (1) for Id D and F are identified by these isomorphisms.Together with (6) the above lemma assigns a homomorphism H ∗ ( D ) → H ∗ ( C ) to a fullyfaithful tensor functor F : C → D and a homomorphism H ∗ ( C ) → H ∗ ( D ) to a full tensor functor F : C → D essentially surjective on objects. 5
Schur-Weyl categories and their deformation cohomology
Here we recall the definitions from [5] and set our notations.A multiplicative sequence of algebras is a collection of associative unital algebras A ∗ = { A n | n > } (with A = k ) equipped with a collection of (unital) algebra homomorphisms µ m,n : A m ⊗ A n → A m + n , m, n > , satisfying the following associativity axiom: for any l, m, n > A l ⊗ A m ⊗ A n µ l,m ⊗ I / / I ⊗ µ m,n (cid:15) (cid:15) A l + m ⊗ A nµ l + m,n (cid:15) (cid:15) A l ⊗ A m + n µ l,m + n / / A l + m + n . (7)commutes.We denote by µ m ,...,m n : A m ⊗ ... ⊗ A m n → A m + ... + m n the unique composition of homomor-phisms µ m,n .A model example of a multiplicative sequence of algebras is provided by the following con-struction. Example . Let C be a (strict) tensor category such that End C ( I ) = k , where I denotes the unitobject of C . Given an object X of C , the sequence A ∗ with A n = End C ( X ⊗ n ) is multiplicativewith respect to the homomorphisms µ m,n given by the tensor product on morphisms End C ( X ⊗ m ) ⊗ End C ( X ⊗ n ) → End C ( X ⊗ m + n ) . Moreover, any multiplicative sequence can be obtained in this way. Indeed, starting with amultiplicative sequence A ∗ , define its Schur-Weyl category SW ( A ∗ ) with objects [ n ] parameterizedby natural numbers, with no morphisms between different objects and with the endomorphismalgebras End SW ( A ∗ ) ([ n ]) = A n . Define tensor product on the objects of SW ( A ∗ ) by [ m ] ⊗ [ n ] =[ m + n ]. The multiplicative structure of the sequence A ∗ yields the tensor product on morphisms: End SW ( A ∗ ) ([ m ]) ⊗ End SW ( A ∗ ) ([ n ]) = A m ⊗ A n µ m,n / / A m + n = End SW ( A ∗ ) ([ m + n ]) . Note that the Schur-Weyl category SW ( A ∗ ) is a strict (and skeletal) tensor category. Remark . Let C be a non-strict monoidal linear category with the associativity constraint α .For X ∈ C define X ⊗ n inductively by X ⊗ n = X ⊗ X ⊗ n − with X ⊗ = I . Define isomorphisms α m,n : X ⊗ m ⊗ X ⊗ n → X ⊗ m + n inductively by X ⊗ m ⊗ X ⊗ n α m,n / / X ⊗ m + n ( X ⊗ X ⊗ m − ) ⊗ X ⊗ n α X,X ⊗ m − ,X ⊗ n / / X ⊗ ( X ⊗ m − ⊗ X ⊗ n ) ⊗ α m − ,n / / X ⊗ X ⊗ m + n − The homomorphisms µ m,n : End C ( X ⊗ m ) ⊗ End C ( X ⊗ n ) → End C ( X ⊗ m + n ) . can now be defined by µ m,n ( a, b ) = α m,n ( a ⊗ b ) α − m,n . It follows from Mac Lane coherence that the homomorphisms µ m,n make End C ( X ⊗ n ) a multi-plicative sequence of algebras. 6 emark . The notation used for this construction in [5] was SW ( A ∗ ) and SW ( A ∗ ) was used forthe k -linear abelian envelope of SW ( A ∗ ). Since here SW ( A ∗ ) is the more basic object we simplifyits notation.Let f ∗ = { f n | n > } be a sequence of algebra homomorphisms f n : A n → B n between thecorresponding algebras of two multiplicative sequences A ∗ and B ∗ . We call f ∗ a homomorphismof multiplicative sequences if for any m, n the following diagram commutes: A m ⊗ A n f m ⊗ f n / / µ m,n (cid:15) (cid:15) B m ⊗ B nµ m,n (cid:15) (cid:15) A m + n f m + n / / B m + n . (8)We will say that f ∗ is an epimorphism , if all homomorphisms f n are surjective.A model construction of a homomorphism of multiplicative sequences is provided by the fol-lowing Example . Let F : C → D be a tensor functor between (strict) tensor categories (with
End C ( I ) = End D ( I ) = k ). Given an object X ∈ C , define a sequence of homomorphisms f ∗ : A ∗ → B ∗ with A n = End C ( X ⊗ n ) and B n = End D ( F ( X ) ⊗ n ) by f n ( a ) = φ n ( F ( a )) φ − n , where isomorphisms φ n : F ( X ⊗ n ) → F ( X ) ⊗ n are defined inductively by F ( X ⊗ n ) φ n / / F ( X ) ⊗ n F ( X ⊗ X ⊗ n − ) F X,X ⊗ n − / / F ( X ) ⊗ F ( X ⊗ n − ) ⊗ φ n − / / F ( X ) ⊗ F ( X ) ⊗ n − Moreover, any homomorphism of multiplicative sequences can be obtained in this way. In-deed, the construction of Schur-Weyl categories is functorial with respect to homomorphismsof multiplicative sequences: a homomorphism f ∗ : A ∗ → B ∗ defines a strict tensor functor F ( f ∗ ) : SW ( A ∗ ) → SW ( B ∗ ).A gaude transformation between homomorphisms of multiplicative sequences f ∗ , g ∗ : A ∗ → B ∗ is a collection invertible elements c ( n ) ∈ B × n such that c ( n ) f n ( a ) = g n ( a ) c ( n ) for any a ∈ A n andsuch that c ( m + n ) = µ m,n ( c ( m ) ⊗ c ( n )) , ∀ m, n . The following is straightforward.
Proposition 3.5.
Strict tensor functors F : SW ( A ∗ ) → SW ( B ∗ ) between Schur-Weyl categoriessuch that F ([1]) = [1] are in 1-1 correspondence with homomorphisms A ∗ → B ∗ of multiplicativesequences.Tensor natural transformations between tensor functors corresponding to twisted homomorphisms f ∗ , f ( ∗ , ∗ ) and g ∗ , g ( ∗ , ∗ ) are gaude transformation between these twisted homomorphisms. Proposition 3.6.
The deformation complex of the Schur-Weyl category SW ( A ∗ ) is E n ( SW ( A ∗ )) = M m ,...,m n C A m ... + mn ( A m ⊗ ... ⊗ A m n ) , where the direct sum is taken over n -tuples of positive integers.The cosimplicial differentials restricted to C A m ,...,mn ( A m ⊗ ... ⊗ A m n ) are given by the following mbeddings:the zero differential is the direct sum (over m ) of C A m ... + mn ( A m ⊗ ... ⊗ A m n ) → C A m m ... + mn ( A m ⊗ A m ⊗ ... ⊗ A m n ) ; for < i < n + 1 the differential ∂ i is the direct sum (over m ′ i and m ′ i such that m ′ i + m ′′ i = m i )of C A m ... + mn ( A m ⊗ ... ⊗ A m n ) → C A m ...m ′ i + mi ”+ ... + mn ( A m ⊗ ...A m ′ i ⊗ A m ′′ i ⊗ ... ⊗ A m n ) ; the differential ∂ n +1 is the direct sum (over m n +1 ) of C A m ... + mn ( A m ⊗ ... ⊗ A m n ) → C A m ... + mn + mn +1 ( A m ⊗ ... ⊗ A m n ⊗ A m n +1 ) . Proof.
By the definition E n ( SW ( A ∗ )) is the space of natural in X , ..., X n ∈ SW ( A ∗ ) collectionsof endomorphisms a X ,...,X n : X ⊗ ... ⊗ X n → X ⊗ ... ⊗ X n . By naturality any such collection isdetermined by the collection of its values for X = [ m ] , ..., X n = [ m n ]. Being an endomorphismof [ m ] ⊗ ... ⊗ [ m n ] = [ m + ... + m n ] a value a [ m ] ,..., [ m n ] is an element of A m + ... + m n . Naturalityof the collection a is equivalent to the condition that values a [ m ] ,..., [ m n ] belong to the centralisers C A m ... + mn ( A m ⊗ ... ⊗ A m n ).Denote C ( m , ..., m n ) = C A m ... + mn ( A m ⊗ ... ⊗ A m n ) . (9)Below is the picture of the first four layers of the deformation cohomology complex. C (1) C (2) C (3) C (4) C (1 , ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (1 , , / / C (1 , , , + + ❳❳❳❳❳❳❳❳❳❳ C (1 , , + + ❳❳❳❳❳❳❳❳❳❳❳❳ / / C (2 , / / ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (2 , , + + ❳❳❳❳❳❳❳❳❳❳❳❳ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (3 , + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳ / / ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ C (1 , ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ C (1 , , + + ❳❳❳❳❳❳❳❳❳❳❳❳ % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ C (2 , + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ C (1 , / / % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ (cid:24) (cid:24) ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ (cid:24) (cid:24) ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ (cid:27) (cid:27) ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ (cid:21) (cid:21) ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱ Arrows represent components of the differential. Arrows in a layer (homomorphisms between cen-tralisers C ( m , ..., m n ) preserving the sum m + ... + m n ) correspond to the differentials ∂ i for0 < i < n + 1. Arrows between layers are components of the differentials ∂ and ∂ n +1 . Note8hat some of those arrows correspond to two different homomorphisms, e.g. C (1 , → C (1 , , ∂ and ∂ .Define a decreasing filtration on E ∗ ( SW ( A ∗ )) by F p E n ( SW ( A ∗ )) = M m + ... + m n >p C A m ... + mn ( A m ⊗ ... ⊗ A m n ) . (10)Clearly F p E n ( SW ( A ∗ )) is a subcomplex of E n ( SW ( A ∗ )). We call the associated graded complex E np = F p E n /F p +1 E n the horizontal complex of depth p .The filtration (10) gives rise to the spectral sequence E p,q = H p ( E ∗ p + q ) ⇒ H p + q ( SW ( A ∗ )) (11)converging to the deformation cohomology E n ( SW ( A ∗ )).The first four horizontal complexes are depicted below. C (1) C (2) C (3) C (4) C (1 , − ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (1 , , / / C (1 , , , − + + ❳❳❳❳❳❳❳❳❳❳ C (1 , , + + ❳❳❳❳❳❳❳❳❳❳❳❳ / / C (2 , − / / − ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (2 , , + + ❳❳❳❳❳❳❳❳❳❳❳❳ − ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (3 , − + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳ − / / − ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (1 , − ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (1 , , + + ❳❳❳❳❳❳❳❳❳❳❳❳ C (2 , − + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳ − ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C (1 , − / / Here signs are those coming from the formula (2) for the cochain differential on E ∗ ( SW ( A ∗ )).The following show how quickly the complexity of the cohomology H ∗ ( E ∗ p ) grows with p . Example . The cohomology of the horizontal complex of depth 1 are H ( E ∗ ) = 0 , H ( E ∗ ) = C (1 , C (2) . The cohomology of the horizontal complex of depth 2 are H ( E ∗ ) = 0 , H ( E ∗ ) = C (2 , ∩ C (1 , C (3) , H ( E ∗ ) = C (1 , , C (2 ,
1) + C (1 , . The cohomology of the horizontal complex of depth 3 are H ( E ∗ ) = 0 , H ( E ∗ ) = C (3 , ∩ C (2 , ∩ C (1 , C (4) , H ( E ∗ ) = C (1 , , , C (2 , ,
1) + C (1 , ,
1) + C (1 , , , C (2 , , ∩ C (1 , , C (2 ,
2) + C (3 , ∩ C (1 , / / H ( E ∗ ) / / C (1 , , ∩ (cid:0) C (2 , ,
1) + C (1 , , (cid:1) C (3 ,
1) + C (1 , . Remark . Note that the highest degree (degree p ) cohomology of the horizontal complex of anarbitrary depth p is H p ( E ∗ p ) = C (1 , ..., C (2 , , ...,
1) + C (1 , , , ...,
1) + ... + C (1 , ..., , . (12)In general the degree m component of the horizontal complex of depth p has the form E mp = M | λ | = m C ( λ ) , where C ( λ ) = C ( λ , ..., λ m ) and the sum is taken over all compositions of p + 1 into m parts.By a composition λ we mean an expansion of p as the sum of a sequence of positive integers p = λ + ... + λ m . Every composition of p can be thought of as a set-theoretic partition of the set { , ..., p } into intervals, i.e. parts of the form { λ + ... + λ i , λ + ... + λ i +1 , ..., λ + ... + λ i +1 − } . Notealso that every composition λ of p corresponds to a binary vector x ( λ ) = ( x , ..., x p − ) ∈ { , } p − of length p −
1. Under this correspondence x j is 1 if j and j + 1 are in different parts of λ and 0otherwise. This correspondence shows that the combinatorial shape of the horizontal complex ofdepth p is the one of the ( p − C ( λ ) sits in the vertex with thecoordinates x ( λ ), while the edges of the cube correspond to the components of the differential.Note that the sign with which a differential corresponding to the cube edge x → x + e i appears inthe chain differential (2) is ( − P is =1 x i .Define a cubic diagram as a collection of subspaces Q x one for each x ∈ { . } n such that forany x, y ∈ { , } n Q xy ⊂ Q x ∩ Q y . Here xy is the component-wise product of binary vectors. We will also use the labelling bycomposition Q ( λ ) such that Q ( λ ) = Q x ( λ ) .A cubic daigram Q ∗ gives rise to a cochain complex C ∗ = C ∗ ( Q ∗ ) with C k = M l ( λ )= k − Q ( λ ) = M P i x i = k Q x , where the first sum is taken over all compositions in to k − x ∈ { . } n with the coordinate sum being k . The differential d : C k → C k +1 is thedirect sum of signed embeddings Q ( λ ) → Q ( µ ), where µ is the result of subdivision of one ofthe parts of λ into two. Alternatively the differential d : C k → C k +1 is the direct sum of signedembeddings Q x ⊂ Q x + e i , whenever x i = 0. The sign of the embedding Q ( λ ) → Q ( µ ) is ( − j if µ is the division of the j -th part of λ , while the sign of the embedding Q x ⊂ Q x + e i is ( − P is =1 x i . Example . The horizontal complex of depth 3 has the following cubic arrangement C (1 , C (1 , , C (2 , C (2 , , C (4) C (3 , C (1 , , C (1 , , , − / / − O O − O O / / − = = ⑤⑤⑤⑤⑤ / / − = = ⑤⑤⑤⑤⑤ − O O − O O − / / = = ⑤⑤⑤⑤⑤ = = ⑤⑤⑤⑤⑤ Q (1 , , Q (1 , , Q (0 , , Q (0 , , Q (0 , , Q (0 , , Q (1 , , Q (1 , , − / / − O O − O O / / − = = ⑤⑤⑤⑤⑤ / / − = = ⑤⑤⑤⑤⑤ − O O − O O − / / = = ⑤⑤⑤⑤⑤ = = ⑤⑤⑤⑤⑤ A ∗ , which somewhat simplifies thecomputation of the deformation cohomology H n ( SW ( A ∗ )).We say that a multiplicative sequence A ∗ is generated by its first two members A and A if theimages of the homomorphisms µ ,..., , , ,..., : A ⊗ m ⊗ A ⊗ A ⊗ n − m − → A n jointly generate A n as an algebra for any n .Note that in this case the centralisers (9) satisfy the property C ( λ ) ∩ C ( µ ) = C ( λ ∪ µ ) . (13)Here λ and µ are compositions of n and λ ∪ µ stands for their union (the composition correspondingto the union of the equivalence relations corresponding to λ and µ ). For example C (2 , ∩ C (1 ,
2) = C (3). Example . The first deformation cohomology of a Schur-Weyl category SW ( A ∗ ) is H ( SW ( A ∗ )) = { a ∈ Z ( A ) | a ⊗ ⊗ ... ⊗ ⊗ a ⊗ ⊗ ... ⊗ ... + 1 ⊗ ... ⊗ ⊗ a ∈ Z ( A n ) ∀ n } . If A ∗ is a multiplicative sequence generated by its first two members A and A then H ( SW ( A ∗ )) = { a ∈ Z ( A ) | a ⊗ ⊗ a ∈ Z ( A ) } . Finally we give an answer for the deformation cohomology under some very restrictive condi-tions, which will applicable later.
Theorem 3.11.
Let A ∗ be a multiplicative sequence such that its horizontal complexes are acyclicaway from the top degree. Then the deformation cohomology of the Schur-Weyl category SW ( A ∗ ) coincides with the cohomology of the complex ... / / H p ( E ∗ p ) µ ,p +( − p +1 µ p, / / H p +1 ( E ∗ p +1 ) / / ... (14) Proof.
The zero sheet of the spectral sequence associated to the filtration (10) is E p,q = F p E p + q /F p +1 E p + q = E p + qp . By the assumption its cohomology is only non-zero at E p, = H p ( E ∗ p ) . The differential of the first sheet d p, : E p, → E p +1 , coincides with the map H p ( E ∗ p ) → H p +1 ( E ∗ p +1 ) induced by µ ,p + ( − p +1 µ p, → A p +1 .The spectral sequence degenerates at the second sheet giving the result. Here we compute the deformation cohomology of free symmetric categories generated by oneobject.
We denote by S n the group of permutations on n -symbols, i.e. the n -th symmetric group. Herewe look at cubic diagrams of invariants of symmetric group action.Let M be a linear representation of a symmetric group S n . For a composition n + n + ... + n r = n define Q (( n , n , ..., n r ) to be the subspace of invariants M S n × ... × S nr with respectto the subgroup S n × ... × S n r ⊂ S n . It is straightforward that the collection of subspaces Q (( n , n , ..., n r ) forms a cubic diagram Q ∗ ( M ).11 xample . Let k ( S n ) be the regular S n -representation , i.e. the space of function f : S n → k with the S n -action given by ( σ.f )( x ) = f ( xσ ) , σ, x ∈ S n . The components of the cubic diagram Q ∗ ( k ( S n )) have the form Q (( n , n , ..., n r ) = k ( S n ) S n × ... × S nr = k ( S n /S n × ... × S n r ) . (15)Note that the right S n -action on k ( S n )( f.τ )( x ) = f ( τ x ) , τ, x ∈ S n gives rise to an S n -action on the cubic diagram Q ∗ ( k ( S n )).The cubic diagram Q ∗ ( k ( S n )) is special in the sense that for any linear representation M of thesymmetric group S n Q ∗ ( M ) = ( Q ∗ ( k ( S n )) ⊗ M ) S n , where the S n -invariants are taken with respect to the diagonal action.We will (following the idea of [6, proposition 2.2]) compute the cohomology of such cubicdiagrams by relating them to certain standard (co)simplicial complexes. We start by recallingbasic facts about simplicial sets.Let ∆ be the category of finite linearly ordered sets and order preserving maps. Denote by[ n ] the linearly ordered set { , , . . . , n } . Any order preserving map [ n ] → [ m ] is a composite of d i : [ k − → [ k ] (where d i is the unique injective map that does not take the value i ∈ [ k ]) and s j : [ k ] → [ k −
1] (where s j is the unique surjective map that takes the value j twice).A simplicial set is a functor X : ∆ op → S et . We refer to elements of X ( k ) = X ([ k ]) as k -simplices of X . The maps X ( d i ) : X ( k ) → X ( k −
1) and X ( s j ) : X ( k − → X ( k ) are called the face and degeneration maps correspondingly. A simplex is degenerate if it is in the image of a degenerationmap. Example . The simplicial interval is the simplicial set I = ∆ ( − , [1]) : ∆ op → S et . Its k -simplices I ( k ) = ∆ ([ k ] , [1]) are order preserving maps [ k ] → [1], which will be represented bynon-decreasing binary vectors of length k + 1. In this presentation the face map I ( d i ) removes the i -th coordinate, while the degeneration map I ( s j ) duplicates the j -th coordinate. In particular,a simplex is degenerate if the corresponding binary vector has repeating coordinates. Here is thepicture of non-degenerate simplices of the simplicial interval(0) (01) (1)The product X × Y of two simplicial sets X, Y : ∆ op → S et is the simplicial set( X × Y )( k ) = X ( k ) × Y ( k ) , ( X × Y )( f ) = X ( f ) × Y ( f ) f ∈ ∆ ([ n ] , [ m ]) . Example . The simplicial n -cube I n is the n -fold product I × n . We record its simplices asvertical arrays of simplices of I , i.e. k -simplices I n ( k ) are represented by binary matrices with n rows and k + 1 columns, with non-decreasing rows. The face map I n ( d i ) removes the i -th column,while the degeneration map I n ( s j ) duplicates the j -th column. A simplex of I n is degenerate ifthe corresponding binary matrix has repeating columns. Here is the picture of non-degenerate12implices of the simplicial square I (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ Example . The boundary of the simplicial n -cube ∂I n is the simplicial subset of I n consistingof simplices, which do not have the diagonal as an iterated face. In other words, k -simplices ∂I n ( k ) are represented by binary matrices witheither the first column being not all zeroes or the last column being not all ones. Here is thepicture of non-degenerate simplices of the boundary of the simplicial square ∂I (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) For a simplicial set X denote by C ∗ ( X ) its normalised cochain complex with coefficients k , i.e.the collection of C n ( X ) = { f ∈ S et ( X ( n ) , k ) | f ◦ X ( s j ) = 0 ∀ j } vector spaces of functions on non-degenerate n -simplices of X with the differential ∂ : C n ( X ) → C n +1 ( X ) , ∂ = n X i =0 ( − i ∂ i , where ∂ i ( f ) = f ◦ X ( d i ).Note that a map of simplicial sets X → Y (i.e. a natural transformation of functors X, Y : ∆ op →S et ) gives rise to a homomorphism of cochain complexes C ∗ ( Y ) → C ∗ ( X ).For a simplicial subset X ⊂ Y denote by C ∗ ( Y, X ) =
Ker ( C ∗ ( Y ) → C ∗ ( X )) its relative normalisedcochain complex .Note that the natural S n -action on the simplicial n -cube I n gives rise to an S n -action on therelative normalised cochain complex C ∗ ( I n , ∂I n ).13 emma 4.5. The relative normalised cochain complex C ∗ ( I n , ∂I n ) is isomorphic in an S n -equivariant way to the cochain complex C ∗ ( Q ∗ ( k ( S n )))[1] of the cubic diagram of the regularrepresentation of S n shifted by one in degree.Proof. The cochain complex C ∗ ( I n , ∂I n ) in degree m is the space of functions on non-degenerate m -simplices of I n , which contain the diagonal. In the notations of example 4.3 these simplicescorrespond to n -by-( m + 1) binary matrices with non-decreasing rows, with the first column beingall zeroes and with the last column being all ones. It is straightforward to see that by a permutationof rows (which corresponds to the S n -action on C ∗ ( I n , ∂I n )) any such matrix can be brought toa form . . . . . . (16)where the number of zeroes in each row is not more than in the previous one. Let n m − k be theincrement of the number of ones in the ( k + 2)-th column compared to ( k + 1)-th column. Clearly n + n + ... + n r = n is a composition and S n × ... × S n r ⊂ S n is the S n -stabiliser of the matrix(16). Thus the subspace C m ( I n , ∂I n ) of functions supported on the S n -orbit of (16) coincideswith the component (15) of the cubic diagram Q ∗ ( k ( S n )).The differential of C ∗ ( I n , ∂I n ) is the alternating sum of maps of functions induced by columnerasing maps. It is easy to see that they correspond to the embeddings of the cubic diagram Q ∗ ( k ( S n )).Here is our main technical tool for computing the deformation cohomology of free symmetriccategories. Proposition 4.6.
Let M be a linear representation of a symmetric group S n . The cohomologyof the cochain complex C ∗ ( Q ∗ ( M )) of the cubic diagram Q ∗ ( M ) of invariants is zero apart fromdegree n − , where it has the form H n − ( Q ∗ ( M )) = M P n − i =1 (1 + t i )( M ) . Proof.
By example 4.1 the cochain complex C ∗ ( Q ∗ ( M )) coincides with ( Q ∗ ( k ( S n )) ⊗ M ) S n . Bylemma 4.5 the cochain complex C ∗ ( Q ∗ ( k ( S n ))) is isomorphic to the degree shift of the relativenormalised cochain complex C ∗ ( I n , ∂I n )[ − I n is homeomorphic to the n -disk the cohomology H ∗ ( I n ) is concentrated in degree zero.Similarly, the geometric realisation of the boundary ∂I n of the simplicial cube is homeomorphic tothe ( n − H ∗ ( I n ) is non-zero only in degree zero and n −
1. Thelong exact sequence of the relative cohomology shows that the cohomology H ∗ ( I n , ∂I n ) is non-zeroonly in degree n . Finally remark 3.8 gives the answer for the top cohomology H n − ( Q ∗ ( M )). Here we assume that the characteristic of the ground field k is odd. Recall from [5] that theSchur-Weyl category SW ( A ∗ ) of the multiplicative sequence A n = k [ S n ] is the free symmetric14ategory S on one object X . In this case the centralisers C ( λ ) can be written as the subalgebrasof invariants k [ S n ] S λ × ... × S λm with respect to the conjugation action. In other words the horizontalcomplexes correspond to cubic diagrams of S n -invariant of the adjoint action on k [ S n ]. Accordingto proposition 4.6 the horizontal complex of depth n is acyclic away from the top degree n and its n -th cohomology is H n ( E ∗ n ) = k [ S n ] / (cid:0) C k [ S n ] ( t ) + C k [ S n ] ( t ) + ... + C k [ S n ] ( t n − ) (cid:1) . (17)Assuming that the ground field k is of odd characteristic each centraliser C k [ S n ] ( t i ) can be writtenas { x + t i xt i | x ∈ k [ S n ] } . Thus a linear function on H n ( E ∗ n ) is a function f : S n → k , which iszero on x + t i xt i for all x ∈ k [ S n ] and all i = 1 , ..., n −
1. In other words, the dual space of H n ( E ∗ n )is the vector space of functions f : S n → k such that f ( σπσ − ) = sign ( σ ) f ( π ) ∀ σ, π ∈ S n (18)Denote it V n . Proposition 4.7.
Let k be a field of odd characteristic. Then the deformation cohomology of thefree symmetric category S is H n ( S ) = k [ S n ] / (cid:0) C k [ S n ] ( t ) + C k [ S n ] ( t ) + ... + C k [ S n ] ( t n − ) (cid:1) . Proof.
By theorem 3.11 the deformation cohomology H ∗ ( S ) coincides with the cohomology of thecomplex (14). Its dual complex (according to proposition 4.8) is ... V p o o V p +1 µ ∗ ,p +( − p +1 µ ∗ p, o o ... o o Note that for any σ ∈ S p one has µ ,p ( σ ) = τ µ p, ( σ ) τ − , where τ = (1 , , ..., p + 1) ∈ S p +1 is a( p + 1)-cycle. Since sign ( τ ) = ( − p the property (18) implies that µ ∗ ,p ( f ) = f ◦ µ ,p = ( − p f ◦ µ p, = ( − p µ ∗ p, ( f )for any f ∈ V p +1 . Thus µ ∗ ,p + ( − p +1 µ ∗ p, = 0 and H p ( S ) = H n ( E ∗ n ). Lemma 4.8.
Let s n be the dimension of V n . Then X n ≥ s n t n = Y m ≥ (1 + t m − ) . Proof.
Note that a function f ∈ V n must be supported on a union of conjugacy classes of S n .Moreover the centraliser of any conjugacy class in the support should consist of just even permu-tations. It is straightforward to see that such classes correspond to partitions of n into distinct oddparts. The generating function for the numbers of such partitions is clearly Q m ≥ (1 + t m − ).The first few terms of the generating series are X n ≥ s n t n = 1 + t + t + t + t + t + t + 2 t + 2 t + 2 t + 2 t + 3 t + ... Example . For an odd n define the function f n ∈ V n by f n ( t t ...t n − ) = 1. For example, when n = 3 the function f ( t t ) = − f ( t t ) = 1 spans V .For f ∈ V n define c f = P σ ∈ S n f ( σ ) σ ∈ k [ S n ]. Denote e i − = c f i − . For example e = 1 , e = t t − t t . We will also denote by e i − the corresponding class in H i − ( S ). Theorem 4.10.
Let k be a field of odd characteristic. Then the deformation cohomology of thefree symmetric category S is the exterior algebra H ∗ ( S ) = Λ( e , e , e , ... ) , deg ( e i − ) = 2 i − generated by elements defined in example 4.9.Proof. The cup-product (3) defines the homomorphism Λ( e , e , e , ... ) → H ∗ ( S ). It follows fromthe proof of lemma 4.8 that this homomorphism is surjective. Finally dimension comparison showsthat this is an isomorphism. 15 .3 Free symmetric categories Let A be a commutative algebra. Denote by A ⊗ n ∗ S n the skew group algebra with respect tothe permutation S n -action on A ⊗ n . The Schur-Weyl category C = SW ( A ∗ ) of the multiplica-tive sequence A n = A ⊗ n ∗ S n is the free symmetric category S ( A ) on one object X with theendomorphism algebra End C ( X ) = A (see [5] for details).Denote by S m ( A ) = ( A ⊗ m ) S m the m -th symmetric power of a vector space A . Lemma 4.11.
Let A be a commutative algebra without zero divisors, which is more than onedimensional. Then C ( n , ..., n r ) = S n ( A ) ⊗ ... ⊗ S n r ( A ) . Proof.
We first consider the largest centralizer C (1 , , ...
1) = C A ⊗ n ∗ S n ( A ⊗ n ). An element a = P a σ σ ∈ A ⊗ n ∗ S n belongs to C (1 , , ...
1) if for any b ∈ A ⊗ n we have0 = hX a σ ∗ σ, b i = X [ a σ ∗ σ, b ] = X a σ [ σ, b ] = X a σ ( σ ( b ) − b ) σ . Thus a σ ( σ ( b ) − b ) = 0. Note that if σ is non-trivial we can find such b ∈ A ⊗ n that σ ( b ) − b .Indeed, since dim ( A ) ≥
2, we can choose two linearly independent elements of c, d ∈ A c and set b = c ⊗ c ⊗ c ⊗ d ⊗ c...c ⊗ c , where the position of d is where σ has acts nontrivially. Hence a σ = 0for any non-trivial σ and so we get that C (1 , , ...
1) = A ⊗ n .Now note that C ( n , n , ..., n r ) = C (1 , , ... S n × ... × S nr , so C ( n , n , ..., n r ) = ( A ⊗ n ) S n × ... × S nr = S n ( A ) ⊗ ... ⊗ S n r ( A ) . Denote by Λ n ( A ) the exterior power of A . Thus we recover the following (originally [6, propo-sition 2.2]). Lemma 4.12.
Let A be a commutative algebra without zero divisors, which is more than onedimensional. Then the horizontal complex of SW ( A ∗ ) of depth n is acyclic away from the topdegree, where its cohomology is Λ n ( A ) .Proof. This is a direct consequence of proposition 4.6, since M P n − i =1 (1 + t i )( M ) = Λ n ( A ) . Example . For a ∈ A define the endomorphism ψ ( a ) ∈ End ( Id S ( A ) ) by ψ ( a ) [ n ] = a ⊗ ⊗ ... ⊗ ⊗ a ⊗ ⊗ ... ⊗ ... + 1 ⊗ ... ⊗ ⊗ a ∈ A ⊗ n ⊂ End S ( A ) ([ n ]) . It follows from example 3.10 that the assignment a ψ ( a ) gives an isomorphism H ( S ( A )) = A . Theorem 4.14.
Let A be a commutative algebra without zero divisors, which is more than onedimensional. Then the deformation cohomology of the free symmetric category S ( A ) is the exterioralgebra of the first cohomology H ( S ( A )) = AH ∗ ( S ( A )) ≃ Λ ∗ ( A ) . Proof.
The cup-product (3) defines the homomorphism Λ ∗ ( A ) = Λ ∗ ( H ( S ( A ))) → H ∗ ( S ( A )),which is an isomorphism by lemma 4.12 and theorem 3.11. Remark . The theorem 4.11 say that the deformation cohomology of the free symmetriccategory S ( A ) is generated by the space P rim ( S ( A )) of primitive endomorphisms of the identityfunctor.Explicitly an endomorphism α ( a ) corresponding to a ∈ A has the following specialisations α ( a ) [ n ] = a ⊗ ⊗ ... ⊗ ⊗ a ⊗ ... ... + 1 ⊗ ... ⊗ a ∈ A ⊗ n ∗ S n . .4 Degenerate affine Hecke category The degenerate affine Hecke algebra Λ n is the unital associative algebra generated by elements t , . . . , t n − and y , . . . , y n subject to the relations t i = 1 , t i t i +1 t i = t i +1 t i t i +1 , t i t j = t j t i for | i − j | > , (19) y i t i − t i y i +1 = 1 , y i y j = y j y i . (20)The assignments t i ⊗ t i , ⊗ t j t j + m ,y i ⊗ y i , ⊗ y j y j + m (21)define algebra homomorphisms Λ m ⊗ Λ n → Λ m + n . (22)It is easy to see that these homomorphisms satisfy the associativity axiom thus giving rise to themultiplicative sequence of algebras Λ ∗ = { Λ n | n > } The Schur-Weyl category C (Λ ∗ ) of themultiplicative sequence Λ ∗ was called the degenerate affine Hecke category L in [5].Define the length l of a permutation σ as the smallest number of neighbouring transpositions t i required to write σ as a product. We extend the length function to elements of k [ S n ] by takingthe maximum over all non-zero summands: l ( P a τ τ ) = max τ : a τ =0 l ( τ ).For i = 1 , ..., n define ∂ i : S n → Λ n by ∂ i ( σ ) = x i σ − σx σ − ( i ) . Lemma 4.16.
The map ∂ i obeys the following twisted Leibniz formula ∂ i ( στ ) = ∂ i ( σ ) τ + σ∂ σ − ( i ) ( τ ) Moreover, ∂ i ( σ ) ∈ k [ S n ] and l ( ∂ i ( σ )) < l ( σ ) in other terms, ∂ i ( σ ) is a linear combination ofpermutations that are less in length than σ .Proof. We first prove the Leibniz formula: x i στ = ( σx σ − ( i ) + ∂ i ( σ )) τ = σ ( τ x τ − ( σ − ( i )) + ∂ σ − ( i ) ( τ )) + ∂ i ( σ ) τ == στ x τ − ( σ − ( i )) + σ∂ σ − ( i ) ( τ ) + ∂ i ( σ ) τ . Hence, ∂ i ( στ ) = x i στ − στ x ( στ ) − ( i ) = σ∂ σ − ( i ) ( τ ) + ∂ i ( σ )( τ )We prove that ∂ i ( σ ) ∈ k [ S n ] by induction on length. In the base case, when l ( σ ) = 1 we have ∂ i ( t j ) = 0 for | i − j | >
1, and ∂ i ( t i ) = 1, ∂ i +1 ( t i ) = −
1, all of which belong to k [ S n ]. Nowassume that ∂ j ( σ ) ∈ k [ S n ] for all j and σ where l ( σ ) < l . For l ( σ ′ ) = l write σ ′ = t j σ for σ ,where l ( σ ) = l −
1. By the Leibniz formula ∂ i ( σ ′ ) = ∂ i ( t i σ ) = ∂ i ( t j ) σ + t j ∂ t − j ( i ) ( σ ), which clearlybelongs to k [ S n ].We also prove the decreasing length property by induction on length. Again in the base case ∂ i ( t j ) = 0 for | i − j | >
1, and ∂ i ( t i ) = 1, ∂ i +1 ( t i ) = −
1, which fits the property l ( ∂ i ( σ )) < l ( σ )since scalars have zero length. Again taking l ( σ ′ ) = l and writing it as σ ′ = t j σ with l ( σ ) = l − ∂ i ( σ ′ ) = ∂ i ( t j σ ) = ∂ i ( t j ) σ + t j ∂ t − j ( i ) ( σ ). Since l ( σ ) < l ( t j σ ) and ∂ i ( t j ) is just a scalaras seen above, the first term has length less than t j σ . By induction for the second term we have l ( ∂ t − j ( i ) ( σ )) < l ( σ ), hence l ( t j ∂ t − j ( i ) ( σ )) < l ( t j σ ). Thus the second term also satisfies the propertyand we are done.The following is a generalisation of the Bernstein’s lemma, computing the centre of a degenerateaffine Hecke algebra. Lemma 4.17.
The centralisers C ( m , ..., m n ) = C Λ m ... + mn (Λ m ⊗ ... ⊗ Λ m n ) have the form C ( m , ..., m n ) = S m ( k [ x ]) ⊗ ... ⊗ S m n ( k [ x ]) . roof. The proof is very similar to the one of lemma 4.11. We first show that C (1 , , ...,
1) = k [ x ] ⊗ n . Note that an element a ∈ Λ n can be written as a unique combination a = P σ ∈ S n a σ σ with a σ ∈ k [ x ] ⊗ n . Note also that a belongs to C (1 , , ..
1) iff the commutator [ a, x i ] is zero. Writing[ a, x i ] = X σ a σ x i σ − X σ a σ ( x σ ( i ) σ + ∂ σ ( i ) ( σ )) = X σ a σ ( x i − x σ ( i ) ) σ + X σ a σ ∂ σ ( i ) ( σ )we get X σ a σ ∂ σ ( i ) ( σ ) = X σ a σ ( x σ ( i ) − x i ) σ Take σ to be a permutation of maximum length such that a σ = 0. By lemma 4.16 we have that l ( ∂ σ ( i ) σ ) < l ( σ ) < l ( σ ), thus l ( P σ a σ ( x i − x σ ( i ) ) σ ) < l ( σ ), which implies that the coefficient of σ in P σ a σ ( x i − x σ ( i ) ) σ is zero, i.e. a σ ( x i − x σ ( i ) ) = 0. Since a σ = 0 we have x i − x σ ( i ) = 0for all i , which implies σ = e . Thus a ∈ k [ x ] ⊗ n as desired.To treat the general case of C ( m , ..., m n ) recall a well known formula at i − t i t i ( a ) = a − t i ( a ) x i − x i +1 , where a is an element of k [ x ] ⊗ n . By the definition, an element a of C (1 , , ...,
1) = k [ x ] ⊗ n belongsto C ( m , ..., m n ) iff the commutator [ a, t i ] is zero for any t i ∈ S m × ... × S m n . Writing at i − t i a = at i − t i t i ( a ) − t i ( a − t i ( a )) = a − t i ( a ) x i − x i +1 − t i ( a − t i ( a ))we get a − t i ( a ) = 0. Thus C ( m , ..., m n ) = C (1 , , ..., S m × ... × S mn = S m ( k [ x ]) ⊗ ... ⊗ S m n ( k [ x ]) . Theorem 4.18.
The deformation cohomology of the degenerate affine Hecke category L is theexterior algebra of k [ x ] H ∗ ( L ) ≃ Λ ∗ ( k [ x ]) . Proof.
The proof is identical to the proof of theorem 4.11 for A = k [ x ]. Denote by R ep ( g ) the symmetric tensor category of representations of a Lie algebra g . R ep ( g ) Here we recall (from [2]) some basic facts about deformation cohomology of categories of modulesover Hopf algebras in general and of representations of Lie algebras in particular.Let H be a Hopf algebra. Denote by H - M od the tensor category of its (left) modules. Denoteby F : H - M od → V ect the forgetful functor.The collection of tensor powers H ⊗ n has a structure of cosimplicial complex with coface maps ∂ i : H n −→ H n +1 ∂ i ( h ⊗ ... ⊗ h n ) = ⊗ h ⊗ ... ⊗ h n , i = 0 h ⊗ ... ⊗ ∆( h i ) ⊗ ... ⊗ h n , ≤ i ≤ nh ⊗ ... ⊗ h n ⊗ , i = n + 1and codegeneracy maps σ i ( h ⊗ ... ⊗ h n +1 ) = h ⊗ ... ⊗ ε ( h i ) ⊗ ... ⊗ h n +1 ))The associated (unnormalised) cochain complex C ∗ coHoch ( H ) is called the co-Hochschild complex and its cohomology - the co-Hochschild cohomology of Hopf algebra H .18 roposition 5.1. The deformation complex of the forgetful functor F : H - M od → V ect isisomorphic to the co-Hochschild complex of H .The deformation complex of the tensor category H - M od is isomorphic to the subcomplex of adjoint H -invariants of the co-Hochschild complex of H .Remark . The isomorphism in the above proposition is realized by two mutually inverse maps:
End ( F ⊗ n ) −→ H ⊗ n which sends the endomorphism to its specialization on the objects H, ..., H , and H ⊗ n −→ End ( F ⊗ n ) , which associates to the element x ∈ H ⊗ n the endomorphism of multiplying by x .Note that the deformation complexes of the forgetful functor and of the category of modulesover a Hopf algebra depend only on the coalgebra structure.For a Lie algebra g the category of representations R ep ( g ) coincides with the category ofmodules U ( g )- M od over the universal enveloping algebra U ( g ). The following was also establishedin [2]. We add a sketch of the proof here. Theorem 5.3.
The deformation cohomology of the forgetful functor F : R ep ( g ) → V ect is theexterior algebra of g H ∗ ( F ) ≃ Λ ∗ ( g ) . The deformation cohomology of the tensor category of representations of the Lie algebra g is thesubalgebra of invariants H ∗ ( R ep ( g )) ≃ Λ ∗ ( g ) g of the exterior algebra of g .Proof. By Poincare-Birkhoff-Witt theorem U ( g ) is isomorphic to the symmetric algebra S ∗ ( g ) as acoalgebra. The grading of S ∗ ( g ) induces a grading on the co-Hochschild complex C ∗ coHoch ( S ∗ ( g )).The graded component of degree n of C ∗ coHoch ( S ∗ ( g )) is the cubic diagram Q ∗ ( g ⊗ n ) of invariantsthe natural S n -module g ⊗ n . The first part of the theorem now follows from proposition 4.6.By proposition 5.1 the deformation complex C ∗ ( R ep ( g )) is isomorphic to the subcomplex C ∗ coHoch ( U ( g )) g of adjoint g -invariants. Since the Poincare-Birkhoff-Witt isomorphism S ∗ ( g ) → U ( g ) is equivariantwith respect to the adjoint g -action the latter is isomorphic to the subcomplex C ∗ coHoch ( S ∗ ( g )) g .By the above the cubic diagram Q ∗ ( g ⊗ n ) is quasi-isomorphic to its cohomology Λ n ( g )[ n ], with thequasi-isomorphism given by the projection g ⊗ n → Λ n ( g ). Since the projection is g -equivariant thesubcomplex Q ∗ ( g ⊗ n ) g is also quasi-isomorphic to its cohomology Λ n ( g ) g [ n ]. R ep ( gl ( V )) Let V be a vector space over k of dimensional dim ( V ) = d . Here V ∗ stands for the dual vectorspace. Denote by gl ( V ) = V ⊗ V ∗ the general linear Lie algebra of V .We will use Penrose’s calculus [10] to represent tensors graphically. Our diagrams are to beread top down. We will use orientation to distinguish between V and V ∗ . For example, theidentity morphisms on V and V ∗ arecorrespondingly. The canonical pairing V ∗ ⊗ V → k and the canonical element k → V ⊗ V ∗ havethe form 19he following is the graphical presentation of the adjunction between the canonical pairing andthe canonical element = =The transposition of tensor factors is depicted asFor example, the picture =represents the canonical element k → V ∗ ⊗ V . Another example is the identity... σ ... = ... σ − ... (23)where σ ∈ S m is a permutation.As above alt m = ( m !) − P σ ∈ S m sign ( σ ) σ stands for the projector on the m -th exterior power.For a permutation σ ∈ S m denote by σ (2) : ( V ⊗ V ∗ ) ⊗ m → ( V ⊗ V ∗ ) ⊗ m the automorphism permut-ing the factors V ⊗ V ∗ . Graphically... σ (2) ... = ... σ σ ... (24)We denote by alt (2) m = ( m !) − P σ ∈ S m sign ( σ ) σ (2) the projector on Λ m ( V ⊗ V ∗ ).The following is a classical result [9] (see also [7]). Proposition 5.4.
The algebra of invariants Λ ∗ ( gl ( V )) gl ( V ) ≃ Λ( x , x , ..., x d − ) is the exterior algebra with generators x m = alt (2) m ... ∈ Λ m ( V ⊗ V ∗ ) . (25) Remark . The elements x m defined in (25) make sense for all values of m . It can be verified(see [7]) that x m = 0 for even m or for m > d − gl ( V )-action on V , written as V ⊗ V ∗ ⊗ V → V , takes the shapeThe gl ( V ) ⊗ m -action on V ⊗ m , written as ( V ⊗ V ∗ ) ⊗ m ⊗ V ⊗ m → V ⊗ m , has the graphical form... ......In particular, the graphical presentation of the action of x m is... alt (2) m ... ... (26)The next proposition shows that the x m -action on V ⊗ m coincides up to a scalar multiple with theeffect of the element e m ∈ k [ S m ] from example 4.9. Proposition 5.6.
The action of x m on V ⊗ m coincides with (( m − − e m .Proof. The x m -action (26) is the alternating sum of terms... σ (2) ... ...for σ ∈ S m , which according to (24) can be rewritten as... ... σ σ ... ... = ... ... σ σ ... ...21he identity (23) allows us to rewrite it further as follows... σ − σ ...... = ... σ − ... σ ...Thus the x m -action (26) is the alternating sum of the conjugates of the long cycle in S m . Sincethe centraliser of the long cycle in S m has order m we get the answer (( m − − e m .The assignment [1] V extends to a symmetric tensor functor SW : S → R ep ( gl ( V )) , the Schur-Weyl functor (see [5] for details). According to (6) we have a pair of homomorphisms H ∗ ( S ) → H ∗ ( SW ) ← H ∗ ( R ep ( gl ( V ))) . The following theorem shows that the first homomorphism is surjective and the second homomor-phism is an isomorphism.
Theorem 5.7.
The deformation cohomology of the Schur-Weyl functor SW is the exterior algebra H ∗ ( SW ) = Λ( e , e , ..., e d − ) . The homomorphism H ∗ ( S ) → H ∗ ( SW ) is the quotioning by the ideal generated by e s , s > d − .The homomorphism H ∗ ( R ep ( gl ( V ))) → H ∗ ( SW ) is an isomorphism sending x m to (( m − − e m .Proof. The classical Schur-Weyl duality implies that the Schur-Weyl functor is full. Moreover itidentifies the kernel of (i.e. the ideal of morphisms in S annihilated by) the Schur-Weyl functor SW with the tensor ideal J generated by alt d +1 . By lemma 2.3 the deformation cohomology H ∗ ( SW ) of the Schur-Weyl functor SW coincides with the deformation cohomology H ∗ ( S /J ) ofthe quotient category S /J of S by this ideal.The quotient S /J is a Schur-Weyl category SW ( A ∗ ) of the multiplicative sequence A n = k [ S n ] /J n .The methods of section 4.1 also apply to this sequence. Namely the cohomology of horisontal com-plexes is concentrated in the top degree and coincides with the deformation cohomology H ∗ ( S /J ).In particular the homomorphism H ∗ ( S ) → H ∗ ( S /J ) = H ∗ ( SW ) is surjective.Proposition 5.6 says that the image of s s ∈ H ∗ ( S ) is H ∗ ( SW ) coincides up to a scalar with the im-age of x s ∈ H ∗ ( R ep ( gl ( V ))). This shows that the homomorphism H ∗ ( R ep ( gl ( V ))) → H ∗ ( SW )is an isomorphism. Finally remark 5.5 describes the kernel of the homomorphism H ∗ ( S ) → H ∗ ( SW ). R ep ( gl ( V ) ⊗ A ) Let A be a commutative algebra and g be a Lie algebra. The tensor product g ⊗ A has s structureof Lie algebra [ x ⊗ a, y ⊗ b ] = [ x, y ] ⊗ ab, x, y ∈ g , a, b ∈ A .
The following is an auxiliary result. For a ∈ A we will denote by a ( i ) = 1 ⊗ ... ⊗ ⊗ a ⊗ ... ⊗ A ⊗ n with all by i -th components being one.22 emma 5.8. Let A be a commutative algebra with dim ( A ) > and such that A ⊗ n has no zero-divisors. Let M be a free A ⊗ n -module. Let m i ∈ M be such that P ni =1 m i a ( i ) = 0 for any a ∈ A .Then m i = 0 for any i = 1 , ..., n .Proof. It is enough to prove the lemma for M = A ⊗ n . Take a ∈ A to be non-scalar. The system ofequations P ni =1 m i ( a s ) ( i ) = 0 for s = 1 , ..., n can be considered as a linear system with unknowns m i . By multiplying with the adjugate matrix we can see that m i are annihilated by the determinantof the system. This determinant is the Vandermonde determinant Q i Let A be a commutative algebra with dim ( A ) > and such that A ⊗ n has nozero-divisors for any n . Let g be a Lie algebra and let z ( g ) be its centre. Then the algebra ofexterior invariants of g ⊗ A is isomorphic to the exterior algebra of z ( g ) ⊗ A Λ ∗ ( g ⊗ A ) g ⊗ A ≃ Λ ∗ ( z ( g ) ⊗ A ) . Proof. Using the isomorphism ( g ⊗ A ) ⊗ n ≃ g ⊗ n ⊗ A ⊗ n we can identify Λ ∗ ( g ⊗ A ) with the subspaceof S n -anti-invariants in g ⊗ n ⊗ A ⊗ n . An element u ∈ g ⊗ n ⊗ A ⊗ n is g ⊗ A -invariant if n X i =1 ad ( i ) x ( u ) a ( i ) = 0for any x ∈ g and a ∈ A . Here ad x ( y ) = [ x, y ] for y ∈ g and ad ( i ) x ( u ) is the result of applying ad x to the i -th component (in the g ⊗ n part) of u . The product ad ( i ) x ( u ) a ( i ) is in the sense of thenatural A ⊗ n -module structure on g ⊗ n ⊗ A ⊗ n . Now the proposition follows from lemma 5.8. Remark . By the above proposition (assuming that A is a commutative algebra with dim ( A ) > gl ( V ) ⊗ A is isomorphic tothe exterior algebra of A Λ ∗ ( gl ( V ) ⊗ A ) gl ( V ) ⊗ A ≃ Λ ∗ ( A ) . The natural gl ( V )-module V give rise to a gl ( V ) ⊗ A -module structure on V ⊗ A ( x ⊗ a ) . ( v ⊗ b ) = x ( v ) ⊗ ab, x ∈ gl ( V ) , v ∈ V, a, b ∈ A . As before the assignment [1] V extends to a symmetric tensor functor SW : S ( A ) → R ep ( gl ( V ⊗ A )) , (27)the Schur-Weyl functor (see [5] for details). Again according to (6) we have a pair of homomor-phisms H ∗ ( S ( A )) → H ∗ ( SW ) ← H ∗ ( R ep ( gl ( V ⊗ A ))) . (28)The following theorem shows that the first homomorphism is surjective and the second homomor-phism is an isomorphism. Theorem 5.11. Let A be a commutative algebra with dim ( A ) > and such that A ⊗ n has nozero-divisors for any n . Then the deformation cohomology of the Schur-Weyl functor (27) is theexterior algebra H ∗ ( SW ) = Λ ∗ ( A ) and the homomorphisms (28) are isomorphisms.Proof. The proof is similar to the proof of theorem 5.7. The Schur-Weyl duality implies that theSchur-Weyl functor is full. By lemma 2.3 the deformation cohomology H ∗ ( SW ) of the Schur-Weylfunctor (27) coincides with the deformation cohomology H ∗ ( S ( A ) /J ) of the quotient category S ( A ) /J of S by the kernel of the Schur-Weyl functor.23he quotient S ( A ) /J is a Schur-Weyl category SW ( A ∗ ) of the multiplicative sequence A n =( A ⊗ n ∗ S n ) /J n . The methods of section 4.1 also apply to this sequence. Namely the cohomologyof the horisontal complexes is concentrated in the top degree and coincides with the deformationcohomology H ∗ ( S ( A ) /J ). In particular the homomorphism H ∗ ( S ( A )) → H ∗ ( S ( A ) /J ) = H ∗ ( SW )is surjective.By theorem 4.11 the cohomology H ∗ ( S ( A )) is generated by degree one elements and so is H ∗ ( SW ).Now by remark 5.10 the cohomology H ∗ ( R ep ( gl ( V ⊗ A ))) is also generated by degree one elementsand the space of generators is the same as for S ( A ). Comparing (with the help of remark 4.15)the effect of generators on V ⊗ A ∈ R ep ( gl ( V ⊗ A )) we get the theorem. References [1] M. Batanin, A. Davydov, Brackets on deformation cohomology of tensor categories, in prepa-ration.[2] A. Davydov, Twisting of monoidal structures. Preprint of MPI, MPI/95-123, arXiv:q-alg/9703001[3] A. Davydov, Bogomolov multiplier, double class-preserving automorphisms and modular in-variants for orbifolds, Journal of Mathematical Physics , 55 (2014), 9, arXiv:1312.7466.[4] A. Davydov, L. Kong and I. Runkel, Functoriality of the center of an algebra, Advances inMathematics , Volume 285, 5 (2015), 811-876.[5] A. Davydov, A. Molev, A categorical approach to classical and quantum Schur-Weyl duality, Contemporary Math. , 537 (2011), 143-171.[6] V. G. Drinfeld, Quasi-Hopf algebras, Algebra i Analiz , 1:6 (1989), 114-148.[7] M. Itoh, Invariant theory in exterior algebras and Amitsur-Levitzki type theorems, Adv. Math. ,288 (2016), 679-701.[8] M. Gerstenhaber, The cohomology of an associative ring, Ann. of Math. , 78 (2) (1963), 267-288.[9] B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory,