Braid group representations from twisted tensor products of algebras
aa r X i v : . [ m a t h . QA ] J un BRAID GROUP REPRESENTATIONS FROM TWISTED TENSORPRODUCTS OF ALGEBRAS
PAUL GUSTAFSON , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG Abstract.
We unify and generalize several approaches to constructing braid grouprepresentations from finite groups, using iterated twisted tensor products. Our resultshint at a relationship between the braiding on the G -gaugings of a pointed modularcategory C ( A, Q ) and that of C ( A, Q ) itself. Introduction
Braid group representations are plentiful: for example, from any object X in a braidedfusion category C one obtains a sequence of braid group representations ρ X : B n → End( X ⊗ n ). Braided vector spaces ( R, V ) [1] (i.e., matrix solutions to the Yang-Baxterequation) are also a very rich source, as are families of finite dimensional quotients ofthe braid group algebra F ( B n ), such as Hecke algebras, Temperley-Lieb algebras [24]and BMW-algebras [3, 28]. That the braided fusion category construction essentiallysupersedes these sources is well-known, but explicit matrices for the generators arenot easy to come by–one typically needs the associativity constants (6 j -symbols or F -matrices) in addition to the R -symbols, and these are only available for a few familiesof categories. Generally, the irreducible representations of B n are only classified fordimensions at most n [12] and for B for dimensions up to 5 [36]. Braided vector spaces( R, V ) are only classified for dim( V ) = 2 [21].In this article we outline an approach to finding families of braid group representationsfrom twisted tensor products of algebras. We motivate our approach with the followingtwo “proof of principle” examples.In [18] the following representations of the braid group B n were described: set q = e πi/m for m odd and let A n ( Z m ) be the C -algebra generated by u i for 1 ≤ i ≤ n − • u mi = 1 Date : June 20, 2019.The authors gratefully acknowledge support under USA NSF grant DMS-1664359. We also thankC. Galindo, JM Landsberg, Z. Wang and S. Witherspoon for valuable insight. ER is also partiallysupported by a Texas A&M Presidential Impact Fellowship. , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG • u i u i +1 = q u i +1 u i • u i u j = u j u i for | i − j | 6 = 1The algebra A n ( Z m ) is denoted by ES ( m, n −
1) in [26] and by T mn ( q ) in [34]. Ourperspective is to regard A n ( Z m ) as an iterated twisted tensor product of the groupalgebra C [ Z m ]: A n ( Z m ) = C [ Z m ] ⊗ ϑ C [ Z m ] ⊗ ϑ · · · ⊗ ϑ C [ Z m ] , where ϑ is the twisting map corresponding to the second relation above. Defining ρ n ( σ i ) = R i := 1 √ m m X j =0 q j u ji we obtain a representation B n → A n ( Z m ). These representations are known to havefinite image, see [16, 18]. Moreover, we may obtain a matrix representation by defining U ∈ End( C m ⊗ C m ) by U ( e i ⊗ e j ) = q j − i e i +1 ⊗ e j +1 and assigning u i Id ⊗ i ⊗ U ⊗ Id ⊗ n − i − . From this representation one obtains a braided vector space as R := ρ ( σ ) on V = C m .Another example of braid group representations related to twisted tensor productsof algebras is found in [32] (due to Jones) where the quaternion group Q appears. For1 ≤ i ≤ n − A n ( Q ) be the algebra generated by u i , v i satisfying:(1) u i = v i = − i ,(2) [ u i , v j ] = − | i − j | < u i , u j ] = [ v i , v j ] = 1,(4) [ u i , v j ] = 1 if | i − j | ≥ A n ( Q ) is not, strictly speaking, a twisted tensor product of group algebraswe nonetheless obtain braid group representations via σ i (1 + u i + v i + u i v i ) . In this article we initiate the general problem of finding braid group representationsin twisted tensor products of (group) algebras, unifying the two examples just outlined.Our study is not just motivated by idle curiosity. In the last section we explore somerelationships between these twisted tensor products of group algebras and G -gaugingsof pointed modular categories, laying the groundwork towards understanding braidgroup representations associated with weakly group theoretical modular categories andthe property F conjecture. This conjecture [29] predicts that the braid group imagesobtained from any weakly integral braided fusion category C have finite image, i.e., C has property F . By taking Drinfeld centers one may reduce this conjecture to thecase where C is a modular category. The property F conjecture has been verified for n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 3 many classes of braided fusion categories: for example, group-theoretical categories [11],quantum group categories [24, 14, 27, 32, 34], and certain metaplectic categories [20].The paper is organized as follows: in section 2 we set down the general frameworkfor our problem, which is explicitly described and analyzed in section 3. We carryout several case studies for both abelian and non-abelian cases in section 4 while theconnections to categories obtained by gauging symmetries of pointed modular categoriesare speculated upon in section 5, followed by a short section of conclusions. An appendixcontains some MAGMA code for some explicit examples.2.
Preliminaries
We first describe the general algebraic ingredients for the problem we are interestedin.2.1.
Twisted Tensor Products.
The treatment of twisted tensor products in [6] ismost suitable for our purposes: Let A and B be F -algebras with multiplication maps µ A , µ B respectively, and a map ϑ : B ⊗ A → A ⊗ B , i.e. ϑ is F -linear map with ϑ ( b ⊗
1) = (1 ⊗ b ) and ϑ (1 ⊗ a ) = ( a ⊗ µ ϑ : A ⊗ B → A ⊗ B defined by µ ϑ = ( µ A ⊗ µ B ) ◦ ( Id A ⊗ ϑ ⊗ Id B )defines an associative multiplication if and only if:(2.1) ϑ ◦ ( µ B ⊗ µ A ) = µ ϑ ◦ ( ϑ ⊗ ϑ ) ◦ ( Id B ⊗ ϑ ⊗ Id A ) . The corresponding algebra, denoted A ⊗ ϑ B will be called a twisted tensor product of A and B , and the map ϑ will be called a unital twisting map. We shall be mostinterested in the case where A = B .To iterate this process, we rely on results of [23]. Given 3 algebras A, B and C andunital twisting maps ϑ : B ⊗ A → A ⊗ B , ϑ : C ⊗ B → B ⊗ C and ϑ : C ⊗ A → A ⊗ C each of which satisfy eqn. (2.1) one can define two maps T = ( Id A ⊗ ϑ ) ◦ ( ϑ ⊗ Id B ) on C ⊗ ( A ⊗ ϑ B ) and T = ( ϑ ⊗ Id C ) ◦ ( Id B ⊗ ϑ ) on ( B ⊗ ϑ C ) ⊗ A which are potentiallyunital twisting maps. [23, Theorem 2.1] show that these are both unital twisting mapsif and only if the compatibility condition:(2.2) ( Id A ⊗ ϑ )( ϑ ⊗ Id B )( Id C ⊗ ϑ ) = ( ϑ ⊗ Id C )( Id B ⊗ ϑ )( ϑ ⊗ Id A )is satisfied. Moreover, the two iterated twisted tensor products ( A ⊗ ϑ B ) ⊗ T C and A ⊗ T ( B ⊗ ϑ C ) constructed from these twisting maps are isomorphic algebras. One mayinductively define twisted tensor products for any number of algebras A i provided theanalogous compatibility conditions are satisfied. Again, we will be especially interestedin the case where A = A i = A j , and ϑ = ϑ i,i +1 : A i +1 ⊗ A i → A i ⊗ A i +1 for adjacentcopies of A and σ = ϑ i,j : A j ⊗ A i → A i ⊗ A j for | i − j | > PAUL GUSTAFSON , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG σ ( a ⊗ b ) = b ⊗ a . In fact, for all of our examples we will have ϑ ( a ⊗ b ) = τ ( a, b ) b ⊗ a forsome function τ : A ⊗ A → F . One then easily sees that 2.2 is satisfied:( Id ⊗ ϑ )( σ ⊗ Id )( Id ⊗ ϑ )( a ⊗ b ⊗ c ) and ( ϑ ⊗ Id )( Id ⊗ σ )( ϑ ⊗ Id )( a ⊗ b ⊗ c )are both equal to τ ( a, b ) τ ( b, c )( c ⊗ b ⊗ a ). Moreover, condition (2 .
1) and unitality areequivalent to τ : A ⊗ A → F being a bihomomorphism of F -algebras: τ ( a a , b b ) = τ ( a , b ) τ ( a , b ) τ ( a , b ) τ ( a , b ) , and unitality implies τ (1 , a ) = τ ( a,
1) = 1, while bilinearity is immediate.2.2.
Braid group representations and property F . Our goal is to study families ofrepresentations of the braid group B n . In particular we are interested in representationsthat are related in the following way: Definition 2.1.
An indexed family of complex B n -representations ( ρ n , V n ) is a sequenceof braid representations if there exist injective algebra homomorphisms ι n : C ρ n ( B n ) → C ρ n +1 ( B n +1 ) such that the following diagram commutes: C B n / / (cid:127) _ (cid:15) (cid:15) C ρ n ( B n ) (cid:127) _ ι n (cid:15) (cid:15) C B n +1 / / C ρ n +1 ( B n +1 )where the left-hand side of the square is induced by the inclusion B n ֒ → B n +1 given by σ i σ i .Our examples are typically of the following form: let 1 ∈ A ⊂ A ⊂ · · · ⊂ A n ⊂ · · · be a tower of finite dimensional semisimple algebras, and ρ n : C B n → A n algebrahomomorphisms that respect the inclusions A n ⊂ A n +1 . Then the canonical faithfulrepresentation of A n provides a sequence of representations.For example, we obtain a sequence of B n -representations from any braided vectorspace ( R, V ) i.e., an invertible operator R ∈ Aut( V ⊗ ) that satisfies the Yang-Baxterequation ( R ⊗ I )( I ⊗ R )( R ⊗ I ) = ( I ⊗ R )( R ⊗ I )( I ⊗ R ) ∈ Aut( V ⊗ ) . Explicitly we have B n → Aut( V ⊗ n ) via σ i → Id ⊗ i − V ⊗ R ⊗ Id ⊗ n − i − V .Other standard examples come from the Temperley-Lieb, Hecke and BMW-algebrasmentioned in the Introduction.Some conjectures on the images of such representations are found in [33, 15, 16]. Forexample it is an open question whether unitary braided vector spaces have virtualyabelian images, but there is strong evidence that this is so. n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 5 Twisted Tensor Products of Algebras and Yang-Baxter Operators
The problem that we propose to study is the following:
Problem . Find and classify braid group representations inside twisted tensor prod-ucts of (group) algebras, generalizing the well-known Gaussian solutions.3.1.
Twisted tensor products of group algebras.
First we describe the twistedtensor products of (group) algebras we will study. Fix a finite group G and q ∈ U (1).On the group algebra Q ( q )[ G ] we would like to find a unital twisting map ϑ : Q ( q )[ G ] ⊗ Q ( q )[ G ] → Q ( q )[ G ] ⊗ Q ( q )[ G ] such that ϑ ( g ⊗ h ) = τ ( g, h ) h ⊗ g on basis elementswhere τ ( g, h ) = q α ( g,h ) . We use subscripts to emphasize that g ∈ G is a basis elementof the first factor and h ∈ G is a basis element of the second factor. If ϑ is a unitaltwisting map then τ ( g, h ) : G × G → U (1) is a bicharacter of G . Since q α ( g k ,h ) = q kα ( g,h ) we assume that q is an m th root of unity (where m | exp ( G )), and that α : G × G → Z m is a bihomomorphism. Now define σ : Q ( q )[ G ] ⊗ Q ( q )[ G ] → Q ( q )[ G ] ⊗ Q ( q )[ G ] to bethe usual flip map g ⊗ h = h ⊗ g . It is routine to check that (2.2) is satisfied by ϑ and σ .With these verifications we can define a finite dimensional semisimple algebra A n ( G, τ )as an iterated twisted tensor product of C [ G ] follows: as a Q ( q ) vector space A n ( G, τ ) = Q ( q )[ G ] ⊗ n − . For each 1 ≤ i ≤ n − g ∈ G we define elements g i = 1 ⊗ i − ⊗ g ⊗ ⊗ n − i − . We can then dispense with the ⊗ symbol altogether, and write monomialsas g ( i )1 · · · g ( i n − ) n − where g ( i j ) ∈ G . The multiplication on A n ( G, τ ) have the followingstraightening rules on the generators g i : g i h j = h j g i , | i − j | > q ± α ( g,h ) h i ± g i j = i ± gh ) i j = i where the bihomomorphism α : G × G → Z m determines τ . The following is presumablywell-known but can be proved directly using classical techniques, which we provide forthe reader’s amusement. Proposition 3.2.
The algebra A n ( G, τ ) is semisimple of dimension | G | n − over Q ( q ) .Proof. Notice that the set of monomials in normal form M := { q ℓ g ( i )1 · · · g ( i n − ) n − : g ( i j ) ∈ G, ℓ ∈ Z m } form a basis for A n ( G, τ ) over Q since A n ( G, τ ) is Q ( q )[ G ] ⊗ n − as a vectorspace. To show that A n ( G, τ ) is semisimple, let X ⊂ A n ( G, τ ) be a submodule, and π : A n ( G, τ ) → X any vector space projection. Note that any t ∈ M has an inversein M , since the straightening rules allow us to write t − in the normal form of M . We PAUL GUSTAFSON , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG then use the standard averaging trick to find an A n ( G, τ )-module projection onto X : T π ( y ) := 1 m | G | n − X t ∈ M t π ( t − y ) . One can readily check that T π is a surjective A n ( G, τ )-module homomorphism so thatthe kernel of T π provides a direct complement to X in A n ( G, τ ), proving that A n ( G, τ )is semisimple. (cid:3)
Connection to group extensions.
We also note the following alternative construc-tion of the algebra A n ( G, τ ) via group extensions. In this section, we show that, as a Q -algebra, A n ( G, τ ) is isomorphic to the group algebra over Q of a central extension of G n − .More concretely, let G be a finite group and m | exp( G ). Let α : G × G → Z m be abihomomorphism. For n ≥
2, let c : G n × G n → Z m be the bihomomorphism definedby c ( g, h ) = − n − X i =1 α ( h i , g i +1 ) . Since c is a bihomomorphism, it satisfies the 2-cocycle condition. We define G × α n tobe the central extension of G n corresponding to the 2-cocycle c ∈ Z ( G n , Z m ). Proposition 3.3.
Let q be a primitive m -th root of unity. There is an isomorphism of Q -algebras A n +1 ( G, τ ) ∼ = Q ( G × α n ) , where τ : Q ( q )[ G ] ⊗ Q ( q )[ G ] → Q ( q )[ G ] ⊗ Q ( q )[ G ] is the same twisting map definedabove, i.e. τ ( g ⊗ h ) = q α ( g,h ) h ⊗ g on basis elements.Proof. We represent G × α n as the set Z m × G n where the multiplication is given by( x × g ) · ( y × h ) = ( c ( g, h ) + x + y ) × gh. Let φ : A n +1 ( G, τ ) → Q ( G × α n ) be the Q -linear bijection defined by q j g ⊗ · · · ⊗ g n − j × ( g , . . . g n − ). To see that φ is an algebra map, we need to verify that it preservesthe straightening relations. If i − j = 1, we have c ( φ ( g i ) , φ ( h j )) = 1. If i − j = 1, wehave c ( φ ( g i ) , φ ( h j )) = α ( h, g − ). Thus,[ φ ( g i ) , φ ( h i +1 )] = φ ( g i ) φ ( h i +1 ) φ ( g − i ) φ ( h − i +1 )= φ ( g i )( α ( g, h ) × ( e, . . . , e, g − i , h i +1 , e, . . . , e )) φ ( h − i +1 )= α ( g, h ) × e = φ (cid:0) q α ( g,h ) (cid:1) . n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 7
Thus, the straightening relations are preserved by φ . It follows that φ is an isomorphismof Q -algebras. (cid:3) Braid group representations.
The second part of the problem is to look forand classify the representations of the braid group inside the algebras A n ( G, τ ). Fix apair (
G, α ) where G is a finite group and α : G × G → Z m is a bihomomorphism and thecorresponding twisted tensor power A n ( G, τ ). We are interested in finding invertible: r = X g ∈ G f ( g ) g ∈ C [ G ]so that for i = 1 , r i := P g ∈ G f ( g ) g i ∈ A ( G, τ ) ⊗ Q ( q ) C satisfy the braid relation r r r = r r r . We shall call such solutions r A ( G, τ ) -Yang-Baxter operators(YBOs) . Since the braid equation may be written as a linear combination of monomials g ( i )1 g ( i )2 ∈ A ( G, τ ) with coefficients in Q ( q )[ x , . . . , x | G | ] where x i := f ( g ( i ) ), we mayassume that the function f takes values in Q ( q ) = Q i.e., the algebraic closure of Q ( q ).For the sake of notation we will usually just consider scalars in the complex field C .This should be compared with the problem of finding Yang-Baxter operators on avector space V . In our case we seek invertible r ∈ C [ G ] so that ρ n ( σ i ) = r i is ahomomorphism ρ n : B n → A n ( G, τ ) (suitably complexified). As A n ( G, τ ) ⊗ C is a finitedimensional semisimple C -algebra, one can obtain B n -representations by pull-back onany A n ( G, τ ) ⊗ C -module. For example, one might use the regular representation toget a sequence of braid group representations ( ρ n , V n ), as defined in [33]. Howeverone cannot, in general, turn such a homomorphism into a solution to the Yang-Baxterequation. There is one situation where one can perform such a transformation: if thesequence of braid group representations ( ρ n , V n ) is localizable in the sense of [33]. Forexample, suppose A n ( G, τ ) has a representation ϑ n of the form V ⊗ n with ϑ ( g i ) actinglocally: ϑ n ( g i )( v ⊗ · · · v i ⊗ v i +1 ⊗ · · · ⊗ v n ) = ( v ⊗ · · · ϑ ( g )( v i ⊗ v i +1 ) ⊗ · · · ⊗ v n )where ϑ : G → Aut( V ⊗ ) is a G -representation. Then ϑ ( r ) will be a Yang-Baxter oper-ator, and ϑ n ( r i ) = ( Id V ) ⊗ i − ⊗ ϑ ( r ) ⊗ ( Id V ) ⊗ n − i − is a localization of the correspondingbraid group representation.We may also put a ∗ -structure on A n ( G, τ ) as follows: define g ∗ i = g − i and q ∗ =1 /q = q and then extend to an antiautomorphism on products and linearly on sums inthe usual way. This makes A n ( G, τ ) a ∗ -algebra. In this way we can discuss unitary A ( G, τ )-YBOs, as those r with r ∗ r = 1.3.3. Equivalence Classes of A n ( G, τ ) . For a fixed G , different choices of τ give iso-morphic algebras. We identify a few of these isomorphisms in order to reduce thecomplexity of our main goal. PAUL GUSTAFSON , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG One equivalence of A n ( G, τ ) comes from the choice of Galois conjugates of q . For( s, m ) = 1 defining τ s ( g, h ) = q sα ( g,h ) obviously gives us A n ( G, τ ) ∼ = A n ( G, τ s ) by Galoisconjugation.Another equivalence comes from automorphisms of G . If ψ ∈ Aut( G ) then β ( g, h ) := α ( ψ ( g ) , ψ ( h )) gives us a new τ ψ ( g, h ) = τ ( ψ ( g ) , ψ ( h )) and gives us A n ( G, τ ) ∼ = A n ( G, τ ψ ).An important problem is to understand the orbits under these actions. The Galoissymmetry amounts to replacing α with sα . Now observe that since α : G × G → Z m hasabelian co-domain, it is determined by its values on the abelianization G ab := G/ [ G, G ].So we may assume G = A is abelian for the purposes of determining the orbits. It isclear that the bicharacters A × A → U (1) for a finite abelian group A form an abeliangroup under pointwise addition. In fact, this group is isomorphic to Hom( A, A ∗ ) where A ∗ = Hom( A, U (1)) is the group of characters. Indeed, if χ : A × A → U (1) is abicharacter then define F χ ∈ Hom(
A, A ∗ ) by F χ ( a )( b ) = χ ( a, b ). Since χ is a bicharacter F χ is a Z -module map with values in Hom( A, U (1)). Since f ∈ Hom(
A, A ∗ ) determinesa unique bicharacter χ f ( a, b ) = f ( a )( b ), the map χ F χ is clearly a bijection, and F χ + F η = F χ + η . As the bihomomorphisms α : G × G → Z m are in one-to-one correspondencewith bicharacters, this determines all such bihomomorphisms α . For elementary abelian p -groups A = ( Z p ) k a bihomomorphism to Z p can be represented as a k × k matrix X with i, j entry α ( e i , e j ) ∈ Z p where e i is the generator (0 , . . . , , , , . . . ,
0) of the i thfactor. That is, α ( g, h ) = g T Xh where we identify g ∈ A with column vectors. Of coursean automorphism Ψ ∈ Aut( A ) ∼ = GL k ( Z p ) as well, so we may sweep out orbits of α under Aut( G ) as Ψ T X Ψ, since α (Ψ( g ) , Ψ( h )) = g T Ψ T Xh . Although we will not need itin what follows, one can handle general abelian p -groups in a similar way by identifying Z p a with a subgroup of Z p b for a ≤ b . Even more generally, bihomomorphisms on afinite abelian group can be factored by restricting to p -Sylow subgroups.3.3.1. Forbidden Symmetries.
Notice we have not considered applying different auto-morphisms of G to each factor, as this will conflict with our goal of finding A ( G, τ )-YBOs–they will not have a uniform description, i.e., the form of r i not be independentof i . Even ignoring this constraint, if we apply different automorphism to each tensorfactor of A n ( G, τ ) the twisting will no longer be uniform across the iterated twistedtensor product. However, in the following special cases, uniformity is preserved.
Proposition 3.4.
Suppose G = Z km for an odd integer m and τ ( x, y ) = q α ( x,y ) for anon-degenerate symmetric or skew-symmetric bihomomorphism α : G × G → Z p . Thenthere is an isomorphism of algebras A n ( G, τ ) ∼ = A n ( G, χ ) , where χ ( x, y ) = q x T y .Proof. Observe that for G = Z kp with p odd, nondegenerate bihomomorphisms α : G × G → Z m are the same as nondegenerate bilinear forms, all of which are of the form α ( x, y ) = x T Ay for some matrix A ∈ GL k ( Z m ). n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 9
First assume α ( x, y ) = x T Sy a non-degenerate symmetric bilinear map G × G → Z m ,let A, B ∈ GL k ( Z m ) be such that I = ASB (the Smith normal form of S ). Let η ( x, y ) := x T y be the corresponding twist. We claim that the map φ : G n − → G n − defined by φ ( g i ) = ( (( A T ) − g ) i , i odd( B − g ) i , i eveninduces an isomorphism A n ( G, τ ) ∼ = A n ( G, η ). Indeed, for x, y ∈ G , we have η (( A T ) − x, B − y ) = (( A T ) − x ) T ( ASB )( B − y ) = x T Sy.
On the other hand, η ( B − x, ( A T ) − y ) = ( B − x ) T ( A T ) − y = x T ( B − ) T ( B T S T A T )( A T ) − y = x T Sy.
Now since the Smith normal form of a non-degenerate symmetric matrix over Z m isdiagonal and we may rescale each entry by isomorphisms described above, we obtainan isomorphism with A n ( G, χ ) as promised.Now suppose that k is even, and S is invertible and skew-symmetric so that wemay assume S = (cid:18) I − I (cid:19) where I = Id ( Z m ) k/ and α ( x, y ) = x T Sy the associatedbihomomorphism. Notice that S = − I and S T = − S . Then define φ ( g i ) = ( S i − g ) i and η ( x, y ) = − x T y . To see that φ is an algebra homomorphism A n ( G, τ ) ∼ = A n ( G, χ )we compute: α ( S i x, S i +1 y ) = x T ( − S ) i ( S ) S i +1 y = ( − i x T S i +2 y = − x T y = η ( x, y ) . Since φ is clearly bijective, we have shown that it is an algebra isomporphism. Since η is symmetric we may use the above to obtain an isomorphism A n ( G, τ ) ∼ = A n ( G, χ ) aspromised. (cid:3)
Symmetries of ( A n ( G, τ ) , r ) . For a fixed G and τ , the set R of A ( G, τ )-YBOscould be quite large: they are determined by the functions f : G → C with r = P g ∈ G f ( g ) g ∈ R . To reduce the search space we can make use of various symmetries,identifying function f in the same orbit. Informally we will say that two A n ( G, τ )-YBOs r and s are equivalent if ρ r , ρ s : B n → A n ( G, τ ) have the same image, projectively.One obvious symmetry comes from the homogeneity of the braid equation r r r = r r r : we can rescale any solution by z ∈ C × and if our solution is unitary, then we , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG can rescale by z ∈ U (1). This corresponds to identifying f and zf , since the B n imagesare projectively equivalent.We also have rescaling symmetries of the form g i q s ( g ) g i . Since the straighteningrelations in A n ( G, τ ) are homogeneous it is only necessary to check that χ : G → U (1) by χ ( g ) = q s ( g ) is a linear character. This automorphism of C [ G ] lifts to an automorphismof A n ( G, τ ), which carries r i = P g ∈ G f ( g ) g i to r χi := P g ∈ G f ( g ) q s ( g ) g i and hence theimages are isomorphic. Therefore we can identify the solutions that are in the orbit of f under f q s ( g ) f .Denote by Aut( G, α ) the group of automorphisms ψ ∈ Aut( G ) such that α ◦ ( ψ × ψ ) = α . Any ψ ∈ Aut(
G, α ) lifts to an automorphism of A n ( G, τ ). For such ψ ∈ Aut(
G, α )the ψ ( r i ) = P g ∈ G f ( g ) ψ ( g ) i for i = 1 , ψ − ( r ) = P g ∈ G f ( ψ − ( g )) g will also be a A ( G, τ )-YBO. Moreover, this obviously in-duces an isomorphism between ρ r ( B n ) and ρ ψ ( r ) ( B n , so we will thus identify all solutionsin the orbit of f under this symmetry f ψ ∗ f for ψ ∈ Aut(
G, α ).3.4.1.
Symmetry induced by Inversion. . An important special case of symmetry in-duced by automorphisms is the following: If G is abelian , the inversion automorphism ι : g g − on G lifts to an automorphism of A n ( G, τ ) by defining ι ( q ) = q , and ι ( g i h j ) = ι ( g i ) ι ( h j ) = g − i h − j on products and extending linearly. We will carefullycheck the defining relations are preserved. Firstly, ι ( g i ) ι ( h i +1 ) = g − i h − i +1 = ( h i +1 g i ) − = ( q − α ( g,h ) g i h i +1 ) − = q α ( g,h ) ι ( h i +1 ) ι ( g i )so that ι ( g i h i +1 ) = q α ( ι ( g ) ,ι ( h )) ι ( h i +1 g i ). Secondly since G is abelian, ι (( gh ) i ) = ι ( g i h i ) = g − i h − i = ( g − ) i ( h − ) i = ( g − h − ) i ⋆ = (( gh ) − ) i = ( ι ( gh )) i , where the equality denoted ⋆ = uses the G abelian assumption. Finally note that if g i and h j commute then so do ι ( g i ) and ι ( h j ). If r = P g ∈ G f ( g ) g is an A ( G, τ )-YBO thenso is ι ( r ) = P g ∈ G f ( g ) g − , hence r ′′ = P g ∈ G f ( g − ) g is as well.If G is abelian there is an additional symmetry of the braid relation r r r = r r r that we may use to show that r = P g ∈ G f ( g ) g and r ′ := P g ∈ G f ( g ) g have isomorphicimages. The map σ on A ( G, τ ) given by σ ( g ) = g and σ ( xy ) = σ ( y ) σ ( x ) for x, y ∈ A ( G, τ ) and σ ( q ) = q − = q is an anti-automorphism of A ( G, τ ), since σ ( g h ) = h g = q α ( h,g ) g h = σ ( q α ( g,h ) h g )and σ (( gh ) ) = ( gh ) = g h = h g = σ ( g h ) . This implies that r ′ is a A ( G, τ )-YBOsince r ′ r ′ r ′ = σ ( r r r ) = σ ( r r r ) = r ′ r ′ r ′ . n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 11 Case Studies
In practice we take the following approach, using symbolic computation software suchas Magma and Maple.(1) Fix G and α , and present the corresponding finitely generated algebra A ( G, τ ) × Q ( q )[ x g : g ∈ G ], using generators and relations, with the x g being commutingvariables.(2) Define r i = P g ∈ G x g g i for i = 1 ,
2, and use non-commutative Gr¨obner bases towrite r r r − r r r in its normal form i.e., as a polynomial in the g ( i )1 g ( i )2 withcoefficients in Q ( q )[ x g : g ∈ G ].(3) Compute a commutative Gr¨obner basis for ideal generated by the the coefficientsusing pure lexicographic order to find the ideal of solutions.(4) Use symmetries to describe families of related solutions.Often we find that there are finitely many solutions, so that we can give a completedescription of them.4.1. Abelian Groups.
Prime Cyclic Groups G = Z p . We first apply our approach to a well-known caseboth as a proof of principle and a template for further study.Let p ≥ q a primitive p th root of unity. A nontrivial bicharacter Z p × Z p → U (1) must take values in µ p = { q j : 0 ≤ j ≤ p − } , so that any bicharactercorresponds to a bihomomorphism α ∈ Hom( Z p × Z p , Z p ), which is determined by α (1 , α : Z p × Z p → Z p by α (1 ,
1) = 2. The orbitof α under automorphisms of Z p give half of all non-trivial bihomomorphisms, since α ( ϕ k (1) , ϕ k (1)) = 2 k , which is a square modulo p if and only if 2 is. Galois symmetry ψ ( q ) = q s maps the bicharacter τ ( x, y ) = q α ( x,y ) to τ ψ ( x, y ) = q sα ( x,y ) . Thus we mayassume that our bicharacter is associated with the bihomomorphism α ( x, y ) = 2 xy .The reader may wonder why we do not choose α ′ ( x, y ) = xy instead–we will see laterthat this simplifies the form of our A n ( Z p , τ )-YBOs. Indeed, this choice of α recoversthe Q ( q )-algebra A n ( Z p ) described in the introduction, with generators u , . . . , u n − satisfying: u i u i +1 = q u i +1 u i and u i u j = u j u i for | i − j | > u pi = 1. The goal nowis to find invertible A ( Z p )-YBOs r = γ P p − j =0 f ( j ) u j ∈ C [ Z p ].To reduce redundancy we will normalize f (0) = 1 (the solutions where f (0) = 0 donot seems to be interesting). The symmetries of these solutions again come in severalforms. Firstly, since each automorphism of Z p that leaves α invariant leads to anautomorphism of A n ( Z p ) we may identify the corresponding solutions. For α ( x, y ) =2 xy only inversion x → − x leaves α invariant, which means we may freely identify f and f ′ ( j ) = f ( − j ). We have an additional symmetry in A n ( Z p ) given by u i → q s u i , , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG since the first two defining relations are homogeneous and ( q s u i ) p = u pi = 1. Thiscorresponds to identifying f with f s ( j ) := f ( j ) q js . Finally, complex conjugation is asymmetry of the braid equation r r r = r r r , so that we may identify f and itscomplex conjugate f .4.1.2. The Gaussian Solution.
One unitary A ( Z p )-YBO is the Gaussian solution r = √ p P p − j =0 q j u , i.e. f ( j ) = q j [18, 26, 16] and γ = √ p . Complex conjugation gives usthe solution f ( j ) = q − j and the rescaling symmetry gives us f s ( j ) = q j + sj , giving 2 p distinct solutions.In [16] it is shown that the braid group representation ρ n : B n → A n ( Z p ) given by σ i → r i has finite image. In fact, one has: r i u i +1 r − i = qu − i u i +1 r i u i − r − i = q − u i − u i , so that the conjugation action on A n ( Z p ) provides a homomorphism of ρ n ( B n ) intomonomial matrices, with kernel a subgroup of the center of A n ( Z p ). For n odd thenormal form for A n ( Z p ) allows one to show that the center consists of scalars, and sinceany ρ n ( β ) in the center of A n ( Z p ) has determinant a root of unity (under the regularrepresentation of A n ( Z p )) the kernel of the conjugation action above is finite, for n odd.Since ρ n ( B n ) ⊂ ρ n +1 ( B n ), this is sufficient.The algebras A n ( Z p ) have a local representation (see [33]). Let V = C p and definean operator on V ⊗ by U ( e i ⊗ e j ) = q j − i e i +1 ⊗ e j +1 where { e i } p − i =0 is a basis for V with indices taken modulo p . Then Φ n : u i → ( Id V ) ⊗ i − ⊗ U ⊗ ( Id V ) ⊗ n − i − defines arepresentation A n ( Z p ) → End( V ⊗ n ). In particular Φ ( r ) is an honest p × p YBO.
Example 4.1.
We use MAGMA [5] to work two explicit examples. First consider thecase G = Z , and suppose r = 1 + au + bu is a A ( Z )-YBO. All solutions satisfy a = b = 1 and a = b , so that there are exactly 6 distinct solutions (up to rescaling),all of which are obtained from the Gaussian solution via the symmetries describedabove. In particular the solutions are all unitary when appropriately normalized.Similarly for p = 5,under the additional assumption that a, b, c, d are 5th roots ofunity, we find that there are exactly 10 non-trivial solutions r = 1 + au + bu + cu + du (up to rescaling), all of which are obtained from the Gaussian solution via the abovesymmetries. These solutions are unitary when appropriately normalized. There are 10other non-trivial solutions, however, none of them are (projectively) unitary.4.2. G = Z p × Z p . Let p be an odd prime, and let G = Z p × Z p . To classify A n (( Z p ) , τ )we first look at orbits of bihomomorphisms α : ( Z p ) → Z p . Such bihomomorphismsare determined by the values on pairs of generators (1 , , (0 ,
1) of Z p × Z p , encoded in n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 13 a matrix A α := (cid:18) a bc d (cid:19) ∈ M ( Z p ), so that α ( x, y ) = x T A α y . Under automorphisms X ∈ GL ( Z p ) of ( Z p ) the orbit of α is represented by the matrices { X T A α X : X ∈ GL ( Z p ) } . From [37] we know that there are p + 7 orbits. Example 4.2.
The case of p = 3 can be completely analyzed computationally asfollows: from a representative of each of the 10 orbits of bihomomorphisms Z × Z → Z we use MAGMA [5] to search for non-degenerate, unitary solutions r = γ P i,j f ( i, j ) u i v j to the corresponding A ( Z × Z , τ )-YBE. Here by non-degenerate we mean that it doesnot degenerate to the Z -case. The results of these computations are: • Non-degenerate unitary solutions only exists for the classes represented by: A = (cid:18) (cid:19) , A = (cid:18) − (cid:19) and A = (cid:18) − (cid:19) . • In all cases, after applying an appropriate symmetry of ( A n ( Z × Z , τ ) , r ), thenon-degenerate unitary solutions factor as a product of Gaussian A n ( Z )-YBOs,and hence have finite images.From this example we expect that the most interesting ones correspond to non-degenerate symmetric or skew-symmetric bilinear forms on Z p . We also allow ourselvesto rescale α by a constant. Thus we focus on A = (cid:18) (cid:19) , A = (cid:18) − (cid:19) and A = (cid:18) x (cid:19) where x is a non-square modulo p . The appearance of the scalar 2 issimply for convenience when we make contact with the Gaussian solution.We consider each of these cases in turn. We will distinguish the symmetric cases A , A by noting that A corresponds to an elliptic form, while A corresponds toa hyperbolic form. For A α = A i the corresponding algebras A n ( Z p × Z p , τ i ) havegenerators u , v , . . . , u n − , v n − with the multiplicative group h u i , v i i ∼ = Z p × Z p . Allgenerators commute except for:(1) For A : u i u i ± = q ± u i ± u i and v i v i ± = q ± v i ± v i .(2) For A : u i v i ± = q ± v i ± u i and v i u i ± = q ∓ v i ± u i .(3) For A : u i u i ± = q ± u i ± u i and v i v i ± = q ± x v i ± v i .We pause to describe the structure of the algebras A n ( Z m × Z m , τ i ) for arbitrary odd m . Since the monomials in the u i , v i form a basis, we see that dim Q ( q ) A n ( Z m × Z m , τ i ) = m n − . The following proposition explores the structure of A n ( Z m × Z m , τ i ) and the subal-gebra of fixed points under the automorphism ι described in Subsection 3.4.1 given bylifting u i u − i , v i v − i to A n ( G, τ i ). We describe inclusions of algebras in terms of , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG Bratteli diagrams (see [19]): generally, to a tower of multi-matrix algebras with com-mon unit 1 ∈ A ⊂ · · · A n ⊂ A n +1 ⊂ · · · we associate a graph with vertices labeled bysimple A k -modules M k,i with d k − ,i,j edges between M k,i and M k − ,j if the restrictionof M k,i to A k − contains M k − ,j with multiplicity d k − ,i,j . Proposition 4.3.
Let m be odd and G = Z m × Z m . Consider the algebra A n ( G, τ i ) with the twists τ i given by A i , ≤ i ≤ . Then (1) The center of A n ( G, τ i ) is 1 dimensional if n is odd and is m dimensional if n is even. Moreover, when n is odd A n ( G, τ i ) ∼ = M m n − ( Q ( q )) is simple while for n even A n ( G, τ i ) decomposes as a direct sum of m simple algebras of dimension m n − . Moreover, the Bratteli diagram of · · · ⊂ A n ( G, τ i ) ⊂ · · · is given inFigure 1. (2) Consider the fixed point subalgebra C n ( G, τ i ) for the automorphism ι induced byinversion on A n ( G, τ i ) . Then for n ≥ odd, C n ( G, τ i ) is a direct sum of twomatrix algebras of dimensions (cid:18) m n − ± (cid:19) . For n ≥ and even, C n ( G, τ i ) has m +32 simple summands: m − of dimension m n − and two others of di-mensions (cid:18) m n − ± (cid:19) . Moreover, the Bratteli diagram for · · · ⊂ C n ( G, τ i ) ⊂ C n +1 ( G, τ i ) ⊂ · · · is given by Figure 2, where the nodes are labelled by the di-mensions of the distinct simple modules.Proof. We first note that, by its construction, the isomorphism A n ( G, τ ) ∼ = A n ( G, χ ) ofProposition 3.4 restricts to a bijection on the set h u j , v j i for each j . It follows that itrestricts to a bijection on each H j , hence an isomorphism C n ( G, τ ) ∼ = C n ( G, χ ). Thus,it suffices to prove the proposition for τ ( x, y ) = χ ( x, y ) := q x T y .In this case, we have an isomorphism ρ : A n ( Z m × Z m ) , χ ) → A n ( Z m ) ⊗ A n ( Z m ) where ρ ( x, y ) = q xy given by mapping u j to the j -th group generator in the first tensor factorand v j to the one in the second tensor factor. The structure of A n ( Z m ) is well-known (seeeg., [25]) from which we can derive the structure of A n ( Z m × Z m ) , χ ). Indeed, the Brattelidiagram of A n ( Z m ) can be easily worked out: it has the same general shape as in Figure1 except with m summands for n even and with dim A n ( Z m ) = m n − . In particular thesimple direct summands of A n ( Z m × Z m ) , χ ) are of the form [ A n ( Z m )] i ⊗ [ A n ( Z m o )] j where [ A n ( Z m )] i denotes the i th direct summand of A n ( Z m ). In particular when n isodd, A n ( Z m × Z m ) , χ ) ∼ = M m n − ( Q ( q )) is simple and has a 1-dimensional center whereasfor n even A n ( Z m × Z m ) , χ ) has an m dimensional center and decomposes as a directsum of m simple algebras of dimension m n − . n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 15 m · · · m m m · · · m m ... Figure 1.
Bratteli diagram for A n ( Z m × Z m , τ i ) for m odd.11 1 1 m +12 m − m +12 m m m − m +12 m − ... ... ······ ······ ······ ··· Figure 2.
Bratteli diagram for C n ( Z m × Z m , τ i ) for m odd. , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG We compute that dim C n ( Z m × Z m , χ ) = m n − +12 as follows. Any fixed point of ι hasthe form P x , y ∈ ( Z m ) n − c x , y ( a x b y + a − x b − y ) where a x = u x · · · u x n − n − and b y = v y · · · v y n − n − .Thus we see that the m n − − linearly independent binomials of the form a x b y + a − x b − y together with 1 form a basis for C n ( Z m × Z m , χ ).We first consider n odd. Define d ( n ) = m n − . Since A n ( Z m , ρ ) ∼ = M d ( n ) ( Q ( q )) issimple of dimension m n − it has a unique irreducible faithful module M of dimension d ( n ). Now the action of the automorphism ι on A n ( Z m ) is an inner automorphism, andhence corresponds to conjugation by some diagonal matrix J . Without loss of generalitywe may assume J has non-zero entries ± C n ( Z m , ρ ): for n odd the restriction of M to C n ( Z m , ρ ) decomposes asa direct sum of two irreducible modules M + and M − , of dimensions d ( n )+12 and d ( n ) − .Notice that ι ( J ) = J so that M + and M − are simply the two eigenspaces of J actingon M .Now since A n ( Z m × Z m , χ ) ∼ = A n ( Z m ) ⊗ A n ( Z m ), the action of ι is (in some basis)conjugation by J ⊗ J ∈ A n ( Z m ) ⊗ A n ( Z m ). Moreover, as a C n ( Z m × Z m , χ ) module M ⊗ M decomposes as the direct sum of the two J ⊗ J eigenspaces. These are precisely M ⊗ ⊕ M ⊗ − (the +1 eigenspace) and M + ⊗ M − ⊕ M − ⊗ M + (the -1 eigenspace) asvector spaces. By the double commutant theorem the faithful C n ( Z m × Z m , χ )-module M ⊗ M decomposes into a direct sum of two simple modules, which are precisely the ± J ⊗ J . Since (cid:16) d ( n )+12 (cid:17) + (cid:16) d ( n ) − (cid:17) = d ( n ) +12 and 2 (cid:16) ( d ( n )+1)( d ( n ) − (cid:17) = d ( n ) − the dimensions are as stated.The case of n even is similar, but slightly more complicated since A n ( Z m ) is notsimple for n even. The Bratteli diagram is found in [25], and the technique is straight-forward so we sketch it here. Restricting the simple d ( n +1)+12 -dimensional C n +1 ( Z m )-module M + to C n ( Z m ) decomposes as N + ⊕ L m − j =1 L j and restricting the simple d ( n +1) − -dimensional C n +1 ( Z m )-module M − to C n ( Z m ) decomposes as N − ⊕ L m − j =1 L j wheredim( L j ) = d ( n ) and dim( N ± ) = d ( n ) ± . Indeed, since on A n ( Z m ) ∼ = L m − j =0 M d ( n ) ( Q ( q ))the automorphism ι either interchanges pairs of simple summands or restricts to anautomorphism on them. In this case only one simple summand is ι -invariant while ι induces isomorphisms between the other m − pairs. The ι -invariant factor then splits asa as direct sum of two C n ( Z m )-modules N ± of dimensions d ( n ) ± , and we get one distinct C n ( Z m )-module L j for each ι -orbit of size 2. Now we apply the restriction techniqueabove to the two simple C n ( Z m × Z m , χ )-modules M ⊗ ⊕ M ⊗ − and M + ⊗ M − ⊕ M − ⊗ M + together with the restriction of ι to the simple direct summands of A n ( Z m × Z m , χ ) toresolve the count of components, from which the result is derived. n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 17 (cid:3)
We now return to the problem of finding solutions r = γ X ≤ j,k ≤ p − f ( j, k ) u j v k to the A ( Z p × Z p , τ i )-YBE for i = 1 , f ( j, k ) = f u ( j ) f v ( k ) after applying the symmetries of Subsection 3.4, and can verify computa-tionally that all solutions factor as products of Gaussian-type solutions in the case for p = 3. Thus we focus on such solutions. All of the solutions that follow will havefinite braid group image when properly normalized to be unitary, which can be easilyverified using the finiteness of the Gaussian representation images. The eigenvalues of r = P ≤ j,k ≤ p − q j εxk u j v k for q = e πi/p and ε = ± A ( G, τ i ) are: λ ε,x ( s, t ) = X j q j + sj ! X k q εxk + tk ! . Up to an overall normalization factor, the eigenvalues and their multiplicities can becomputed using standard Gaussian quadratic form techniques, and only depend on thesign ± and whether − x are squares or non-squares modulo p . We have that themulti-set [ λ ε,x ( s, t )] has:(1) 1 with multiplicity 1 and e πij/p with multiplicity p + 1 for each 1 ≤ j ≤ p − ε, (cid:0) − p (cid:1) , (cid:0) xp (cid:1) ) ∈ { (1 , − , , ( − , − , − , (1 , , − , ( − , , − } and(2) 1 with multiplicity 2 p − e πij/p with multiplicity p − ≤ j ≤ p − ε, (cid:0) − p (cid:1) , (cid:0) xp (cid:1) ) ∈ { (1 , − , − , ( − , − , , (1 , , , ( − , , } .4.2.1. Elliptic Symmetric Case.
First consider the case A . We can deduce some solu-tions t ( u, v ) = X ≤ j,k ≤ p − F ( j, k ) u j v k to the A n ( Z p × Z p , τ i )-YBE from the Gaussian solutions. Indeed, if f, h : Z p → C and are such that r ( u ) = P p − j =0 f ( j ) u j and s ( v ) = P p − j =0 h ( j ) v j are solutions to the A n ( Z p )-YBE then setting t ( u, v ) = r ( u ) s ( v ) we can easily verify: t t t = ( r s )( r s )( r s ) = ( r r r )( s s s ) = t t t since r i := r ( u i ) commutes with s i := s ( v j ). If both f and h correspond to Gaussiansolutions, we may rescale u and v independently followed by complex conjugation toassume that f ( j ) = q j and h ( k ) = q ± k . The choice of sign indeed gives two distinct , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG solutions. The additional symmetry that we have not used comes from the group of τ -invariant G -automorphism, i.e., { X ∈ GL , ( Z p ) : X T A X = A } , which is a groupof order 2( p −
1) in this case.4.2.2.
Skew-Symmetric Case.
Next we consider the case A . Suppose that our solution t ( u, v ) = X ≤ j,k ≤ p − F ( j, k ) u j v k factors as t ( u, v ) = r ( u ) s ( v ) where r ( u ) = P p − j =0 f ( j ) u j and s ( v ) = P p − j =0 h ( j ) v j aresolutions to the A n ( Z p )-YBE. Again, setting r i = r ( u i ) and s i = s ( v i ) we observe that[ r , r ] = 1 and [ s , s ] = 1, so that t t t = r s r s r s = ( s r s )( r s r ). Fromthis we deduce that we should take r ( u ) = s ( u ) and r ( v ) = s ( v ), i.e. h = f so that t ( u, v ) = r ( u ) r ( v ). Now we can use symmetry to choose f ( j ) = h ( j ) = q j . In this casethe group of automorphisms of Z p × Z p that preserve α is SL ( Z p ), a group of order( p − p )( p + 1).4.2.3. Hyperbolic Symmetric Case.
As the details are similar to the elliptic symmetriccase we are content to provide the factored solution t ( u, v ) = X ≤ j,k ≤ p − q j ± xk u j v k . It is an easy exercise to show that this is the unique factorizable solution up to sym-metries. The group of automorphisms of A ( G, τ ) that preserve preserve α has order2( p + 1).4.3. Non-commutative cases.
To illustrate our methods for non-abelian groups wefirst apply them to the case of the symmetric group S and the algebra A n ( Q ) fromthe introduction.4.3.1. Symmetric group S . For S the bihomomorphisms α : S × S → Z m are deter-mined by the abelianization Z × Z → Z m so that we may take m = 2. In particular wehave the following description of A n ( S , τ ) for the non-trivial choice α ((1 2) , (1 2)) = 1.We take generators u = (1 2) and v = (1 2 3) for S and corresponding generators of A n ( S , τ ) u , v , . . . , u n − , v n − with relations: • u i v i = v i u i and u i = v i = 1 ( S relations) and • u i u i +1 = − u i +1 u i and v i v j = v j v i for all i, j , and • u i v j = v j u i for i = j .We seek (invertible) solutions r = γ (1 + au + bv + cv + duv + euv ) ∈ C [ S ] to the A ( S , τ )-YBE, where γ is a normalization factor chosen to give r finite order. AppendixI contains the details of the computation, the upshot of which is that b = c = 0 is n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 19 a consequence of invertibility and to have solutions r that are unitary with respectto the standard ∗ -operation we should take γ = and ( a, d, e ) = (i x, i y, i z ) with( x, y, z ) ∈ R on the intersection of the surface given by xy + xz + yz = 0 with the unitsphere x + y + z = 1. Since ( x + y + z ) = 1 modulo the ideal generated by thesetwo polynomials we conclude that the solutions are the points on the intersection of thetwo planes ( x + y + z ) = ± r = 1, with eigenvalues 1 , − i. This suggests that thisrepresentation is related to the Ising theory, see [13].4.3.2. Quaternionic Algebra A n ( Q ) . Recall the algebra A n ( Q ) described in the intro-duction, generated by u i , v i satisfying:(1) u i = v i = − i ,(2) [ u i , v j ] = − | i − j | < u i , u j ] = [ v i , v j ] = 1,(4) [ u i , v j ] = 1 if | i − j | ≥ i the pair u i , v i generates a group isomor-phic to Q . Notice, however, that h u i , v i i ∩ h u i , v i i = {± } so that A n ( Q ) is not atwisted tensor product of group algebras: indeed it is not C [ Q ] ⊗ n − as a vector space.The algebra is closely related to group algebras, in at least two ways. Firstly, supposethat Q = h u, v i where uv = zvu with u = v = z central of order 2. Then we maydefine the quotient T = C [ Q ] / h z + 1 i and then A ( Q ) above is a twisted tensor prod-uct of n − T with the tensor product twist given as above, determined by τ ( u, v ) = − τ ( u, u ) = τ ( v, v ) = 1 since u i v i +1 = − v i +1 u i .Alternatively, we can consider the twisted group algebra C ν [ Z × Z ] associated withthe cocycle ν ∈ Z ( Z × Z , U (1)) defined by: • ν ((1 , , (0 , − ν ((0 , , (1 , • ν ((1 , , (1 , ν ((0 , , (0 , − C ν [ Z × Z ] given by g ⋆ ν h = ν ( g, h ) gh for g, h ∈ Z × Z . Then A n ( Q ) = C ν [ Z × Z ] ⊗ τ C ν [ Z × Z ] ⊗ τ · · · ⊗ τ C ν [ Z × Z ] , where τ is the twisting corresponding to the relations above.We look for A ( Q )-YBOs of the form r = 1 + au + bv + cuv . We find 8 non-trivial solutions namely a, b, c ∈ {± } , normalized to a unitary solution. These areall related by symmetry since we may rescale a → − a and b → − b independently,permute the a, b, c freely using the fact that τ is invariant under permuting u, v, uv andinversion corresponds to simultaneously changing all signs. The Magma code is foundin Appendix A. , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG Categorical Connections
The class of weakly integral modular categories, i.e. those for which FPdim( C ) ∈ Z isnot well-understood. However, a long-standing question [10, Question 2] asks if the classof weakly integral fusion categories coincides with the class of weakly group-theoreticalfusion categories, i.e., those that are Morita equivalent to a nilpotent fusion category.Recently Natale [30] proved that any weakly group-theoretical modular category is a G -gauging of either a pointed modular category (all simple objects are invertible) ora Deligne product of a pointed modular category and an Ising-type modular category[8, Appendix B]. These latter categories are well-known to have property F , whichreduces the verification of the property F conjecture for weakly integral braided fusioncategories to verifying that G -gauging preserves property F and that weak integralityis equivalent to weak group-theoreticity.The difficulty with verifying property F for a given category is that one rarely has asufficiently explicit description of the braid group representations ρ X associated with anobject X ∈ C . The braiding c X,X on X ⊗ X provides a map C B n → End( X ⊗ n ) whichthen acts on each Hom( Y, X ⊗ n ) for simple objects Y by composition, but an explicitbasis for Hom( Y, X ⊗ n ) is lacking. In all cases where the property F conjecture has beenverified for a weakly integral braided fusion category [11, 29, 33, 24, 14] the first stepis a concrete description of the centralizer algebras End( X ⊗ n ), and the correspondingmodules Hom( Y, X ⊗ n ) which are obtained by studying a specific realization of C , asa subquotient category of representations of a quantum group, for example. Fromthis description one extracts a sufficiently explicit B n representation to facilitate theverification of property F .One approach to a uniform proof of (one direction of) the property F conjectureis to understand the connection between the centralizer algebras of pointed modularcategories and those of its G -gaugings. Pointed modular categories are in one to onecorrespondence with metric groups, i.e., pairs ( A, Q ) where A is a finite abelian groupand Q is a non-degenerate quadratic form on A . We denote by C ( A, Q ) the correspond-ing modular category. Pointed modular categories and their products with Ising-typecategories are well-known to have property F [29]. If it could be proved that G -gaugingpreserves property F then we could reduce this direction of the property F conjectureto [10, Question 2].For a general mathematical reference on G -gaugings, see [7], the notation of whichwe will adopt here. Let A be an abelian group, and Q a non-degenerate quadraticform on A , and C ( A, Q ) the corresponding pointed modular category, with twists givenby θ a = Q ( a ) and braiding by c a,b = β ( a, b ) σ where σ is the usual flip map and β ( a, b ) := Q ( a + b ) − Q ( a ) − Q ( b ). A G symmetry of C ( A, Q ) is a group homomorphism ρ : G → Aut br ⊗ ( C ( A, Q )) ∼ = O ( A, Q ). Provided certain cohomological obstructions n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 21 vanish one may construct (potentially several) modular categories by gauging the G symmetry. In the case of an elementary abelian p -group for p an odd prime all of theobstructions vanish by [9, Theorem 6.1].We expect there to be a connection between the algebras A n ( G, τ ) described aboveand the H -gaugings of pointed modular categories, i.e., categories C ( G, Q ) × ,HH where H ⊂ Aut br ⊗ ( C ( G, Q )). Indeed, in the case G = Z p and H = Z acting by inversionthese categories are called p - metaplectic and we we have the following, using results of[34, 20, 25, 2] and some careful adjustment of parameters: Theorem 5.1.
Let Z act on C := C ( Z p , Q ) by inversion, and let D = C × , Z Z be any ofthe corresponding gaugings, and X a simple object of dimension √ p . Then End( X ⊗ n ) ∼ = h u + u − , . . . , u n − + u − n − i ⊂ A n ( Z p ) . In fact, this result is key to verifying the property F conjecture for p -metaplecticcategories.A similar relationship exists between a Z -gauging of the so-called three fermion theory C ( Z × Z , Q ) where Q ( x ) = − x = (0 ,
0) and the algebra A n ( Q ) describedabove. In this case C ( Z × Z , Q ) × , Z Z ∼ = SU (3) for one choice of Z -gauging, wherethe action of Z at the level of object is given by cyclic permutation of the threenon-trivial simple objects (see [7]). Now for a generating 2-dimensional object X it isshown in [32] that the subalgebra C n ( Q ) of A n ( Q ) generated by ( u i + v i + u i v i ) for1 ≤ i ≤ n − X ⊗ n ), which is also isomorphic to a quotient of theHecke algebra specialization H n (3 , Z actionon Q given by cyclic permutation of u, v and uv lifts to an automorphism of A n ( Q )and C n ( Q ) is the fixed point subalgebra. Finally, we remark that the image of thebraid group representation on End( X ⊗ n ) is finite–it factors through the representationsfound above: σ i (1 + u i + v i + u i v i ).This inspires the following: Principle 5.2. If G ⊂ Aut(
A, Q ) is a gaugeable action on C ( A, Q ) then there is a(quotient of an) iterated twisted tensor product A n ( A, τ ) of C [ A ] and an object X ∈ C ( A, Q ) × ,GG so that End( X ⊗ n ) is isomorphic to the fixed point subalgebra C n ( A, τ ) ofthe automorphism induced by the action of G on A . Moreover there is a A ( A, τ )-YBO r supported in C n ( A, τ ) such that the B n representation on End( X ⊗ n ) factors throughthe B n representation defined by r .We do not have a general proof of this principle for all groups. In the case of Z p × Z p with Z acting by inversion we give some compelling evidence for this principle.Now suppose that | A | = m = 2 k + 1 is odd and ρ : Z → Aut br ⊗ ( C ( A, Q )) is the actionby inversion. The Z -extensions are Tambara-Yamamgami categories T Y ( A, χ, ± ) [35], , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG Z Y Y k Z Z Y Y k X − Z Z ······ ······ ······ ··· Figure 3.
Bratteli diagram for C ( A, Q ) × , Z Z for | A | odd.and their equivariantizations are found in [17] (see also [22]). There are two distinct Z -gaugings D ± := C ( A, Q ) × , Z Z of the inversion action ρ . Each modular category D ± has dimension 4 | A | . It has the following simple objects:two invertible objects, = X + and X − , m − two-dimensional objects Y a , a ∈ A − { } (with Y − a = Y a )two √ m -dimensional objects Z l , l ∈ Z .The fusion rules of D ± are given by: X − ⊗ X − = X + , X ± ⊗ Y a = Y a , X + ⊗ Z l = Z l ,X − ⊗ Z l = Z l +1 , Y a ⊗ Y b = Y a + b ⊕ Y a − b , Y a ⊗ Y a = X + ⊕ X − ⊕ Y a ,Y a ⊗ Z l = Z ⊕ Z , Z l ⊗ Z l = X + ⊕ ( ⊕ a Y a ) , Z l ⊗ Z l +1 = X − ⊕ ( ⊕ a Y a ) , where a, b ∈ A ( a = b ) and l ∈ Z . All objects of D ± are self-dual. Here the addition a + b takes place in A . We see that X − must be a boson, in the sense that thesubcategory h X − i ∼ = Rep( Z ) as a braided fusion category. Indeed, as D ± is a non-degenerate braided fusion category it is faithfully Z -graded with the trivial componenthaving the m +12 simple objects Y a , X ± , and non-trivial component having the two simpleobjects Z l .In particular the algebras End( Z ⊗ n ) ⊂ End( Z ⊗ n +10 ) have the Bratteli diagram ofFigure 3, where we have labeled the objects Y a by an arbitrary choice Y i for 1 ≤ i ≤ k .The categories D ± described above for the group G = Z p × Z p were explored in [17],and found to be non-group-theoretical in one case and group-theoretical in the other. n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 23
For the case p = 3, the group-theoretical cases are equivalent to Rep( D ω S ) where ω isa 3-cocycle on S . Up to equivalence there is one non-trivial choice for ω .We expect that:(1) End( Z ⊗ n ) ∼ = C n ( Z p × Z p , τ ) for some choice of τ .(2) Under the above isomorphisms the image of the braid group generators aredescribed by the A ( Z p × Z p , τ )-YBOs determined above.The two pieces of evidence are as follows:(1) The Bratteli diagrams for End( Z ⊗ n ) and C n ( Z p × Z p , τ i ) coincide and(2) The eigenvalue profile of c Z ,Z and P j,k q j ± xk u j v k coincide, for some choice of ± x where x is either 1 or any non-square modulo p .The S and T matrices of all 4 of these categories are given in [17], as they areequivalent to Z -equivariantizations of Tambara-Yamagami categories. From [4, Prop.2.3] we may deduce the eigenvalues of the braiding for the object Z of dimension p .5.0.1. A special case: p = 3 . Let q = e π i / . The two group-theoretical categoriesRep( D ω S ) can be obtained by gauging the Z inversion symmetry on C ( Z × Z , Q )where Q ( x, y ) = q x − y is hyperbolic. For the elliptic quadratic form Q ( x, y ) = q x + y the two inequivalent Z -gaugings are non-group-theoretical. Each of these categories canbe tensor generated by a simple object Z of dimension 3. The two group theoretical-categories Rep( D ω G ) have property F ([11]), but it is currently open whether thenon-group-theoretical cases have property F .On the other hand, we can have completely determined all unitary solutions to the A ( Z × Z , τ )-YBE for the bicharacters τ associated with the 3 matrices A , A and A ,up to the usual symmetries in Example 4.2.One more piece of circumstantial evidence is that the results of [20] show that thebraid group representations associated with a modular category only nominally dependon the finer structures such as the associativity constraints: for odd primes p , the imagesof the braid group representations for p -metaplectic modular categories are projectivelyequivalent. Since the property F conjecture depends only on the dimensions of objects,which are determined by fusion rules, it would perhaps not be so surprising if the fusionrules essentially determine the braid group images. A related result in [31] impliesthat the images of the braid group representations associated with different modularcategories with the same underlying fusion category are either all finite or all infinite.6. Conclusions
In this paper we have unified some explicit constructions of braid group representa-tions that come from finite groups in a fairly direct way. We have also provided strong , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG evidence that twisted tensor products of group algebras simplify the analysis of gaug-ings of pointed modular categories. In particular, the data describing A ( A, τ )-YBOs,simply a function on A , is much simpler than the construction of the R -matrices of agauged modular tensor category. However, beyond the Gaussian case, the connectionbetween the two braid group representations remains at the level of Bratteli diagramsand eigenvalues.It would be of great interest to formulate precise intertwining operators betweenbraid group representations in centralizer algebras of gaugings of pointed modular tensorcategories and those from A ( A, τ )-YBOs. This was accomplished with great difficulty inthe Gaussian case [33]. Ideally, we would like to find a uniform framework generalizingthis construction to all gaugings of pointed modular tensor categories.
Appendix I: computations for G = S and A n ( Q ) . In what follows we provide some details classifying solutions to the A ( S , τ ) and A ( Q )-YBE. Symmetric group S . We let u, v be the generators for S with u = v = 1 and uvu = v . For example we could take u = (1 2) and v = (1 2 3) By the theory above,we initialize with the following MAGMA code to find conditions on a, b, c, d, e ∈ C sothat r = 1 + au + bv + cv + duv + euv is an A ( S , τ )-YBO. F
The ideal of solutions is generated by the coefficients of the monomials in u i , v j . Weenforce invertibility of r by assuming the determinant of the image of r under thefaithful S representation on C is non-zero. The output of the Gr¨obner basis is thefollowing set of polynomials: { c, b, e ( a + d + e +1) , ad + ae + de, a + a e +2 ae + de + e + a + e, − a e + ae + d +2 de + d } Notice that c = b = 0, in all cases. If e = 0 then ad = 0, and a + a = d + d = 0, whichare degenerate solutions of the form 1 + xu that can be obtained from Z (see [13]).If e = 0 we find that e is a free parameter, and the following code shows that wemay normalize to get r = 1. There is a 1-parameter family of solutions for ( a, d, e ).Moreover one sees that if we require a unitary solution each of a, d, e should be pureimaginary, and consequently the equation a + d + e + 1 = 0 implies that ( a/ i , d/ i , e/ i)is a point on the unit sphere. Geometrically this is the intersection of the unit spherewith the surface given by xy + xz + yz = 0. F
Quaterionic Algebra A n ( Q ) . For the case of the algebra A n ( Q ) we use Magmato classify A ( Q )-YBOs. The following is the final code, where the last polynomialrelations are the coefficients obtained from an initial run of the normal form command , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG on an initial run (i.e., without the last set of relations). One finds that the non-trivialsolutions for ( a, b, c ) are all ±
1, so that if we want unitary solutions, the inverse of R1is of the form given as R1i since u ∗ = u − = − u etc. We conclude that all unitarysolutions are equivalent to the choice ( a, b, c ) = (1 , , F
References [1] Nicol´as Andruskiewitsch and Hans-J¨urgen Schneider. Pointed Hopf algebras. In
New directionsin Hopf algebras , volume 43 of
Math. Sci. Res. Inst. Publ. , pages 1–68. Cambridge Univ. Press,Cambridge, 2002.[2] Eddy Ardonne, Meng Cheng, Eric C. Rowell, and Zhenghan Wang. Classification of metaplecticmodular categories.
J. Algebra , 466:141–146, 2016. n REPRESENTATIONS FROM TWISTED TENSOR PRODUCTS 27 [3] Joan S. Birman and Hans Wenzl. Braids, link polynomials and a new algebra.
Trans. Amer. Math.Soc. , 313(1):249–273, 1989.[4] Parsa Bonderson, Colleen Delaney, Csar Galindo, Eric C. Rowell, Alan Tran, and Zhenghan Wang.On invariants of modular categories beyond modular data.
Journal of Pure and Applied Algebra ,223(9):4065 – 4088, 2019.[5] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The userlanguage.
J. Symbolic Comput. , 24(3-4):235–265, 1997. Computational algebra and number theory(London, 1993).[6] Andreas Cap, Hermann Schichl, and Jiˇr´ı Vanˇzura. On twisted tensor products of algebras.
Comm.Algebra , 23(12):4701–4735, 1995.[7] Shawn X. Cui, C´esar Galindo, Julia Yael Plavnik, and Zhenghan Wang. On gauging symmetry ofmodular categories.
Comm. Math. Phys. , 348(3):1043–1064, 2016.[8] Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. On braided fusion cate-gories. I.
Selecta Math. (N.S.) , 16(1):1–119, 2010.[9] Pavel Etingof and C´esar Galindo. Reflection fusion categories.
J. Algebra , 516:172–196, 2018.[10] Pavel Etingof, Dmitri Nikshych, and Victor Ostrik. Weakly group-theoretical and solvable fusioncategories.
Adv. Math. , 226(1):176–205, 2011.[11] Pavel Etingof, Eric Rowell, and Sarah Witherspoon. Braid group representations from twistedquantum doubles of finite groups.
Pacific J. Math. , 234(1):33–41, 2008.[12] Edward Formanek, Woo Lee, Inna Sysoeva, and Monica Vazirani. The irreducible complex repre-sentations of the braid group on n strings of degree ≤ n . J. Algebra Appl. , 2(3):317–333, 2003.[13] Jennifer M. Franko, Eric C. Rowell, and Zhenghan Wang. Extraspecial 2-groups and images ofbraid group representations.
J. Knot Theory Ramifications , 15(4):413–427, 2006.[14] Michael H. Freedman, Michael Larsen, and Zhenghan Wang. A modular functor which is universalfor quantum computation.
Comm. Math. Phys. , 227(3):605–622, 2002.[15] C´esar Galindo, Seung-Moon Hong, and Eric C. Rowell. Generalized and quasi-localizations ofbraid group representations.
Int. Math. Res. Not. IMRN , (3):693–731, 2013.[16] C´esar Galindo and Eric C. Rowell. Braid representations from unitary braided vector spaces.
J.Math. Phys. , 55(6):061702, 13, 2014.[17] Shlomo Gelaki, Deepak Naidu, and Dmitri Nikshych. Centers of graded fusion categories.
AlgebraNumber Theory , 3(8):959–990, 2009.[18] David M. Goldschmidt and V. F. R. Jones. Metaplectic link invariants.
Geom. Dedicata , 31(2):165–191, 1989.[19] Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones.
Coxeter graphs and towersof algebras , volume 14 of
Mathematical Sciences Research Institute Publications . Springer-Verlag,New York, 1989.[20] Paul Gustafson, Yuze Ruan, and Eric Rowell. Metaplectic categories, gauging and property F.
Tohoku Math. J. to appear, arXiv:1808.00698.[21] Jarmo Hietarinta. All solutions to the constant quantum Yang-Baxter equation in two dimensions.
Phys. Lett. A , 165(3):245–251, 1992.[22] Masaki Izumi. The structure of sectors associated with Longo-Rehren inclusions. II. Examples.
Rev. Math. Phys. , 13(5):603–674, 2001.[23] Pascual Jara Mart´ınez, Javier L´opez Pe˜na, Florin Panaite, and Freddy van Oystaeyen. On iteratedtwisted tensor products of algebras.
Internat. J. Math. , 19(9):1053–1101, 2008. , ANDREW KIMBALL , ERIC C. ROWELL , QING ZHANG [24] V. F. R. Jones. Braid groups, Hecke algebras and type II factors. In Geometric methods inoperator algebras (Kyoto, 1983) , volume 123 of
Pitman Res. Notes Math. Ser. , pages 242–273.Longman Sci. Tech., Harlow, 1986.[25] V. F. R. Jones. Notes on subfactors and statistical mechanics. In
Braid group, knot theory andstatistical mechanics , volume 9 of
Adv. Ser. Math. Phys. , pages 1–25. World Sci. Publ., Teaneck,NJ, 1989.[26] V. F. R. Jones. On knot invariants related to some statistical mechanical models.
Pacific J. Math. ,137(2):311–334, 1989.[27] Michael J. Larsen and Eric C. Rowell. An algebra-level version of a link-polynomial identity ofLickorish.
Math. Proc. Cambridge Philos. Soc. , 144(3):623–638, 2008.[28] Jun Murakami. The Kauffman polynomial of links and representation theory.
Osaka J. Math. ,24(4):745–758, 1987.[29] Deepak Naidu and Eric C. Rowell. A finiteness property for braided fusion categories.
Algebr.Represent. Theory , 14(5):837–855, 2011.[30] Sonia Natale. The core of a weakly group-theoretical braided fusion category.
Internat. J. Math. ,29(2):1850012, 23, 2018.[31] Dmitri Nikshych. Classifying braidings on fusion categories. In
Tensor categories and Hopf alge-bras , volume 728 of
Contemp. Math. , pages 155–167. Amer. Math. Soc., Providence, RI, 2019.[32] Eric C. Rowell. A quaternionic braid representation (after Goldschmidt and Jones).
QuantumTopol. , 2(2):173–182, 2011.[33] Eric C. Rowell and Zhenghan Wang. Localization of unitary braid group representations.
Comm.Math. Phys. , 311(3):595–615, 2012.[34] Eric C. Rowell and Hans Wenzl. SO( N ) braid group representations are Gaussian. QuantumTopol. , 8(1):1–33, 2017.[35] Daisuke Tambara and Shigeru Yamagami. Tensor categories with fusion rules of self-duality forfinite abelian groups.
J. Algebra , 209(2):692–707, 1998.[36] Imre Tuba and Hans Wenzl. Representations of the braid group B and of SL(2 , Z ). Pacific J.Math. , 197(2):491–510, 2001.[37] William C. Waterhouse. The number of congruence classes in M n ( F q ). Finite Fields Appl. , 1(1):57–63, 1995. Department of Electrical Engineering, Wright State University, Dayton, OH 45435,U.S.A.
E-mail address : [email protected] Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A.
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