On the Representation theory of the Infinite Temperley-Lieb algebra
OOn the Representation theory of the InfiniteTemperley-Lieb algebra
Stephen T. Moore
Dept. of Mathematics, Ben Gurion University,Beer-Sheva, Israel,[email protected]
April 30, 2019
Abstract
We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, thenintroduce infinite link state representations and classify when they are ir-reducible or indecomposable. We also define a construction of projectiveindecomposable representations for
T L n that generalizes to give extensionsof T L ∞ representations. Finally we define a generalization of the spin chainrepresentation and conjecture a generalization of Schur-Weyl duality. The Temperley-Lieb algebras [15] are a family of finite dimensional algebras thatfirst appeared in relation to statistical mechanics models, before being rediscov-ered by Jones in the standard invariant of subfactors [9], who used them to definethe Jones polynomial knot invariant [10]. They were further shown to be theSchur-Weyl dual of the quantum group U q ( sl ) by Jimbo and Martin [8, 13]. Therepresentation theory of the Temperley-Lieb algebra was first studied by Martin[12] who used combinatorial techniques to classify irreducibles and constructedprojective representations as ideals generated by projections. Shortly after, Good-man and Wenzl [7] studied path idempotents on Bratelli diagrams to classify theblocks of the algebra. A more recent study was done using cellular algebra tech-niques by Ridout and Saint-Aubin [14] based on earlier work by Westbury [16].A follow-up paper by Belletˆete, Ridout, and Saint-Aubin [2] fully classified theindecomposable representations of the Temperley-Lieb algebra.Whilst current research has focused on the finite dimensional Temperley-Liebalgebras, their infinite dimensional generalization also has potential applications.It has been conjectured based on physical considerations that in the n → ∞ limit,the representation theory of the Temperley-Lieb algebra should approach that ofthe Virasoro algebra. An attempt to realize this based on an inductive limit of RepT L n was given in [6], however we shall see that there are large families ofirreducible T L ∞ representations not appearing in this construction. The infinite1 a r X i v : . [ m a t h . QA ] A p r emperley-Lieb algebra has also appeared in the study of the periplectic Lie su-peralgebra [1] where a categorical action was defined.The paper is organized as follows: In Section 2 we review the representationtheory of the finite Temperley-Lieb algebras. In Section 3 we begin the study ofthe infinite Temperley-Lieb algebra, and fully classify its finite dimensional repre-sentations in Theorem 3.1. We then introduce infinite link state representations ,which generalize standard T L n representations and can be considered analoguesof Verma modules, and classify when these are irreducible and isomorphic in The-orems 3.2 and 3.3, as well as showing that they have at most two irreduciblesubquotients. Next in Definition 3.3 we define a construction of projective inde-composable representations for T L n that generalizes to give extensions of certain T L ∞ representations. Finally in Section 3.3 We relate a bilinear form definedbased on cellular algebras to one defined on the spin chain representation. Wealso define a generalization of the spin chain representation which gives a functor F : RepU q ( sl ) → RepT L ∞ . We conjecture this functor defines an equivalence ofcategories between the category of finite dimensional U q ( sl ) representations andcertain subcategories of RepT L ∞ . T L n . The Temperley-Lieb algebra
T L n ( δ ) is a finite dimensional algebra with generators { , e , ..., e n − } and relations: e i = δe i e i e i ± e i = e i e i e j = e j e i , | i − j | > δ := q + q − , q ∈ C \ { } . This algebra has a natural diagrammaticdescription, due to Kauffman [11], where basis elements of the algebra consist ofrectangles with n marked points along the top and bottom, and non-intersectingstrings connecting these points. Multiplication is given by vertical concatenationof diagrams, and removing a closed loop corresponds to multiplying by δ . Inthis description, the identity element is a diagram with n vertical strings, andthe generator e i is given by a cup connecting the i th and ( i + 1)th points alongthe top, a cap connecting the same points along the bottom, and vertical stringseverywhere else. We will sometimes need to use certain polynomials in q , knownas quantum integers : Definition 2.1.
The quantum integer [ n ] is given by [ n ] := q n − q − n q − q − . Alternatively, [ n ] = n (cid:80) i =0 q n − i − , or [ n ] = δ [ n − − [ n − , with [1] = 1 and [2] = δ . We denote [ n ]! := 1 × [2] × ... × [ n ] . , e ∈ T L , and the relation e = δe .The representation theory of the Temperley-Lieb algebras was first developedby Martin [12] using combinatorial techniques, and Goodman and Wenzl [7] basedon path idempotents for Bratelli diagrams. However for our purposes we followthe more recent version by Ridout and Saint-Aubin [14] based on cellular algebramethods. Definition 2.2.
A (n-p)-link state, with ≤ p ≤ (cid:98) n (cid:99) , is a line with n markedpoints on it, labelled left to right from to n , with non-intersecting strings attachedto each point, such that p strings connect two points together, and the other n − p strings are only connected to a single point. We refer to the strings connected to two points as cups , and the strings con-nected to a single points as a string . A cup connecting the points i and i + 1 isreferred to as a simple cup at the i th point.Figure 2: A (5 ,
2) link state.The number of ( n, p )-link states is given by d n,p := (cid:18) np (cid:19) − (cid:18) np − (cid:19) . Definition 2.3.
The
T L n ( δ ) standard representation V n,p is the representationwith basis given by the set of ( n, p ) -link states, with T L n acting by concatenation,where the action of a T L n generator is zero if it increases the number of cups ofthe link state. There is a symmetric invariant bilinear form (cid:104)· , ·(cid:105) n,p on V n,p defined as follows.Reflect the left entry along its line then join on top of the right entry. It is zeroif every string on one side isn’t connected to a string on the other. Otherwise itgives δ k , where k is the number of closed loops. For example, the bilinear form on V , is as follows: 3sing this bilinear form, we can define two further families of T L n representa-tions: Definition 2.4.
The kernel of the bilinear form, R n,p := { x ∈ V n,p : (cid:104) x, y (cid:105) =0 ∀ y ∈ V n,p } is a subrepresentation of V n,p , called the radical. We denote thequotient representation by L n,p := V n,p / R n,p . Given ( n, p )-link states x, y , we can form the
T L n element (cid:107) xy (cid:107) by reflecting y and joining it to the bottom of x . This leads to the following lemma: Lemma 2.1.
For x, y, z ∈ V n,p , (cid:107) xy (cid:107) z = (cid:104) y, z (cid:105) x . This can be viewed as saying that if x, z / ∈ R n,p , then there are
T L n elementsmapping x to z and z to x . This lemma is essential for the proof of the followingproposition: Proposition 2.1. If (cid:104)· , ·(cid:105) n,p is not identically zero, then V n,p is cyclic and inde-composable, and L n,p is irreducible. The only case in which the bilinear form is identically zero is when δ = 0 and n = 2 p , so in this case R p,p = V p,p . In general, the radical is described by thefollowing: Proposition 2.2.
The radical R n,p is irreducible. Hence the standard representations are either irreducible or indecomposablewith two irreducible components. Determining when the radical is non-trivialis done by studying the Gram matrix of the bilinear form. Namely, take thematrix whose entries are given by (cid:104) x, y (cid:105) for link states x, y ∈ V n,p . Then when thedeterminant of this matrix is zero, the radical is non-trivial. The result of this canbe summarized as follows:
Theorem 2.1.
Let l ∈ N be minimal such that q l = 1 . • If no such l exists, l = 1 , or l = 2 and n is odd, then R n,p = 0 . • If n − p + 1 = kl for some k ∈ N , k (cid:54) = 0 , then R n,p = 0 . • If l = 2 (i.e. δ = 0 ) and n = 2 p , then R p,p = V p,p . • If ≤ n − p + 1 < l then R n,p = 0 . • Otherwise R n,p (cid:54) = 0 , V n,p . Generally, the representation theory of
T L n can be split into the cases of q being generic and q a root of unity , where we consider q = ± n, p ) as being critical whenever n − p + 1 = kl . Astraightforward consequence of the above theorem is the following: Corollary 2.1.
T L n is semisimple when q is generic or l = 2 and n is odd. Thestandard representations V n,p , ≤ p ≤ (cid:98) n (cid:99) then form a complete set of irreducibles. , | (2 ,
1) (2 , , | (3 , ,
2) (4 ,
1) (4 , | (5 , | (5 ,
1) (5 , ,
3) (6 ,
2) (6 , | (6 , , | (7 ,
2) (7 ,
1) (7 , | (8 ,
4) (8 ,
3) (8 , | (8 ,
1) (8 , , | (9 ,
3) (9 ,
2) (9 , | (9 , l = 3.The possible labels ( n, p ) of standard representations can be arranged into aBratelli diagram. For roots of unity, the critical labels form into critical lines. If( n, p ) and ( n, p (cid:48) ) are located the same distance on either side of a given criticalline then we refer to ( n, p ) , ( n, p (cid:48) ) as a symmetric pair . An example of the Bratellidiagram for l = 3 is given in Figure 2. Examples of symmetric pairs in this caseinclude (3 , , (3 ,
1) and (8 , , (8 , Proposition 2.3.
End ( V n,p ) (cid:39) C for all values of q . Hom ( V n,p , V n,p (cid:48) ) = 0 unless q is a root of unity, p (cid:48) > p , and ( n, p ) , ( n, p (cid:48) ) is a symmetric pair, inwhich case Hom ( V n,p , V n,p (cid:48) ) (cid:39) C . There is an exceptional case for δ = 0 , with Hom ( V , , V , ) (cid:39) C . For p, p (cid:48) such that
Hom ( V n,p , V n,p (cid:48) ) (cid:39) C , we denote by φ n,p ∈ Hom ( V n,p , V n,p (cid:48) )the unique such map up to scalars. An explicit construction of this map was givenin [4] and is given as follows: Definition 2.5.
For ( n, , ( n, p ) a symmetric pair, the map φ n, : V n, → V n,p is given by φ n, ( | n ) = (cid:80) a i y i where y i are link states in V n,p , and the sum isover all such link states. For the coefficients a i , given a link state, y i , each line h in y i splits y i into two regions. Let n ( h ) be the number of lines (including h itself ) in the region of y i not containing the right hand edge of the diagram. Then a i = ( − e ( y i ) [ n − p ]! (cid:81) h [ n ( h )] , where the product is over all lines h in y i , and e ( y i ) is thenumber of even quantum integers in the coefficient. The maps φ n,p : V n,p → V n,p (cid:48) can be constructed from φ n − p, by joining theimage of φ n − p, to the bottom of link states in V n,p . The maps φ n,p allow us toalmost fully describe the representations L n,p , R n,p . Namely, for a symmetric pair( n, p ) , ( n, p (cid:48) ), Ker ( φ n,p ) = R n,p , Im ( φ n,p ) = R n,p (cid:48) , so L n,p (cid:39) R n,p (cid:48) . Combined withknowledge of the T L n projective representations, the following can be shown: Proposition 2.4.
For δ = 0 and n even, the representations L n,p , ≤ p ≤ n − form a complete set of irreducibles. Otherwise, the representations L n,p , ≤ p ≤(cid:98) n (cid:99) form a complete set of irreducibles for T L n . L n,p by L n,p . For given n, p, l , we can write n − p +1 = k ( n, p ) l + r ( n, p ) uniquely, where k ( n, p ) , r ( n, p ) ∈ N , 1 ≤ r ( n, p ) ≤ l . For roots ofunity we have the following: Proposition 2.5.
The dimensions of the irreducible quotients L n,p satisfy: L n,p = d n,p r ( n, p ) = lL n − ,p r ( n, p ) = l − L n − ,p + L n − ,p − otherwise with initial conditions L n, = 1 and L p − ,p = 0 . As T L n is semisimple for q generic, the projective indecomposables are just thestandard representations. At roots of unity, denote the projective cover of L n,p by P n,p . Then they are as follows: Proposition 2.6. If ( n, p ) is critical, so L n,p (cid:39) V n,p , then P n,p (cid:39) V n,p . If ( n, p ) isnot critical, and there is no p (cid:48) > p such that ( n, p ) , ( n, p (cid:48) ) is a symmetric pair, then P n,p (cid:39) V n,p . If such a symmetric pair exists, then P n,p is the non-trivial extension → V n,p (cid:48) → P n,p → V n,p → . A construction of the projective representations using idempotents was givenin [12]. We give an alternative construction in Section 3.3.The possible extensions of irreducible and standard representations, as well asa complete list of indecomposable
T L n representations was classified in [14]. Proposition 2.7.
Let ( n, p ) , ( n, p (cid:48) ) , and ( n, p (cid:48) ) , ( n, p (cid:48)(cid:48) ) , p < p (cid:48) < p (cid:48)(cid:48) , be symmetricpairs. Then the non-trivial extensions of irreducible and standard representationsare as follows: • Ext ( L n,p , L n,p (cid:48) ) (cid:39) Ext ( L n,p (cid:48) , L n,p ) (cid:39) C • Ext ( V n,p , L n,p (cid:48) ) (cid:39) C • Ext ( L n,p , V n,p (cid:48)(cid:48) ) (cid:39) C • Ext ( L n,p , V n,p (cid:48) ) (cid:39) C if there is no p − < p such that ( n, p − ) , ( n, p ) is symmet-ric. • Ext ( V n,p , V n,p (cid:48) ) (cid:39) Ext ( V n,p , V n,p (cid:48)(cid:48) ) (cid:39) C • For δ = 0 , Ext ( V p +2 ,p , L p +2 ,p ) (cid:39) C T L ∞ ( δ ) Define the algebra
T L ∞ ( δ ) as generated by { , e i } , i ∈ N , satisfying the usualrelations, where we only consider finite products of generators, and finite sums ofbasis elements. 6 .1 Finite dimensional representations We begin by considering finite dimensional representations of
T L ∞ . The trivialrepresentation is given by e i v = 0 for all i . For δ (cid:54) = ±
1, this is the only one-dimensional representation. However, for δ = ±
1, there is a second given by e i = δv . For higher dimensions, we have the following result: Theorem 3.1.
The only finite dimensional representations of
T L ∞ are direct sumsof one-dimensional representations. To prove this, we first need the following two lemmas:
Lemma 3.1.
Let X be an indecomposable T L n ( δ ) representation. Then either X is one-dimensional, or dimX ≥ n − .Proof. We split this into several parts, and consider the cases l = 2, l = 3, and n ≤ l ≥ n ≥
6. For irreduciblerepresentations L n,p , we denote dim L n,p := L n,p .For q generic, the irreducible representations have dimension d n,p , for which d n, =1, and d n,p ≥ n − p ≥
1. For l = 2, for n odd, the irreducibles are thestandard representations, so L n,p = d n,p . For n even, we have r ( n, p ) = 1, so bythe recursion relation in Proposition 2.5, we have L n,p = d n − ,p .For l = 3, we have the cases r ( n, p ) = 1 ,
2, to consider. We first notethat the case r ( n, p ) = 2 implies the case r ( n, p ) = 1. For the r ( n, p ) = 2case, L n, = d n − , + d n − , = n −
2. For L n,p , 2 ≤ p < (cid:98) n (cid:99) , we can rewrite L n,p = L n − ,p − + d n − ,p it follows from induction that if L n − ,p − ≥ n −
4, then L n,p ≥ n − d n,p ≥
2. For p = (cid:98) n (cid:99) , because of the condition L p − ,p = 0, weinstead get L p,p = L p +1 ,p = 1.For n ≤
6, the cases l = 4 , , , ,
8, can be computed directly and checked. For l ≥
9, we note that as 0 ≤ p ≤ (cid:98) n (cid:99) , we have 1 ≤ r ( n, p ) ≤ l −
2. Hence the onlypart of the recursion relation used is L n,p = L n − ,p + L n − ,p − . It follows that for n ≤ l ≥
9, we have L n,p = d n,p .Now we assume that l ≥ n ≥
6. Assume that n is minimal such thatthere is a 1 ≤ p ≤ (cid:98) n (cid:99) such that L n,p < n −
2. Clearly r ( n, p ) (cid:54) = 0. If r ( n, p ) = l − L n,p = L n − ,p , with r ( n − , p ) = l −
2. If n − p −
1, so that L n − ,p = 0,then r ( n − , p ) = 0, which requires l = 2. Hence L n − ,p is non-zero, and we have L n,p = L n − ,p + L n − ,p − . If n − p −
1, so that L n − ,p = 0, then r ( n − , p ) = 0,which requires l = 3. Hence L n − ,p and L n − ,p − are non-zero. We assumed n wasminimal such that L n,p < n −
2. Hence we must have L n − ,p , L n − ,p − ≥ n −
4, andso L n,p ≥ n −
8, and hence L n,p ≥ n − n ≥ ≤ r ( n, p ) ≤ l −
2, then L n,p = L n − ,p + L n − ,p − . If L n − ,p , L n − ,p − are both non-zero, then as n is minimal, L n − ,p , L n − ,p − ≥ n −
3, and so L n,p ≥ n − ≥ n − n = 2 p , so that L n − ,p = 0, we have r ( n − , p −
1) = 2 ≤ l − l (cid:54) = 2 , L n,p = L n − ,p − = L n − ,p − + L n − ,p − ≥ n − ≥ n − T L n representations have dimensiongreater than or equal to n −
2, except for the trivial representation, and L n, (cid:98) n (cid:99) for7 = 3. Finally we note that the classification of extensions of irreducible represen-tations in Proposition 2.7 that there is a non-trivial extension of two irreduciblerepresentations, if and only if they form a symmetric pair. Hence the only non-trivial extensions of one-dimensional representations are the extensions of L , , L , and L , , L , for l = 3, and both these extensions have dimension ≥ n − Lemma 3.2.
Let A be an n × n diagonal matrix with entries a i ∈ { , } or { , − } .Let B be an arbitrary n × n matrix. If ABA = A , BAB = B , and AB = BA ,then A = B .Proof. Let B := ( b ij ) ≤ i,j ≤ n . Considering AB = BA we get a i b ij = b ij a j . Hence b ij = 0 unless a i = a j . Next, for ABA = A , we have a i b ii = a i and a i a j b ij = 0 for i (cid:54) = j . Hence if a i (cid:54) = 0, then b ii = a i , and if a i a j = 1 for i (cid:54) = j then b ij = 0. Finally,considering BAB = B , we get n (cid:80) j =1 a j b ij b jk = b ik . If a i = a k = 0, but a j (cid:54) = 0, wehave b ij = b jk = 0. Hence b ik = 0, and so we have B = A .Now we can prove Theorem 3.1: Proof.
Assume that there is a finite dimensional
T L ∞ representation X , of di-mension k . By Lemma 3.2, if the generators e and e act as diagonal matriceswith entries { , } or { , − } , then every generator must act as the same diago-nal matrix, and so X must be a direct sum of one-dimensional representations.Assume otherwise, then consider the restriction of X to a T L n representation forsome n > k + 2. Then by Lemma 3.1, the only T L n representations in the de-composition of X are one dimensional, and so the generators e i , 1 ≤ i ≤ n − X is a direct sum of one-dimensional T L ∞ representations. T L ∞ Representations generated by link states.
We now aim to study representations of
T L ∞ by generalizing the standard repre-sentations of T L n . We start by defining an infinite link state as a link state withinfinitely many points labelled by n ∈ N . Given an infinite link state w , we define s ( w ) ∈ N ∪ {∞} as the number of strings on w , and c ( w ) ∈ N ∪ {∞} as the numberof cups on w , we then define X ( w ) as the representation generated by w . We willgenerally prove results about X ( w ) by using results from the T L n representationtheory. To do this, we will need the following definition: Definition 3.1.
Define the restriction to length n of a link-state as the link-stateformed by deleting all points greater than n . Note that we do not allow the restric-tion to cut cups, as this will not always respect the bilinear form, as it introducesan extra string. In this case, we instead restrict to the smallest n (cid:48) > n that doesn’trequire the cutting of a cup. It is clear that if two link states, w, w (cid:48) differ at infinitely many points, thenthe representation generated by them must decompose into a direct sum, i.e. X ( w, w (cid:48) ) (cid:39) X ( w ) ⊕ X ( w (cid:48) ). However it is not obvious if this is true when w and w (cid:48) only differ at finitely many points. To resolve this, we need the following:8 emma 3.3. Let x, y be two finite sums of infinite link states, that only differ upto the first n points. If x and y both restrict to V n (cid:48) ,p , x, y / ∈ R n (cid:48) ,p , for n (cid:48) ≥ n , thenthere are a, b ∈ T L ∞ such that x = ay and y = bx .Proof. This follows by restricting x and y to V n (cid:48) ,p then using Lemma 2.1. Theelements a, b are given by extending the appropriate elements from Lemma 3 . T L ∞ by adding infinitely many strings to the right hand side of the diagrams.In general, restrictions of link states are never in R n,p , with the exception when δ = 0 and s ( w ) = 0. However in this case we can still find T L ∞ elements that maplink states to each other, as R n,p is irreducible. From this, we get the followingproposition: Proposition 3.1.
Let w , w (cid:48) be two infinite link states, and X ( w, w (cid:48) ) the repre-sentation generated by them. If w and w (cid:48) differ at only the first n points, andtheir restrictions are both in V n (cid:48) ,p for n (cid:48) ≥ n , then X ( w, w (cid:48) ) (cid:39) X ( w ) (cid:39) X ( w (cid:48) ) .Otherwise, X ( w, w (cid:48) ) (cid:39) X ( w ) ⊕ X ( w (cid:48) ) . We can define an equivalence relation on link states as follows: We say w ∼ w (cid:48) if they differ at only finitely many points and there is an n ∈ N such that therestrictions of w and w (cid:48) are both in V m,p for all m ≥ n . From now on, we considerlink states under this equivalence.To prove irreducibility of representations, we generalize the method of [14], anddefine a subrepresentation generalizing the radical for each representation. Weshall see that the representations are indecomposable, and their quotient by theradical is irreducible. We define the radical R ( w ) ⊆ X ( w ) as follows: R ( w ) := { x ∈ X ( w ) : there is n ∈ N such that x restricts to R n (cid:48) ,p for all n (cid:48) ≥ n } It is straightforward to see that R ( w ) is a T L ∞ representation, as R n (cid:48) ,p is al-ways irreducible. Using Lemma 3.3, Proposition 3 . Proposition 3.2. X ( w ) is cyclic and indecomposable, and X ( w ) / R ( w ) is irre-ducible. The only case when X ( w ) = R ( w ) is when δ = 0 and s ( w ) = 0, otherwise wehave R ( w ) (cid:40) X ( w ). The structure of R ( w ) is given by the following: Proposition 3.3. R ( w ) is either zero or irreducible.Proof. Given x, y ∈ R ( w ), then for n large enough, x and y both restrict to R n,p .By Theorem 7 . R n,p is either zero or irreducible. If it is zero for n (cid:48) ≥ n ,then x and y must be zero, and so R ( w ) is zero. If R n,p is non-zero for all n ,then as it is irreducible, we can find T L n elements that map the restrictions of x and y to each other. These extend to T L ∞ elements that map x and y to eachother, hence every R ( w ) element generates the representation, and so R ( w ) isirreducible.Combined with Proposition 3.2, it follows that every X ( w ) has at most twoirreducible representations in its composition series. We now want to determinewhen these representations are irreducible. For that, we will first need the followinglemma: 9 emma 3.4. Let x ∈ R n,p , and y ∈ V k,k , so that y only consists of cups. Then x ⊗ y ∈ R n +2 k,p + k , where x ⊗ y is formed by joining y onto the right hand side of x .Proof. As R n,p (cid:54) = 0, we know that ( n, p ) is not critical. Then ( n + 2 k, p + k ) isalso non-critical, and so R n +2 k,p + k (cid:54) = 0. Let p (cid:48) < p be such that ( n, p (cid:48) ), ( n, p )form a symmetric pair, so that Hom ( V n,p (cid:48) , V n,p ) (cid:39) C . The image of this mapcan be constructed explicitly as given in Definition 2.5. Given a link state z ∈V n,p (cid:48) , its image in V n,p is the sum over all link states formed by joining together2( p − p (cid:48) ) strings in z . The image of this, Im ( z ), is then in R n,p . We now note that Hom ( V n +2 k,p (cid:48) + k , V n +2 k,p + k ) (cid:39) C , as ( n + 2 k, p (cid:48) + k ) , ( n + 2 k, p + k ) is a symmetricpair if ( n, p (cid:48) ) , ( n, p ) is. Given z ⊗ y ∈ V n +2 k,p (cid:48) + k , we again construct its image, butthis only depends on joining together strings, so Im ( z ⊗ y ) = Im ( z ) ⊗ y , and hence Im ( z ) ⊗ y ∈ R n +2 k,p + k . It then follows that this is true for any x ∈ R n,p .Note that if instead y contains j strings as well as cups in the above lemma,then it no longer holds, as x ⊗ y ∈ V n +2 k + j,p + k which can be critical, and so wecan’t guarantee that R n +2 k + j,p + k (cid:54) = 0. Given this, we can now classify which ofthe representations are irreducible. Theorem 3.2.
For q generic, all representations X ( w ) generated by an infinitelink state are irreducible. For q a root of unity, with l the smallest positive integersuch that q l = 1 , if s ( w ) = ∞ , s ( w ) = kl − for k ∈ N , or s ( w ) = 0 and l = 2 ,then X ( w ) is irreducible. Otherwise it is indecomposable but not irreducible.Proof. For q generic, R n,p = 0 for all n and p . Hence R ( w ) = 0, and so X ( w ) isirreducible.For q a root of unity, R n,p may now be non-zero. We first look at the case s ( w ) = ∞ . Assume there is some x ∈ R ( w ), x (cid:54) = 0. Then x must restrict to R n,p for n large. However, as s ( w ) = ∞ , we can always find an n (cid:48) > n such that x restricts to V n (cid:48) ,p (cid:48) with ( n (cid:48) , p (cid:48) ) critical. Hence x = 0, and therefore R ( w ) = 0.For s ( w ) = kl −
1, any x ∈ X ( w ) will restrict to V n, n − kl +12 for n large enough,which is critical and so has trivial radical.For l = 2 and s ( w ) = 0, X ( w ) = R ( w ), which we have already shown isirreducible.For the remaining representations, by Proposition 3.2 we know the representa-tions are indecomposable. For a given link state w with s ( w ) = j , given x ∈ X ( w )that restricts to R n, n − j for n large enough, by Lemma 3.4, it will restrict to R n +2 k, n − j + k for k ∈ N . Hence R ( w ) is non-empty. However, not every element of X ( w ) is in R ( w ), namely w / ∈ R ( w ). Therefore the X ( w ) are not irreducible.Now that we have determined when these representations are irreducible, wewant to find out if they are pairwise isomorphic. To do this, we look at whenthere are non-zero morphisms between representations. To start, we first need thefollowing definition: Definition 3.2.
Given two infinite link states, w, z , that differ at only finitelymany points, with s ( w ) and s ( z ) both finite, s ( w ) < s ( z ) , we say w and z form asymmetric pair if ( s ( z ) , , ( s ( z ) , s ( z ) − s ( w )2 ) form a symmetric pair. roposition 3.4. Given two infinite link states, w, z , if w and z are equiv-alent link states, or w and z form a symmetric pair with s ( z ) < s ( w ) , then Hom ( X ( w ) , X ( z )) (cid:39) C . Otherwise, Hom ( X ( w ) , X ( z )) = 0 .Proof. We consider two cases, when w and z differ at finitely many points, andwhen they differ at infinitely many points. When w and z only differ at finitelymany points, assume there is some map φ w,z : X ( w ) → X ( z ). We can restrict thisto some n large enough to get a map φ n,p : V n,p → V n,p (cid:48) . By Proposition 2.3, either φ n,p is the identity and p = p (cid:48) , or φ n,p is as in Definition 2.5 and ( n, p ) , ( n, p (cid:48) ) isa symmetric pair. By considering all possible restrictions to n (cid:48) > n , we see thateither φ w,z is the identity with w and z equivalent, or ( n (cid:48) , p + n (cid:48) − n ) , ( n (cid:48) , p (cid:48) + n (cid:48) − n )are symmetric pairs for all n (cid:48) > n . As the construction of each of the φ n (cid:48) ,p dependonly on the number of strings, we can take p = 0 to get ( s ( z ) , , ( s ( z ) , s ( z ) − s ( w )2 )is a symmetric pair.If w and z differ at infinitely many points, assume there is some map f : X ( w ) → X ( z ), f : w (cid:55)→ (cid:80) a i z i , a i ∈ C , z i ∈ X ( z ). As w and z differ at infinitelymany points, we can find some point i such that there is a simple cup at the i thposition on one of w or f ( w ) but not the other. Considering ( e i − δ f ( w ), wesee that f can’t exist for δ (cid:54) = 0. If δ = 0, we can instead consider e i f ( w ), unlessone of w or z is constructed from the other by replacing multiple simple cupswith two strings. In this case, let i, i + 1 , j, j + 1 be the locations of two adjacentreplacements. We further assume that a cup connects the ( i + 2)th and ( j − e i +1 e j − on w and f ( w ) we then get zero for whichever has stringsat i, i + 1 and j, j + 1, but for whichever has cups we get:Hence f must equal zero. Corollary 3.1. X ( w ) (cid:39) X ( z ) if and only if w and z are equivalent link states. The map φ w,z : X ( w ) → X ( z ) described in Proposition 3.4 for when w and z area symmetric pair is an immediate generalization of the map defined in Definition2.5. As previously noted, Proposition 3.2 and Proposition 3.3 combined give thatwhen X ( w ) is not irreducible, it has exactly two irreducible components in itscomposition series, namely R ( w ) and L ( w ) := X ( w ) / R ( w ). We now want todetermine when these irreducible components are isomorphic. Given an infinitelink state w with s ( w ) = j ∈ N , let w ( j ) be an infinite link state that differs atfinitely many points from w , and s ( w ( j ) ) = j + j , i.e. the link state formed bycutting j cups on w . For 0 ≤ j ≤ l −
2, we then have the following sequence: ... φ −→ X ( w ( j ) ) φ −→ X ( w ( j ) ) φ −→ X ( w )with j k := 2( kl − − j k − , and φ i ∈ Hom ( X ( w ( j i ) ) , X ( w ( j ( i − ) )). By consideringrestricting terms in this sequence to finite n , we get the following: Proposition 3.5.
The above sequence is exact, with Im ( φ k ) (cid:39) R ( w ( j k − ) ) .
11e note that this allows each irreducible representation to be realized as thekernel of the bilinear form, except for irreducibles L ( w ), s ( w ) < l −
1, noting thatfor l = 2, L ( w ) = 0 if s ( w ) = 0. By the proof of Proposition 3.4 we find that theirreducibles are not isomorphic for s ( w ) > l −
1. For s ( w ) < l −
1, we have thefollowing result:
Proposition 3.6.
For l ≥ , given inequivalent infinite link states w, z with s ( w ) < l − and s ( z ) finite, then L ( w ) and L ( z ) are not isomorphic.Proof. If w and z differ at only finitely many points then the proof follows imme-diately from restriction. Hence, we only consider when w and z differ at infinitelymany points. Assume otherwise that L ( w ) (cid:39) L ( z ), with w (cid:48) (cid:55)→ z (cid:48) , where we view w (cid:48) , z (cid:48) as elements of X ( w ), X ( z ) respectively. Choose some n large such that e n w (cid:48) = δw (cid:48) , but e n z (cid:48) (cid:54) = δz (cid:48) , and the n th and ( n + 1)th points of z (cid:48) are connectedto cups, not strings. For this map to give an isomorphism of L ( w ) and L ( z ), wemust have e n z (cid:48) − δz (cid:48) ∈ R ( z ). We can therefore restrict e n z (cid:48) − δz (cid:48) to R n (cid:48) ,p for some n (cid:48) > n . Hence we must have (cid:104) e n z (cid:48) − δz (cid:48) , z (cid:48) (cid:105) = 0, i.e. (cid:104) e n z (cid:48) , z (cid:48) (cid:105) = δ (cid:104) z (cid:48) , z (cid:48) (cid:105) . Asthere isn’t a cup at the n th point of z (cid:48) , but it is connected to a cup, we find that (cid:104) e n z (cid:48) , z (cid:48) (cid:105) = δ − (cid:104) z (cid:48) , z (cid:48) (cid:105) , (see figure below), and so e n z (cid:48) − δz (cid:48) / ∈ R ( z ) unless δ = 1.Hence there are no such isomorphism maps for l ≥ e n affects the bilinear form.For the cases s ( w ) = 0 , l = 3, we have the following result: Corollary 3.2.
For l = 3 , let X be the non-trivial one-dimensional representationgiven by e i ν = δν . If s ( w ) = 0 , , then L ( w ) (cid:39) X .Proof. If l = 3 and s ( w ) = 0 ,
1, then R ( w ) is the set of elements of the form w (cid:48) − δw (cid:48)(cid:48) , where w (cid:48) , w (cid:48)(cid:48) are link states in X ( w ) such that joining together twostrings on w (cid:48) and cutting a cup results in w (cid:48)(cid:48) . Any two link states in X ( w ) canbe transformed into each other by a sequence of cutting cups and joining togetherstrings, hence quotienting by R ( w ) results in a one-dimensional representation,which we can see is not the trivial one.Combining together the results of Propositions 3.4, 3.5, 3.6, and Corollaries3.1 and 3.2, we get the following: Theorem 3.3.
The Representations X ( w ) , X ( w (cid:48) ) are isomorphic if and only if thelink states w, w (cid:48) are equivalent. The non-trivial irreducible quotients L ( w ) , L ( w (cid:48) ) are isomorphic if and only if the link states w, w (cid:48) are equivalent, or l = 3 and s ( w ) = 0 , . X ( w ) representations, wewant to study if there are any possible extensions of them. We define a method ofconstructing the projective indecomposable T L n representations that generalizesto give extensions of certain T L ∞ representations: Definition 3.3.
Let ( n, p ) , ( n, p (cid:48) ) be a symmetric pair, with p < p (cid:48) , and x ∈ V n,p .Let φ be the unique map φ : V n,p → V n,p (cid:48) given by φ : x (cid:55)→ (cid:80) i a i y i , a i ∈ C , where thesum is over all elements y i ∈ V n,p (cid:48) formed by adding p (cid:48) − p cups to x (see Definition2.5). We define an extension V n,p (cid:48) → P n,p → V n,p as follows: If a generator e k increases the number of cups of x , set ˜ e k x := (cid:80) j a j y j , where the sum is over allelements y j ∈ φ ( x ) with a simple cup at the k th position, and coefficients from φ ( x ) . As an example, the map φ : V , → V , exists for l = 4, i.e. δ = ±√
2. Thismap is given as follows:From this, we define the action of the generators in the extension V , → P , →V , as follows: Remark 3.1.
For some extensions, such as V n, → P n, → V n, , the simplerchoice of taking all coefficients to be equal to one is possible, but for higher p thisdoes not always define an extension of representations, for example in the case V , → P , → V , . We shall show that our more complicated choice of coefficientsalways defines a non-split extension. Theorem 3.4.
This extension is non-split, i.e. P n,p (cid:39) P n,p .Proof. We consider the case V n,p → P n, → V n, , as any other case can be con-structed from this by adding cups to both sides. We first check that this newaction defines a representation. The relation e k x = δe k x is immediate. For therelation e k e k ± e k x = e k x , we have e k e k +1 ∪ ↓ = e k ↓ ∪ = ∪ ↓ , where we use ↓ todenote a point connected to an arbitrary string or cup. For the relation e i e j = e j e i if | i − j | >
1, this is immediate if p = 1. For p >
1, fix such a choice of i and j ,then rewrite φ as follows: φ ( | ⊗ n ) = (cid:80) k m a k y k + a k y k + a k y k + a k y k + a k y k ,where each of the y k m are diagrams of the form as follows: y k are diagrams with asimple cup at the i th position but not at the j th position. y k are diagrams with asimple cup at the j th position but not at the i th position. y k are diagrams withsimple cups at both the i th and j th position. y k are diagrams of the form:13he y k are then the diagrams not appearing in any of the previous sums. Underthis relabelling, we have ˜ e i | ⊗ n = (cid:80) k m a k y k + a k y k , ˜ e j | ⊗ n = (cid:80) k m = a k y k + a k y k .We want to show that e j ˜ e i | ⊗ n = e i ˜ e j | ⊗ n , which we can now rewrite as (cid:80) k a k e j y k = (cid:80) k a k e i y k . We now note that e i y k = e j y k ∈ y k , and the y k are the onlydiagrams without a simple cup at i such that e j y k has a simple cup at i , andrespectively for i and j switched. Next we consider e i φ ( | ⊗ n ) = 0, in terms of thesum, this gives: (cid:88) k m δa k y k + a k e i y k + δa k y k + a k e i y k + a k e i y k = 0By separating into diagrams with or without a simple cup at the j th position, weget: (cid:88) k m a k e i y k + δa k y k + a k e i y k = 0Repeating for e j φ ( | ⊗ n ), we get: (cid:88) k m a k e j y k + δa k y k + a k e j y k = 0As (cid:80) k a k e i y k = (cid:80) k a k e j y k , this gives (cid:80) k a k e i y k = (cid:80) k a k e j y k , and hence therelation holds.To show that the extension doesn’t split, we again consider the case V n +1 ,p → P n +1 , → V n +1 , , and show there is no injective map V n +1 , → P n +1 , , with thegeneral case following from this. We start with the case p = 1. Note that the map φ : V n, → V n, can be written as φ ( | ⊗ n ) = n − (cid:80) i =1 ( − i +1 [ i ] c i , where c i is the diagramwith a single cup at the i th position, so ˜ e i | ⊗ n = ( − i +1 [ i ] c i in P n +1 , . Assume thereis some non-zero map Ψ : | ⊗ n +1 (cid:55)→ ˜ | ⊗ n +1 + n (cid:80) i =1 b i c i , where we use ˜ | ⊗ n +1 to denote | ⊗ n +1 ∈ P n +1 , , b i ∈ C . By considering the action of e j on Ψ for each j , we findthat the coefficients b i must satisfy the following:1 + δb + b = 0 , ( − i +1 [ i ] + b i − + δb i + b i +1 = 0 , ( − n +1 [ n ] + b n − + δb n = 0The last equation can be considered as the second with the extra condition b n +1 =0. By induction, we find the following general formula for b k : b k = ( − k +1 (cid:98) k − (cid:99) (cid:88) i =0 ( k − i − k − i − + ( − k +1 [ k ] b b n +1 = 0 then gives: b = − (cid:98) n (cid:99) (cid:88) i =0 ( n − i )[ n − i ][ n + 1]For ( n + 1 , , ( n + 1 ,
1) to be a symmetric pair, we must have n + 1 = kl , andso [ n + 1] = 0. Hence there is no solution for the coefficients, unless they can besimplified so that [ n + 1] doesn’t appear on the denominator. Using the solutionfor b , we get that b n is of the form: b n = ( − n [ n + 1] (cid:98) n (cid:99) (cid:88) i =0 ( n − i )[ n − i ][ n ] − (cid:98) n − (cid:99) (cid:88) j =0 ( n − j − n − j − n + 1] By using the relations [ n − i ][ n ] − [ n − i − n +1] = [2 i +1], and j (cid:80) i =0 [2 i +1] = [ j +1] ,this simplifies to give: b n = ( − n [ n + 1] (cid:32) n (cid:88) i =1 [ i ] (cid:33) At roots of unity, we have [ k ] ∈ R , and so n (cid:80) i =1 [ i ] >
0. Hence b n can’t be simplifiedto remove [ n + 1], and so the coefficients have no solution.For p >
1, we first need the following lemma:
Lemma 3.5.
Given the maps φ p : V n, → V n,p , φ p − : V n − , → V n − ,p − . Let ¯ φ p ∈ V n − ,p − be formed from the image of φ p by taking only terms with a cup atthe n th point, and deleting the n th point. Then ¯ φ p = ± φ p − ( | ⊗ n − ) .Proof. Acting e i , 1 ≤ i ≤ n −
2, on the link states contributing to ¯ φ p , we geteither zero or a sum of link states with a string at the n th point. If the resulthas a string at the n th point, then deleting the n th point results in link stateswith p cups which are zero in V n − ,p − . Hence e i ¯ φ p = 0 for 1 ≤ i ≤ n −
2, andso ¯ φ p ∈ R n − ,p − . As R n − ,p − is the image of φ p − , it is one-dimensional, hencewe have ¯ φ p = λφ p − ( | ⊗ n − ) for some non-zero constant λ . To find λ , consider thecoefficients in φ p , φ p − , respectively of the following two diagrams:The coefficients of these diagrams are ( − n − p +1 [ n − p ] , 1. Hence we must have ¯ φ p = ( − n − p +1 [ n − p ] φ p − ( | ⊗ n − ). For ( n, , ( n, p ) to be a symmetric pair, we must have n − p + 1 = kl , and so [ n − p ] = ±
1. 15o show the extension doesn’t split for p >
1, assume there is some minimal p such that the map Ψ p : V n, → P n,p exists. Let { y i } be the link states appearingin the image of Ψ p that have their n th point connected to a cup, and let { b i } bethe corresponding coefficients of the link states. Denote by ↓ y i the link state in V n − ,p − formed from y i by cutting the cup connected to the n th point and thenremoving the n th point. Consider the map θ p − : | ⊗ n − (cid:55)→ ˜ | ⊗ n − ± (cid:80) b i ↓ y i , itfollows from the previous lemma that if Ψ p exists, then θ p − exists, and defines amap θ p − : V n − , → P n − ,p − , but repeating this, we get a map θ : V n − p +1 , → P n − p +1 , , which we have shown doesn’t exist. Hence Ψ p doesn’t exist, and so theextension doesn’t split. As the only non-split extension of symmetric pairs ofstandard representations is the projective indecomposable representation, we find P n,p (cid:39) P n,p .We can use this extension to construct a second extension of representations. Corollary 3.3.
Let ( n, p (cid:48) ) , ( n, p (cid:48)(cid:48) ) and ( n, p ) , ( n, p (cid:48) ) be symmetric pairs, with p (cid:48)(cid:48) >p (cid:48) > p , and φ : V n,p → V n,p (cid:48) . Then there is a non split extension V n,p (cid:48)(cid:48) → Q n,p →V n,p , given by ¯ e i x := ˜ e i φ ( x ) , x ∈ V n,p . It follows immediately from the above construction, that given infinite linkstates w, w (cid:48) , w (cid:48)(cid:48) that differ at only finitely many points, and such that s ( w ) >s ( w (cid:48) ) > s ( w (cid:48)(cid:48) ) are both finite and form two symmetric pairs, there are non-splitextensions of representations X ( w (cid:48) ) → P ( w ) → X ( w ), X ( w (cid:48)(cid:48) ) → Q ( w ) → X ( w ). Proposition 3.7.
Given infinite link states w , z that differ at only finitely manypoints with s ( w ) , s ( z ) finite. Then the above two extensions are the only possibleextensions of X ( w ) , X ( z ) .Proof. Assume otherwise, so that there are link states w, z with a non-split ex-tension X ( w ) → M → X ( z ). As w and z differ at only finitely many points,let xz (cid:48) = w (cid:48) be a relation in the extension, with x ∈ T L ∞ , and w (cid:48) , z (cid:48) viewed aselements of X ( w ), X ( z ) respectively. We can then restrict w (cid:48) , z (cid:48) to finite n-linkstates and x to a T L n element, for some n . As s ( w ) is finite, then either therestriction splits, and so does the extension M , or else the restriction is the ex-tension of standard representations defined above, and so the extension M is thesame construction. Remark 3.2.
We have only considered extensions of link state representationswith finitely many strings. However the case when there are infinitely many stringsis also non-trivial. For example, Let w i be the infinite link state, up to equivalence,with i cups and infinitely many strings. We can construct an extension X ( w ) → M → X ( w ) , by setting e j w := c j , where c j is the infinite link state with a singlecup at the i th position. For T L n , this generally splits, but for T L ∞ , this extensionis non-split for all q , as defining an injective map X ( w ) → M would requirea sum with infinitely many terms. It gets more complicated when we consider X ( w ) → M → X ( w ) . Let c i,j be the diagram given as follows: hen setting e i w := c i,j , with the condition e i e k w = e k e i w for | i − k | > on the coefficients a i,j ∈ C , this defines an extension for each j ≥ . By consid-ering the action of e j +1 e i for different extensions, it suggests that these may notbe isomorphic for different j . Hence in general extensions of link state representa-tions with infinitely many strings appears to be more complicated. We leave a fulldiscussion of their extensions for future research. A standard result due to Jimbo and Martin [8, 13] is that
T L n ( q + q − ) is inSchur-Weyl duality with U q ( sl ) over the spin-chain representation ( C ) ⊗ n . C isviewed as the 2-dimensional U q ( sl ) representation X := { ν − , ν } , with action Kν i = q − i ν i , Eν − = 0, Eν = ν − , F ν − = ν , F ν = 0. The U q ( sl ) actionon ( C ) ⊗ n is then given by use of the coproduct. The Temperley-Lieb actioncommuting with the U q ( sl ) action is given by e ( ν − ⊗ ν − ) = e ( ν ⊗ ν ) = 0, e ( ν − ⊗ ν ) = qν − ⊗ ν − ν ⊗ ν − , e ( ν ⊗ ν − ) = q − ν ⊗ ν − − ν − ⊗ ν . For q generic, the spin chain decomposes as a U q ( sl ) ⊗ T L n ( q + q − ) bimodule as follows:( C ) ⊗ n (cid:39) n +1 (cid:77) i =0 ,i = n +1 mod 2 X i ⊗ V n, n − i +12 where X i is the irreducible ( i + 1)-dimensional U q ( sl ) representation with basis X i := { ν j : − i ≤ j ≤ i, j = i mod 2 } , Kν j = q − j ν j , Eν j = [ i − j +1] ν j − , Eν − i = 0, F ν j = [ i + j +1] ν j +2 , F ν i = 0. There is a U q ( sl ) ⊗ T L n ( q + q − ) invariant orthogonalbilinear form on the spin chain defined as follows: (cid:104) ν i ⊗ ... ⊗ ν i n , ν j ⊗ ... ⊗ ν j n (cid:105) := q ν − ) ν ) δ i ,j ...δ i n ,j n where ν ± ) is the number of ν ± ’s appearing in ν i ⊗ ... ⊗ ν i n , and i k , j k ∈ {± } .We note that the determinant of the Gram matrix of this bilinear form beingnon-zero doesn’t prove semisimplicity of the spin chain representation. This isdue to the failure of lemma 3 . { ν − ⊗ ν − , ν − ⊗ ν , q − ν − ⊗ ν + ν ⊗ ν − , ν ⊗ ν } of ( C ) ⊗ with q = ± i . Then thedeterminant of the Gram matrix with respect to this basis is q . However taking x := ν − ⊗ ν , y := q − ν − ⊗ ν + ν ⊗ ν − , we have (cid:104) x, y (cid:105) x = x , but there is no U q ( sl ) ⊗ T L n ( q + q − ) element that maps y (cid:55)→ x . The relationship between theabove bilinear form and the one on the standard representations is given by thefollowing: Proposition 3.8.
The restriction of the bilinear form on the spin chain to astandard representation induces the bilinear form on the standard representation.Proof.
We assume that q is generic, as the map ( C ) ⊗ n → V n,p is independentof q . Consider the copy of V n,p appearing in the highest weight space of ( C ) ⊗ n ,i.e. ν n − p +1 ⊗ V n,p . A single cup is given by ∪ := q − ν ⊗ ν − − ν − ⊗ ν , and astring by ν . It follows that if the cup is connected to two strings then the bilinearform gives zero. Hence the bilinear form is the same as the one on the standardrepresentation. The case for lower weights follows from the commutativity of theTemperley-Lieb generators with the E action.17imilarly, restricting the bilinear form on the spin chain to one of the U q ( sl )representations in its decomposition induces a bilinear form on X i , given as follows: (cid:104) ν j , ν k (cid:105) := δ j,k [ i + 1]![ i − j + 1]![ i + j + 1]!We want to generalize Schur-Weyl duality to the case of T L ∞ . To do this,we need an appropriate generalization of the spin chain representation. However,simply taking the tensor product of infinitely many copies of X results in issueswith the U q ( sl ) action. Instead we aim to generalize the decomposition of ( C ) ⊗ n .Let w , w be link states with s ( w ) = 0, s ( w ) = 1, and let w (2 i ) k , k = 0 , i ∈ N be a link state up to equivalence that differs from w k at only finitely manypoints and s ( w (2 i ) k ) = k + 2 i . The spin chain ( C ) ⊗ n can be decomposed as givenpreviously, but with the standard representations relabelled in terms of the numberof strings. There is a natural inclusion from n to n + 2 under this labelling, sowe end up with two decompositions in the limit n → ∞ , one odd, one even. Wedefine the spin chain generalization S ( w k ), k = 0 ,
1, by replacing the standardrepresentations labelled by a certain string number in the decomposition with therepresentation X ( w ik ) that has the same string number. For generic q we can givethis generalization explicitly: S ( w k ) := { ν j ⊗ x : x ∈ X ( w (2 i ) k ) , − k − i ≤ j ≤ k + 2 i, j = k mod 2 } t ( ν j ⊗ x ) := ν j ⊗ tx, t ∈ T L ∞ The U q ( sl ) action on this is then given as follows: K ( ν i ⊗ x ) := q − i ν i ⊗ x, E ( ν i ⊗ x ) :=[ s ( x ) − i ν i − ⊗ x, E ( ν − s ( x ) ⊗ x ) :=0 F ( ν i ⊗ x ) :=[ s ( x ) + i ν i +2 ⊗ x, F ( ν s ( x ) ⊗ x ) :=0The case when q is a root of unity is more complicated. In this case, S ( w k ) hasthe same basis, but now both the T L ∞ and U q ( sl ) actions are more difficult togive a general formula for. To describe the actions, we can use the decompositionof the T L n spin chain at roots of unity given in [5], along with the basis of U q ( sl )projective representations given in [3]. As an example, for l = 2, the even spinchain can be decomposed as follows: ... ν − ⊗ V ν − ⊗ V ν ⊗ V ν ⊗ V ν ⊗ V ν − ⊗ V ν ⊗ V ν ⊗ V ν ⊗ V FE F ˜ e i ˜ F ˜ E ˜ FE F ˜ e i ˜ E ˜ E ˜ FE F ˜ e i ˜ E where we have relabelled V n − p := V n,p the standard representations in terms ofthe number of strings. The map ˜ e i denotes the extension defined in Definition 3.3.18he maps ˜ E , ˜ F are defined by ˜ E ( ν i ⊗ x ) := ν i − ⊗ φ ( x ), ˜ F ( ν i ⊗ x ) := ν i +2 ⊗ φ ( x ). Wehave also omitted labelling the action of the divided powers E (2) := E [2]! , F (2) := F [2]! which act as E (2) , F (2) : ν i ⊗ x (cid:55)→ ν i ± ⊗ x for all ν i ⊗ x . Given this decomposition,the spin chain generalization S ( w ) for l = 2 is given as follows:... ν − ⊗ X ( w (4)0 ) ν − ⊗ X ( w (4)0 ) ν ⊗ X ( w (4)0 ) ν ⊗ X ( w (4)0 ) ν ⊗ X ( w (4)0 ) ν − ⊗ X ( w (2)0 ) ν ⊗ X ( w (2)0 ) ν ⊗ X ( w (2)0 ) ν ⊗ X ( w ) ˜ FE F ˜ e i ˜ F ˜ E ˜ F E F ˜ e i ˜ E ˜ E ˜ FE F ˜ e i ˜ E By its construction, the action of
T L ∞ and U q ( sl ) commute on this representation.Given this spin chain generalization, and a choice of link states w , w , we candefine the functor F w ,w : RepU q ( sl ) → RepT L ∞ by F ( − ) := Hom ( − , S ( w ) ⊕S ( w )). Let C ( w , w ) be the Serre subcategory generated by finitely generated T L ∞ ( q + q − ) representations generated by link states that differ from w or w at only finitely many points. We make the following conjecture: Conjecture 3.1.
The functor F w ,w defines an equivalence of abelian categoriesbetween the category of finite dimensional U q ( sl ) representations and C ( w , w ) . Acknowledgements
This work was supported by ISF grants 2095/15 and 711/18 and the Center forAdvanced Studies in Mathematics in Ben Gurion University. The author thanksInna Entova for her comments and for suggesting the project.
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