Fock space representation of the circle quantum group
aa r X i v : . [ m a t h . QA ] A p r IPMU–19–0027
FOCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP
FRANCESCO SALA AND OLIVIER SCHIFFMANNA
BSTRACT . In [SS17] we have defined quantum groups U υ ( sl ( R )) and U υ ( sl ( S )) , which can beinterpreted as continuum generalizations of the quantum groups of the Kac-Moody Lie algebrasof finite, respectively affine type A . In the present paper, we define the Fock space representation F R of the quantum group U υ ( sl ( R )) as the vector space generated by real pyramids (a continuumgeneralization of the notion of partition). In addition, by using a variant version of the “foldingprocedure” of Hayashi-Misra-Miwa, we define an action of U υ ( sl ( S )) on F R . C ONTENTS
1. Introduction 12. Partitions and pyramids 43. Recollections on the standard Fock space representation 84. Fock space representation of the quantum group of the line 125. The circle quantum group and the folding procedure 176. Fock space representation of the circle quantum group 25References 281. I
NTRODUCTION
The present article continues the study of the circle quantum group U υ ( sl ( S )) introduced in[SS17], where S : = R / Z . In that paper we constructed geometrically a family of representations V g , r indexed by a pair of positive integers ( g , r ) , with V being the natural “vector” repre-sentation and V r an analog of the r -fold symmetric power of V
0, 1 . Our goal here is to definecombinatorially a Fock space representation F R of U υ ( sl ( S )) , which may be thought of as alimit of V r as r tends to infinity.Before stating the main results of the paper, let us briefly recall the classical construction ofthe Fock space representation F of U υ ( sl ( ∞ )) and U υ ( b sl ( n )) . Set e Q : = Q [ υ , υ − ] . Then, F is the e Q -vector space with a basis formed by vectors | λ i labelled by partitions λ , i.e., F : = M λ e Q | λ i .Roughly speaking, the action of the positive (resp. negative) part of U υ ( sl ( ∞ )) is given in terms ofthe combinatorial procedure of removing (resp. adding) a box to the Young diagram Y λ associated Mathematics Subject Classification.
Primary: 17B37; Secondary: 22E65.
Key words and phrases.
Quantum groups, continuum quantum groups, circle quantum group, Fock spaces, pyramids.The work of the first-named author is partially supported by World Premier International Research Center Initiative(WPI), MEXT, Japan, by JSPS KAKENHI Grant number JP17H06598 and by JSPS KAKENHI Grant number JP18K13402. In the setting of [SS17], U υ ( sl ( S )) is realized as (reduced) Drinfeld double of the spherical Hall algebra of parabolictorsion sheaves on a genus g curve. The representation V g , r arises from the Hall algebra of rank r parabolic vector bundleson the curve. F. SALA AND O. SCHIFFMANN to the partition λ , while the Cartan subalgebra acts diagonally with a υ -factor depending on thenumber of addable and removable boxes of a Young diagram.The action of U υ ( b sl ( n )) on F was originally defined by Hayashi [Hay90] by using υ -analogsof Clifford and Weyl algebras and their actions on the exterior and polynomial algebras, respec-tively. Miwa-Misra [MM90] gave another interpretation of the action in terms of operations onYoung diagrams. Finally, Varagnolo-Vasserot [VV99] reinterpreted the “folding procedure” interms of Hall algebras associated to nilpotent representations of the infinite quiver A ∞ and theaffine quiver A ( ) n − respectively. Note that F is the level one Fock space with fundamental weight Λ of U υ ( b sl ( n )) .1.1. Main results.
Just as the quantum group U υ ( sl ( S )) is an uncountable colimit of U υ ( b sl ( n )) as n tends to infinity, the Fock space F R is an uncountable colimit of the standard Fock spacerepresentation of U υ ( b sl ( n )) . One main novelty of the limit which we consider is that instead ofpartitions, the Fock space F R has a basis indexed by what we call (real) pyramids (see Section 2).Integral pyramids are close cousins of Maya diagrams and are in bijection with partitions. Forinstance, on the left-hand-side there is the Young diagram of the partition (
5, 4, 4, 3, 1, 1 ) with itsstandard contents written inside the corresponding boxes, and on the right-hand-side there is thecorresponding integral pyramid: − − − − − ←→ − − − − − − − − Pictorially, we pass from Young diagrams to Z -pyramids by tilting 45 degrees to the left (orwriting partitions the Russian way) and letting gravity act. Unlike partitions, pyramids admit anatural extensions to R . For example, the following is a real pyramid: − − −
710 25 e Let
Pyr ( R ) be the set of all R -pyramids. As a vector space, our Fock space is defined as F R : = M p ∈ Pyr ( R ) e Q | p i .In [SS17] we also defined a Lie algebra sl ( R ) — now as a uncountable colimit of the Lie algebras sl ( n ) as n tends to ∞ — and a corresponding quantum group U υ ( sl ( R )) , the line quantum group .It is generated by elements E J , F J , K J for J an interval (cf. Definition 4.2). Let 0 stand for the emptypyramid. Theorem (cf. Theorem 4.6) . The following formulas define an action of the quantum group U υ ( sl ( R )) on F R : for any J = [ a , b [ and p ∈ Pyr ( R ) E J | p i : = ( − υ ( − υ ) p ( b ) − p ( a ) | p − J i if J is a removable interval of p ,0 otherwise , Although we are confident that we have chosen the right terminology, one of the anonymous referees pointed outthat ziggurat could have been a good choice as well.
OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 3 F J | p i : = ( υ ( − υ ) p ( a ) − p ( b ) | p + J i if J is an addable interval of p ,0 otherwise , K J | p i : = υ n J ( p ) | p i , where n J ( p ) = if J is neither addable or removable to p ,1 if J is addable to p , − if J is removable to p . The representation F R is an irreducible highest weight representation with highest weight vector | i . Here, J is the characteristic function of the interval J. As one may see above, the continuum analog of removing or adding a box in a Young diagramis the operation of removing or adding intervals in a pyramid.The circle quantum group U υ ( sl ( S )) was defined in [SS17]. Due to the absence of simple rootsfor the circle Lie algebra sl ( S ) , if we simply substitute the generators of U υ ( b sl ( n )) with those of U υ ( sl ( S )) in Hayashi’s formulas for the action of U υ ( b sl ( n )) on the Fock space, we do not obtainan action of U υ ( sl ( S )) on F R . The way we construct the action of U υ ( sl ( S )) here consists ofgeneralizing the geometric folding procedure due to Varagnolo and Vasserot which is based onthe theory of Hall algebras. Precisely, the folding procedure is induced by an homomorphismbetween the Hall algebra of the continuum finite type A quiver R and the Hall algebra of thecontinuum affine type A quiver S , which realize a positive part of U υ ( sl ( R )) and U υ ( sl ( S )) , re-spectively. The explicit formulas of our folding procedure are quite involved (see Theorem 6.1):in particular, since U υ ( sl ( S )) is generated by elements E J , F J , K J for J an interval in S , the ab-sence of simple roots means that the folding procedure involves breaking up J into finitely manysmaller intervals in all possible ways. We obtain: Theorem (cf. Theorem 6.1) . There exists a natural action of U υ ( sl ( S )) on F R , which strictly containsthe irreducible highest weight representation generated by the vacuum vector | i . As opposed to the case of the Fock space of U υ ( b sl ( n )) (see e.g. [KMS95]), F R contains nohighest weight vector other than the vacuum vector | i . On the other hand, it is not a cyclicrepresentation. It is, however, a cyclic representation of the Hall algebra of S R (which strictlycontains U − υ ( sl ( S )) ). Its precise structure will be studied in a sequel to this paper.1.2. Higher level Fock spaces.
Above, we provide a continuum analog of the level one Fockspace representation with fundamental weight Λ . One can take the ℓ tensor product F R [ ℓ ] of F R . It can be endowed with a well-defined action of U υ ( sl ( R )) . This action is defined via thecoproduct: even though the coproduct is only topological and takes value in a suitable comple-tion of tensor products of U υ ( sl ( R )) , we do not need to complete F R [ ℓ ] as well to get an action.In another direction, one can translate the pyramids along the R -axis: this will give rise to (level ℓ ≥
1) Fock space representations associated to other fundamental weights . Tensor products ofFock spaces with different highest weights are more delicate to handle. We will address the studyof these variations of the construction of the Fock space representation for the line and circlequantum groups in [SS19].1.3. K -Variations. Originally, it is a version of the quantum group associated to the rational circle S Q = Q / Z or of the rationals Q which appears in [SS17]. One can define such quantum groups U υ ( sl ( K )) and U υ ( sl ( S K )) , for any subset of K ⊂ R which is 1-periodic, where S K : = K / Z for Z ( K . When K = Z , our quantum group U υ ( sl ( Z )) coincides with U υ ( sl ( ∞ )) , but it ismuch bigger when K = Q or K = R . Likewise, for K = n Z we have U υ ( sl ( S K )) ≃ U υ ( b sl ( n ))) We believe that, in our theory, weights should correspond to distribution in F ( R ) . F. SALA AND O. SCHIFFMANN but is is much larger in the other two cases. The Fock space construction can be carried out forany such K , and we do it in this generality. For instance, the Fock space representation F Q of U υ ( sl ( S Q )) has a basis indexed by Q -pyramids, which are pyramids whose jumps occur only atrational numbers. When K = Z we recover the usual Fock space action of U υ ( sl ( ∞ )) , up to somescaling factors.1.4. Line and circle quantum groups vs. continuum quantum groups.
In [ASS18], the authors,together with Andrea Appel, define a continuum generalization of the Kac–Moody Lie algebras,associated with a continuum generalization of the notion of a quiver. In particular, the vertexset of a quiver is replaced by an Hausdorff topological space X , and the vertices of the quiverare replaced by connected intervals in X (we refer to loc. cit. for the relevant definitions). Wedenote by g X the corresponding continuum Kac–Moody Lie algebra. It is proved in loc. cit. that g K coincides with sl ( K ) , while the Lie algebra g S K differs from sl ( S K ) by an Heisenberg Liealgebra of order one . In [AS19], the first-named author, together with Andrea Appel, constructed atopological Lie bialgebra structure on the continuum Kac–Moody Lie algebra g X of any space X and defined algebraically the quantization U υ ( g X ) of g X , which is called the continuum quantumgroup of X . It is proved in loc. cit. that U υ ( g K ) coincides with U υ ( sl ( K )) , while the quantumgroup U υ ( g S K ) differs from U υ ( sl ( S K )) by a quantum Heisenberg algebra of order one . Finally, in[AKSS19], we are constructing the quantum group U υ ( g X ) via the theory of Hall algebras. Weexpect to extend some of the techniques of this paper to such a more general setting, in particularthe construction of a Fock space representation for any U υ ( g X ) will be the subject of a sequel tothis paper.1.5. Outline.
The paper is organized as follows. In Section 2, we recall the notions of partitionsand Young diagrams, and describe their topological refinement as rational and real pyramids.Section 3 serves as a reminder about Hayashi’s Fock space and Varagnolo-Vasserot’s geometricinterpretation of the folding procedure. In Section 4, we introduce our Fock space F K and theaction of the quantum group U υ ( sl ( K )) , for K ∈ { Z , Q , R } , on it, while in Section 6, we definethe action of U υ ( sl ( S K )) , for S K : = K / Z and K ∈ { Q , R } , on F K . The proof that such an actionis well-defined is given in Section 6 and it is based on a variant of the “folding procedure” de-pending on the theory of Hall algebras associated with certain topological quivers, as introducedin Section 5. Acknowledgements.
We thank Andrea Appel and Tatsuki Kuwagaki for helpful discussionsand comments. We also thank the anonymous referees for useful suggestions and comments.
Notation and convention.
For any integer n , set [ n ] : = υ n − υ − n υ − υ − and [ n ] ! : = [ n ] [ n − ] · · · [ ] .Set e Q : = Q [ υ , υ − ] and put q = υ .2. P ARTITIONS AND PYRAMIDS
In this section, we introduce integral, rational, and real pyramids and establish their basic prop-erties. Rational and real pyramids are some “continuous” generalization of the notion of parti-tion, while we shall show that integral pyramids coincide with partitions.
OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 5
Recollection on partitions. A partition of a positive integer n is a nonincreasing sequenceof positive numbers λ = ( λ , λ , · · · , λ ℓ ) such that λ i ≥ λ i + for i =
1, . . . , ℓ − | λ | : = ∑ ℓ a = λ a = n . We call ℓ = ℓ ( λ ) the length of the partition λ . Another description of a partition λ of n uses the notation λ = ( m m · · · ) , where m i = { a ∈ N | λ a = i } with ∑ i i m i = n and ∑ i m i = ℓ ( λ ) . We denote by Π ( n ) the set of all partitions of n , and Π : = S n Π ( n ) . On the set Π of all partitions there is a natural partial ordering called dominance ordering : for two partitions µ and λ , we write µ ≤ λ if and only if | µ | = | λ | and µ + · · · + µ a ≤ λ + · · · + λ a for all a ≥
1. Wewrite µ < λ if and only if µ ≤ λ and µ = λ .One can associate with a partition λ its Young diagram , which is the set Y λ = { ( x , y ) ∈ Z > | ≤ y ≤ ℓ ( λ ) , 1 ≤ x ≤ λ y } . Then λ y is the length of the y -th row of Y λ ; we write | Y λ | = | λ | for the weight of the Young diagram Y λ . We shall identify a partition λ with its Young diagram Y λ . Forexample, with the partition λ = (
5, 4, 4, 3, 1, 1 ) we associate the Young diagram Y λ :For a partition λ , the transpose partition λ ′ is the partition whose Young tableau is Y λ ′ : = { ( b , a ) ∈ Z > | ( a , b ) ∈ Y λ } .Finally, we call standard content of s = ( x , y ) ∈ Y λ the quantity c ( s ) : = x − y . We say that box s is of color i if c ( s ) = i . An addable i-box is a box of color i which can be added to Y λ in such away that the new diagram still comes from a partition, similarly a removable i-box is a box of color i which can be removed from Y λ . For i ∈ Z > , define n i ( λ ) : = { addable i -boxes of Y λ } − { removable i -boxes of Y λ } . Remark . Note that n i ( λ ) = i , − i ,0 otherwise . △ Integral pyramids.
We now provide another combinatorial realization of partitions verysimilar to that of Maya diagrams , which we call “integral pyramids”. Below, ’increasing’ and’decreasing’ are meant in the broad sense. Definition 2.2. An integral pyramid is a function p : Z → N satisfying the following properties:i) p ( n ) = | n | ≫ p ( ) = max { p ( n ) | n ∈ Z } ;iii) p is increasing on Z − and decreasing on Z + ;iv) | p ( n ) − p ( n + ) | ≤ n ∈ Z . ⊘ We may represent a pyramid as a box diagram in which we draw p ( n ) boxes over the integer n : for instance, we represent the pyramid such that p ( − ) = p ( − ) = p ( − ) = p ( − ) = A Maya diagram is a sequence { m ( n ) } n ∈ Z which consists of 0 or 1 and satisfies the following property: there exist N , M ∈ Z such that for all n > N (resp. n < M ), m ( n ) = m ( n ) = F. SALA AND O. SCHIFFMANN p ( − ) = p ( ) = p ( ) = p ( ) = p ( ) = p ( ) = p ( ) = p ( ) = p ( n ) = n < − n > − − · · ·· · · The set of pyramids with n boxes will be denoted Pyr ( n ) . Lemma 2.3.
There is a canonical bijection between
Pyr ( n ) and the set Π ( n ) of partitions of n.Proof. Let us define a map f : Π ( n ) → Pyr ( n ) as follows. Let λ be a partition and let Y λ be itsYoung diagram. We fill the boxes of Y λ with the standard content c ( s ) = y − x if s = ( x , y ) . Then f ( λ ) is the pyramid p λ defined by p λ ( n ) : = { s ∈ Y λ | c ( s ) = n } for any integer n . Viceversa, wecan define a map g : Pyr ( n ) → Π ( n ) by assigning to a pyramid p the unique Young diagram forwhich the number of boxes s for which c ( s ) = n is exactly p ( n ) for any n ∈ Z . It is straighforwardto see that g is the inverse of f . (cid:3) Example . For instance, to the partition λ = (
5, 4, 4, 3, 1, 1 ) we associate the pyramid p λ ( − ) = p λ ( − ) = p λ ( − ) = p λ ( − ) = p λ ( − ) = p λ ( ) = p λ ( ) = p λ ( ) = p λ ( ) = p λ ( ) = − − − − − ←→ − − − − − − − − Pictorially, the bijection amounts to folding clockwise by 90 degrees the part of the pyramidwhich lies over − N (and shifting accordingly each layer of the pyramid to the right by a numberof boxes equal to its height). Conversely, starting from a partition, one may tilt its Young diagram45 degrees to the left (so that it stands on its corner) and let gravity act to obtain the associatedpyramid. △ It is easy to translate several classical notions from partitions to pyramids. For instance, ℓ ( λ ) = sup { n | p λ ( − n ) = } , (2.1) p λ ′ = p λ ◦ ι , (2.2)where λ ′ is the transpose partition of λ and ι : Z → Z sends n to − n , and | λ | = Z p λ : = ∑ n p ( n ) . (2.3)The dominance ordering is less pleasant. It translates into the following set of inequalities: p ≥ q ⇔ Z inf ( p , κ n ) ≥ Z inf ( q , κ n ) for all n , (2.4)where κ n ( i ) : = n if i ≥ n + i if − n < i < i ≤ − n . OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 7
Rational and real pyramids.
We now generalize the concept of pyramids to allow for “jumps”of the function p located at non-integral points. More precisely, let us introduce the following def-inition. Definition 2.5. A rational (resp. real ) pyramid is defined to be a function p : Q → N (resp. afunction p : R → N ) satisfying the following properties (we set K = Q , R accordingly):i) p ( x ) = | x | ≫ p ( ) = max { p ( x ) | x ∈ K } ;iii) p is increasing on K − and decreasing on K + ;iv) p is right–continuous and piecewise constant, with finitely many points of discontinuity;v) | p + ( x ) − p − ( x ) | ≤ x ∈ K . ⊘ We extend the notions of length, transpose and size to rational or real pyramids using (2.1),(2.2), (2.3) respectively. The notion of dominance is also extended accordingly, where we nowrequire the inequality (2.4) for any n ∈ K .We may still represent rational or real pyramids as diagrams (now simply the graph of thefunction p ). On the other hand, it is not possible to represent rational or real pyramids as Youngdiagrams anymore, as the operation of “folding clockwise by 90 degrees” does not make sense.We denote by Pyr ( K ) the set of all K -pyramids. We will sometimes also denote by Pyr ( K )( u ) theset of K -pyramids with size u .2.4. Addable and removable intervals.
Fix K ∈ { Z , Q , R } . Definition 2.6.
We call a K -interval a closed-open interval of the form J = [ a , b [ : = { x ∈ R | a ≤ x < b } with a , b ∈ K . We denote by J the characteristic function of J . ⊘ Definition 2.7.
Let p be a K -pyramid. A K -interval J is called addable (resp. removable ) if p + J (resp. p − J ) is still a K -pyramid. ⊘ Remark . When K = Z , as proved in Lemma 2.3 a pyramid p corresponds to a partition λ p . Alength one Z -interval J gives rise to a box s : = ( + p ( a ) , 1 + p ( a ) − a ) . Hence J is addable (resp.removable) if s is addable (resp. removable) in the sense defined in Section 2.1. More generally,when J is of arbitrary length, it corresponds to a connected strip. △ Example . Here is an example of an addable Z -interval and the corresponding strip: − − − − − − − ←→ − − − − − − − − and here is an example of an addable Q (or R )-interval (in this case, J = [ − , [ ): − − −
710 25 Note that we could not have added [ − , [ since there would be a jump of 2 over , nor couldwe have added the interval [ − , 0 [ since 0 would not be the maximum anymore. △ F. SALA AND O. SCHIFFMANN
Definition 2.10.
Let p be a K -pyramid. We put D K ( p ) : = { y ∈ K | y is a point of discontinuity of p } .and call this the set of discontinuities of p . ⊘ Remark . Let J = [ a , b [ be a K -interval and let p be a K -pyramid. Then p + J is a pyramid ifand only if one of the following mutually exclusive cases occurs • a ∈ D K ( p ) if 0 < a ; • a , b / ∈ D K ( p ) if a ≤ < b ; • b ∈ D K ( p ) if b < p − J is a pyramid if and only if one the following mutually exclusive cases occurs • b ∈ D K ( p ) if 0 < a ; • a , b ∈ D K ( p ) if a ≤ < b ; • a ∈ D K ( p ) if b < △ Let p be a K -pyramid and let J be an addable interval. We define the p-height of J as ht p ( J ) : = sup {| p ( x ) − p ( y ) | | x , y ∈ J } .We will also consider the variants ht ± p ( J ) : = sup {| p ( x ) − p ( y ) | | x , y ∈ J ∩ K ± } ,so that ht p ( J ) = max { ht + p ( J ) , ht − p ( J ) } . Example . In the last example above, we have, for J = [ − , [ , ht p ( J ) = ht − p ( J ) = ht + p ( J ) = △
3. R
ECOLLECTIONS ON THE STANDARD F OCK SPACE REPRESENTATION
Type A quantum groups.
Let U υ ( sl ( ∞ )) (resp. U υ ( b sl ( n )) ) be the quantized enveloping algebraof sl ( ∞ ) (resp. of b sl ( n ) ), i.e., the unital associative e Q -Hopf algebra generated by E i , F i , K ± i , for i ∈ Z (resp. i ∈ Z / n Z ), subject to the Drinfeld-Jimbo type relations K i K − i = = K − i K i , K i K j = K j K i , K i E j K − i = υ a i , j E j , K i F j K − i = υ − a i , j F j , [ E i , F j ] = δ i , j K i − K − i υ − υ − , − a i , j ∑ k = ( − ) k E ( k ) i E j E ( − a i , j − k ) i = = − a i , j ∑ k = ( − ) k F ( k ) i F j F ( − a i , j − k ) i if i = j ,where a i , j : = i = j , − | i − j | = For i , j ∈ Z / n Z , we interpret | i − j | modulo n . OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 9 and E ( k ) i : = E ki [ k ] ! and F ( k ) i : = F ki [ k ] ! .The coproduct is ∆ ( K i ) = K i ⊗ K i , ∆ ( E i ) = E i ⊗ + K i ⊗ E i , ∆ ( F i ) = F i ⊗ K − i + ⊗ F i .3.2. Fock spaces for type A quantum groups.
Set F Z : = M λ ∈ Pyr ( Z ) e Q | λ i .The action of U υ ( sl ( ∞ )) on F Z given by E i | λ i : = ( | ν i if Y λ \ Y ν is a box with color i ,0 otherwise , F i | λ i : = ( | µ i if Y µ \ Y λ is a box with color i ,0 otherwise , K i | λ i : = υ n i ( λ ) | λ i for i ∈ Z .Following [Hay90, MM90], the action of U υ ( b sl ( n )) on F Z is defined via the “folding proce-dure” as E ¯ ı | λ i : = ∑ j ∈ Z : j = ¯ ı mod n υ − n − j ( λ ) E j | λ i , F ¯ ı | λ i : = ∑ j ∈ Z : j = ¯ ı mod n υ n + j ( λ ) F j | λ i , K ¯ ı | λ i : = υ n ¯ ı ( λ ) | λ i for ¯ ı ∈ Z / n Z . Here n ¯ ı ( λ ) : = ∑ j = ¯ ı mod n n j ( λ ) , n − j ( λ ) : = ∑ k < j : k = j mod n n k ( λ ) , n + j ( λ ) : = ∑ k > j : k = j mod n n k ( λ ) .3.3. Varagnolo-Vasserot construction.
For future use, we now give, following Varagnolo andVasserot [VV99], a geometric interpretation of the above formulas. This uses the notions of Hallalgebras and quiver representations which we first briefly recall (see [S12] for details).A quiver is an oriented graph Q = ( I , H ) where I stands for the set of vertices and H forthe set of oriented edges; for h ∈ H , we denote by h ′ , resp. h ′′ its source and target vertices.A representation of Q over a field k is by definition the data of an I -graded finite-dimensional k -vector space V = L i V i together with a collection of linear maps x h ∈ Hom ( V h ′ , V h ′′ ) for h ∈ H .The dimension of a representation M = ( V , ( x h ) h ) is defined as dim ( M ) = ( dim ( V i )) i ∈ N I .Representations of a quiver Q over a field k form an abelian category Rep k Q , which is known tobe hereditary , which means that Ext i ( M , N ) = { } for any pair of representations ( M , N ) and any i >
1. The Euler form h· , ·i : K ( Rep k Q ) ⊗ K ( Rep k Q ) → Z , M ⊗ N dim ( Hom ( M , N )) − dim ( Ext ( M , N )) factors through the dimension map and is explicitly given by h· , ·i : Z I ⊗ Z I → Z , h d , d ′ i = ∑ i d i d ′ i − ∑ h d h ′ d ′ h ′′ . To every vertex i there corresponds a simple one-dimensional representation S i , supported at thevertex i and with all linear zero maps x h equal to zero.Another point of view, better suited for our purposes, involves the path algebra k Q : this isthe algebra with basis given by the set of oriented paths and multiplication given by concate-nation of paths (whenever possible, zero otherwise). The algebra k Q has idempotents e i , for i ∈ I , corresponding to length zero paths starting and ending at a vertex i . One can easilycheck that k Q is generated by the collection of idempotents e i , i ∈ I and the length one paths h ∈ H , and that the assignment M ( L i e i M , ρ M ( h ) h ) defines an equivalence between the cat-egory of finite-dimensional k Q -modules and the category of representations of Q over k . Here ρ M : k Q →
End ( M ) is the map defining the k Q -module structure of M .From now on, we fix k to be a finite field F q and set υ : = q . Let M Q denote the set of allisomorphism classes of objects of Rep k Q . As vector space, the Hall algebra H Q is H Q : = { h : M Q → C | | supp ( h ) | < ∞ } .We will denote by [ V ] the characteristic function of an object V . The function dim induces anatural gradation H Q = M d ∈ N I H Q [ d ] (3.1)with finite dimensional graded pieces. Note that when I is infinite, H Q [ d ] = d hasinfinitely many nonzero components. We denote by Z I ⊂ Z I the subset of vectors having finitelymany nonzero components.The multiplication in H Q is defined as follows. For M ∈ H Q [ d ] and M ′ ∈ H Q [ d ′ ] we set [ M ] ⋆ [ M ′ ] : = υ h d , d ′ i ∑ N P NM , M ′ [ N ] where N ranges over the (finite) set of extensions of M ′ by M and where P NM , M ′ : = |{ L ⊂ N | L ≃ M ′ , N / L ≃ M }| .Note that the multiplication is graded with respect to the decomposition (3.1). We let H sph Q be thesubalgebra of H Q generated by the elements [ S i ] for i ∈ I .We will only be interested here in the following three types of quivers : the equioriented finitetype A quivers A n , which has as vertex set 1, . . . , n − i i + i =
1, . . . , n −
2; the infinite quiver A ∞ with vertex set Z and edge set i i +
1; and the cyclically orientedaffine type A quiver A ( ) n which has as vertex set Z / n Z and edge set i i + i ∈ Z / n Z .The following result is well-known and due to Ringel. Theorem 3.1. (i) Assume that Q = A ⋆ with ⋆ ∈ N ∪ { ∞ } . The assignment [ S i ] υ − E i for all vertices idefines an isomorphism of graded algebras H Q ≃ U + υ ( sl ( ⋆ )) .(ii) Assume that Q = A ( ) n . The assignment [ S i ] υ − E i for all vertices i defines an isomorphismof graded algebras H sph Q ≃ U + υ ( b sl ( n )) . We will need a slight variant of the above result, involving U − υ instead of U + υ . Corollary 3.2. (i) Assume that Q = A ⋆ with ⋆ ∈ N ∪ { ∞ } . The assignment [ S i ]
7→ − υ − F i for all vertices idefines an isomorphism of graded algebras H Q ≃ U − υ ( sl ( ⋆ )) .(ii) Assume that Q = A ( ) n . The assignment [ S i ]
7→ − υ − F i for all vertices i defines an isomor-phism of graded algebras H sph Q ≃ U − υ ( b sl ( n )) . OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 11
Proof.
This comes from the algebra isomorphism U + υ ≃ U − υ given by E J
7→ − F J . (cid:3) Because of the relation with quantum groups, it is customary to consider the twisted
Hall alge-bra, which is defined as a semi-direct product H tw Q : = H Q ⋉ H Q ,where H Q : = C [ K ( Rep k Q )] is the group algebra of K ( Rep k Q ) . Thus H Q has a basis { K d } where d runs among the sets Z n , Z Z or Z Z / n Z for Q = A n , Q = A ∞ or Q = A ( ) n respectively. The action of H Q on H Q is givenby K d x K − d = υ ( d , l ) x for all x ∈ H Q [ l ] .We are now ready to recall Varagnolo and Vasserot’s geometric interpretation of Hayashi’sfolding procedure setting of [VV99, Proposition 6.1]. Fix some n >
1. Let us denote by i i ,resp. d d the projections Z → Z / n Z , resp. N Z → N Z / n Z . Following [VV99, Section 6] wewill now construct a family of maps γ d : H A ( ) n [ d ] → H A ∞ [ d ] for every d ∈ N Z . Fix some d ∈ N Z and a Z -graded vector space V = L ı V ı of dimension d .Put: V = M a ∈ Z / n Z V a , V a = M = a V .Thus, even if they may be canonically identified, V is a Z -graded vector space while V is Z / n Z -graded. For any ı ∈ Z set V ≥ ı : = M ≥ ı V ,and denote by V •≥ the associated filtration of V . There is an induced filtration V •≥ of V where V ≥ ı = M a V ≥ ı , a and V ≥ ı , a = M ≥ ı , = a V .The filtrations V • > , V • > are defined in the same way, replacing ≥ by > . The associated gradedspace gr ( V ) : = M ı ( V ≥ ı / V > ı ) is canonically isomorphic to V as a Z -graded vector space.Let E V , E V denote the vector spaces of representations of A ∞ and A ( ) n in V , resp. V . Let also E V , V ⊂ E V be the subset of representations preserving the filtration V •≥ . Let us write j : E V , V → E V for the embedding. Taking the associated graded yields a map p : E V , V → E V .We set γ d : = υ − h ( d ) p ! ◦ j ∗ : H A ( ) n [ d ] → H A ∞ [ d ] ,where h ( d ) : = − ∑ ℓ< h d , τ ℓ ( d ) i = ∑ ı < ı = d ı ( d + − d ) , Note, however, that [VV99] use the opposite multiplication for Hall algebras and work over a field F q . with τ : ( d i ) i ( d i − ) i being the translation operator. Finally, put k ( h , h ′ ) : = ∑ ℓ> ( h , τ ℓ ( h ′ )) = ∑ ı > ı = h ı ( h − h − − h + ) . Proposition 3.3 ([VV99, Proposition 6.1]) . For any pair l , l ′ ∈ N Z / n Z and any d ∈ N Z such thatd = l + l ′ we have ∑ h + h ′ = dh = l , h ′ = l ′ υ k ( h , h ′ ) γ h ( u ) ⋆ γ h ′ ( u ′ ) = γ d ( u ⋆ u ′ ) for any u ∈ H A ( ) n [ l ] , u ′ ∈ H A ( ) n [ l ′ ] . This proposition will be used in the following way. For l ∈ N Z / n Z and x ∈ H A ( ) n [ l ] we set r ( x ) : = ∑ d : d = l K o ( d ) γ d ( u ) where o ( d ) : = ∑ ℓ< τ ℓ ( d ) . Note r ( x ) takes values in a completion H tw , c A ∞ of H tw A ∞ , which we nowdefine. Let H c A ∞ be the group algebra of Z Z . In other words H c A ∞ is generated by elements K d for d ∈ Z Z (possibly having infinitely many nonzero components) subject to the relations K d K l = K d + l , for d , l ∈ Z Z . We set H tw , c A ∞ : = M l ∈ N Z / n Z H tw , c A ∞ [ l ] and H tw , c A ∞ [ l ] : = ∏ d ∈ N Z d = l n H c A ∞ ⋉ H A ∞ [ d ] o .Observe that for any m ∈ Z Z and any d ∈ N Z the Euler form ( m , d ) is well-defined, so that thesemi-direct product above makes sense. Moreover, H tw , c A ∞ is still an algebra since there are onlyfinitely many ways to break up a dimension vector d as a sum d ′ + d ′′ . Corollary 3.4.
The map r : H A ( ) n → H tw , c A ∞ is an algebra homomorphism. Hayashi’s action of U − υ ( b sl ( n )) on the Fock space F Z can now be recovered as follows. ByCorollary 3.2 we embed U − υ ( b sl ( n )) into H A ( ) n and use the homomorphism r to pullback therepresentation of U − υ ( sl ( ∞ )) ≃ H A ∞ on F Z . Note that since the Fock space representation of U − υ ( b sl ( n )) is faithful, the map r is injective; this can also be seen directly.4. F OCK SPACE REPRESENTATION OF THE QUANTUM GROUP OF THE LINE
Fix K ∈ { Z , Q , R } . We denote by F ( K ) the algebra of piecewise constant, right–continuousfunctions f : R → Z , with finitely many points of discontinuity, bounded support and whosepoints of discontinuity belong to K . This means that f ∈ F ( K ) if and only if f = ∑ J c J J ,where the sum runs over all intervals of K and c J ∈ Z is zero for all but finitely many J . Given f , g ∈ F ( K ) , we define h f , g i : = ∑ x f − ( x )( g − ( x ) − g + ( x )) and ( f , g ) : = h f , g i + h g , f i , (4.1)where we have set h ± ( x ) : = lim t → t > h ( x ± t ) . Let F ( K ) + (resp. F ( K ) − ) be the set of functions f ∈ F ( K ) such that f ( x ) ≥ f ( x ) ≤
0) for any x ∈ R . Remark . Let J i = [ a i , b i [ be a K -interval for i =
1, 2. Then • h J , J i = J = J , a = a and b < b , a < a and b = b , a < a < b < b ; • h J , J i = J ∩ J = ∅ , b = a , a = a and b < b , a < a and b = b , a < a < b < b , a < a < b < b ; OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 13 • h J , J i = − b = a or a < a < b < b .Thus, the following holds: • for any K -interval J , we have ( J , J ) = • if J , J are K -intervals of the form J = [ a , b [ and J = [ b , c [ , we get ( J , J ) = ( J , J ) = − • in any other case, (i.e. if I = J are K -intervals such that J + I is not the characteristicfunction of any interval) then ( I , J ) = △ The bialgebra U υ ( sl ( K )) .Definition 4.2 (cf. [SS17, Section 1.1]) . Let U υ ( sl ( K )) be the topological e Q -bialgebra generatedby elements E J , F J , K ± J , where J runs over all K -intervals, modulo the following set of relations: • Drinfeld-Jimbo relations : – for any intervals J , I , I , I , [ K I , K I ] = K I E J K − I = υ ( I , J ) E J , K I F J K − I = υ − ( I , J ) F J ; – if J , J are intervals such that J ∩ J = ∅ , [ F J , E J ] = – for any interval J , [ E J , F J ] = K J − K − J υ − υ − ; • join relations : if J , J are intervals of the form J = [ a , b [ and J = [ b , c [ then K J K J = K J ∪ J ;and E J ∪ J = υ E J E J − υ − E J E J , F J ∪ J = υ − F J F J − υ F J F J ; • nest relations : – if J , J are intervals such that J ∩ J = ∅ , [ E J , E J ] = [ F J , F J ] = – if J , J are intervals such that J ⊆ J , υ h J , J i E J E J = υ h J , J i E J E J , υ h J , J i F J F J = υ h J , J i F J F J .The coproduct is given by: ∆ ( K J ) = K J ⊗ K J , ∆ ( E [ a , b [ ) = E [ a , b [ ⊗ + ∑ a < c < b υ − ( υ − υ − ) E [ a , c [ K [ c , b [ ⊗ E [ c , b [ + K [ a , b [ ⊗ E [ a , b [ , ∆ ( F [ a , b [ ) = ⊗ F [ a , b [ − ∑ a < c < b υ − ( υ − υ − ) F [ c , b [ ⊗ F [ a , c [ K − [ c , b [ + F [ a , b [ ⊗ K − [ a , b [ . Here the sums on the right-hand-side run over all possible K -values c ∈ [ a , b [ . ⊘ Definition 4.3.
Let f = ∑ J c J J ∈ F ( K ) . Then we set K f : = ∏ J K c J J . ⊘ Remark . Let J , J be K -intervals of the form J = [ a , b [ and J = [ b , c [ (so that J ∪ J is again ainterval). Then, by the join and the nest relations we derive: E J E J − [ ] E J E J E J + E J E J = F J F J − [ ] F J F J F J + F J F J = J and J exchanged. In other words, the usual (type A) cubic Serre relationsare implied by the join and nest relations. △ Definition 4.5.
We define a F ( K ) -gradation on U υ ( sl ( K )) by settingdeg ( E J ) = J , deg ( F J ) = − J and deg ( K J ) = K -interval J . ⊘ As usual, we define U + υ ( sl ( K )) (resp. U − υ ( sl ( K )) ) as the subalgebra of U υ ( sl ( K )) generatedby the E J (resp. the F J ) for any K -interval J . Let also U υ ( sl ( K )) be the subalgebra of U υ ( sl ( K )) generated by the K ± J for any K -interval J . Finally, set U ≤ υ ( sl ( K )) : = U − υ ( sl ( K )) · U υ ( sl ( K )) and U ≥ υ ( sl ( K )) : = U υ ( sl ( K )) · U + υ ( sl ( K )) .Then U ≤ υ ( sl ( K )) and U ≥ υ ( sl ( K )) may be endowed with the structure of topological e Q -bialgebras.4.2. The Fock space F K of U υ ( sl ( K )) . Our aim in this section is to define explicitly an action ofthe quantum group U υ ( sl ( K )) on the space F K : = M p ∈ Pyr ( K ) e Q | p i ,generalizing the standard Fock space representation of U υ ( sl ( ∞ )) . Theorem 4.6.
The following formulas define an action of the quantum group U υ ( sl ( K )) on F K : for anyJ = [ a , b [ and p ∈ Pyr ( K ) E J | p i : = ( − υ ( − υ ) p ( b ) − p ( a ) | p − J i if J is a removable interval of p ,0 otherwise , F J | p i : = ( υ ( − υ ) p ( a ) − p ( b ) | p + J i if J is an addable interval of p ,0 otherwise , K J | p i : = υ n J ( p ) | p i , where n J ( p ) = if J is neither addable or removable to p ,1 if J is addable to p , − if J is removable to p . The representation F K is highest weight and irreducible. U υ ( sl ( K )) is a topological coalgebra, i.e. the comultiplication only takes values in a suitable completion, see [SS17];we will not use the coproduct in this paper. OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 15
Proof.
For simplicity, let us put e J , p : = − υ ( − υ ) p ( b ) − p ( a ) and f J , p : = υ ( − υ ) p ( a ) − p ( b ) .We will check the compatibility with all the defining relations directly. Nest relations . Let’s start by verifying the nest relations for F J . We begin with the case of a pairof intervals I , J such that I ∩ J = ∅ . Then it is possible to successively add I and then J to p ifand only if it is possible to successively add J and then I to p . Moreover, in this case we have f J , p + I = f J , p and f I , p + J = f I , p . It follows that F I F J | p i = F J F I | p i as wanted.Now assume that I = [ a , b [ ⊆ J = [ a ′ , b ′ [ . We claim that it is possible to add both I and J to p (ineither order) only if I ⊂ J ◦ , i.e., only if a ′ < a < b < b ′ . Indeed, suppose for instance that a ′ = a .Then ( p + I + J )( a ) = p ( a ) + ( p + I + J ) − ( a ) = p − ( a ) . But then p and p + I + J cannot both be pyramids: if a ≤ p − ( a ) ≤ p ( a ) hence ( p + I + J ) − ( a ) ≤ ( p + I + J )( a ) −
2, violating condition iii) of pyramids; likewise, if a > p ( a ) ≤ p ( a ) + ( p + I + J ) − ( a ) ≤ ( p + I + J )( a ) −
1, again violating condition iii) of pyramids. A verysimilar reasoning takes care of the case b = b ′ . We are thus left with the case a ′ < a < b < b ′ . Inthis situation, it is easy to see that either I and J are both addable to p , in which case they can beadded in either order, or one of them is not addable to p , in which case the two can not be added(in either order). If both may be added, then we have p ( a ) − p ( b ) = ( p + J )( a ) − ( p + J )( b ) and p ( a ′ ) − p ( b ′ ) = ( p + I )( a ′ ) − ( p + I )( b ′ ) ,from which it immediately follows that f I , p + J f J , p = f J , p + I f I , p , i.e., F I F J | p i = F J F I | p i (note that h I , J i = h J , I i =
0, cf. Remark 4.1). The proof of the nest relations for the E J follows similarlyas above. Join relations . Let’s now move to verify the join relations for the F J . Assume that I = [ a , b [ , J =[ b , c [ . There are three mutually exclusive possible situations: it is not possible to add both I and J to p (in either order) and neither I ∪ J ; it is possible to add I , then J , hence also I ∪ J ; it is possibleto add J then I , hence also I ∪ J . In the first case, we have F I F J | p i = = F J F I | p i , hence the joinrelation is proved. In the second case, we have ( p + I )( b ) = p ( b ) , ( p + I )( c ) = p ( c ) ,hence υ − f J , p + I f I , p = υ ( − υ ) p ( a ) − p ( c ) = f I ∪ J , p .In the last case, we have ( p + J )( a ) = p ( a ) , ( p + J )( b ) = p ( b ) + − υ f I , p + J f J , p = − υ ( − υ ) p ( a ) − p ( c ) − = f I ∪ J , p .In both cases, the join relation F I ∪ J | p i = υ F I F J | p i − υ − F J F I | p i is verified. Similarly, weverify the join relation for E J . Drinfeld-Jimbo type relations . Finally, let us verify the Drinfeld-Jimbo type relations. The onlyrelation we have to address carefully is the commutation relation between the E J and the F J . Bythe same argument as at the beginning of this proof, it is not possible to both add and removethe same interval J to a pyramid p (otherwise it would be possible to add twice the interval J to p − J ). Thus either F J E J | p i = E J F J | p i =
0. There are three (mutually exclusive)possibilities: • it is neither possible to add nor remove J . The assertion follows. • it is possible to add J = [ a , b [ . Then E J F J | p i = − υ ( − υ ) ( p + J )( b ) − ( p + J )( a ) υ ( − υ ) p ( a ) − p ( b ) | p i = − υ ( − υ ) − | p i = | p i .Hence, [ E J , F J ] | p i = | p i , as wanted. • it is possible to remove J . Then F J E J | p i = υ ( − υ ) ( p − J )( a ) − ( p − J )( b ) ( − υ )( − υ ) p ( b ) − p ( a ) | p i = − υ ( − υ ) − | p i .Hence, [ E J , F J ] | p i = −| p i , as expected.The rest of the Drinfeld-Jimbo relations are easier to be verified and we leave the check to theinterested reader.To finish, let us check that F K is generated by | i and is irreducible. Let p be a K -pyramid.We may write p as a sum p = ∑ si = I i where I , . . . I s are strictly nested intervals, i.e. I ⊃ I ⊃ · · · ⊃ I s and the endpoints of the I i are all distinct. It is easy to see that, up to a constant, F I s · · · F I · | i = | p i , proving the first assertion. The irreducibility may be proved by reversingthis argument: up to a constant, we have E I · · · E I s · | q i = δ p , q | i if q is any pyramid such that | p | = | q | . (cid:3) Remark . Note that for J = [ a , b [ we have h J , p i = b ∑ x > a ( p − ( x ) − p + ( x )) = p ( a ) − p ( b ) , h p , J i = p − ( b ) − p − ( a ) .Hence e J , p = − υ ( − υ ) −h J , p i , f J , p = υ ( − υ ) h J , p i . △ For completeness, we also state the following result.
Lemma 4.8.
Let J be a K -interval and p be a K -pyramid. Thenn J ( p ) = δ ∈ J − ( J , p ) . Proof.
Let J = [ a , b [ . Note that ( J , p ) = ( p − ( b ) − p ( b )) − ( p − ( a ) − p ( a )) . There are three (mutu-ally exclusive) possibilities to consider: • it is neither possible to add nor remove J . We will check that δ ∈ J − ( J , p ) =
0. There are twosubcases to consider, according to whether 0 ∈ J or 0 J .(a’) 0 J . In that case, either the two endpoints a and b of J do not lie on points of disconti-nuity of p or they both lie on points of discontinuity of p . Thus ( J , p ) = ∈ J . In that case, exactly one of the endpoints a and b of J lie on a point of discontinuityof p . Moreover, a ≤ b > p − ( a ) ≤ p ( a ) while p − ( b ) ≥ p ( b ) . It follows that ( J , p ) = • it is possible to add J . Then we will show that δ ∈ J − ( J , p ) =
1. There are again two subcasesto consider, according to wether 0 ∈ J or 0 J .(b’) 0 J . In that case, either a > a is a point of discontinuity of p but not b , or b < b is a point of discontinuity of p but not a . Hence ( J , p ) = − ∈ J . In that case, neither endpoints a and b of J may lie over a point of discontinuity of p , and thus ( J , p ) = • it is possible to remove J . Then we will show that δ ∈ J − ( J , p ) = −
1. There are two subcasesto consider, according to whether 0 ∈ I or 0 J .(c’) 0 J . Here, either a < a lies over a point of discontinuity of p (but not b ), or a > b lies over a point of discontinuity of p (but not a ). We deduce ( J , p ) = ∈ J . In that case, a ≤ b > a and b lie over points of discontinuity of p , sothat ( J , p ) = (cid:3) OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 17
Generalisation to arbitrary discrete subsets.
Let α : Z → R be any strictly increasing map.We may define a quantum group U υ ( sl ( α ( Z ))) as the subalgebra (but not coalgebra) of U υ ( sl ( R )) generated by all elements E J , F J , K J where the endpoints of J belong to α ( Z ) . There is an obviousisomorphism γ α : U υ ( sl ( Z )) ∼ → U υ ( sl ( α ( Z ))) mapping E J , F J , K J to E α ( J ) , F α ( J ) , K α ( J ) respectively.Let us now assume that 0 ∈ Im ( α ) and let F α ( Z ) be the linear subspace of F R spanned by pyra-mids p satisfying D R ( p ) ∈ Im ( α ) . It is clear that F α ( Z ) is stable under the action of U υ ( sl ( α ( Z ))) and that the following square is commutative: U υ ( sl ( α ( Z ))) End ( F α ( Z ) ) U υ ( sl ( Z )) End ( F Z ) γ α ι α ,where ι α : F Z ∼ → F α ( Z ) is the linear isomorphism induced by α .4.4. Relation with U υ ( sl ( ∞ )) . The quantum group U υ ( sl ( Z )) admits a minimal set of generators { E [ i , i + [ , F [ i , i + [ , K ± [ i , i + [ | i ∈ Z } . Thanks to the nest relation (4.2) and the Serre relations (4.3), theassignment E i E [ i , i + [ , F i F [ i , i + [ , K ± i K ± [ i , i + [ defines an isomorphism of bialgebras U υ ( sl ( ∞ )) → U υ ( sl ( Z )) .The action of U υ ( sl ( Z )) on F Z given in Theorem 4.6 reduces to E [ i , i + [ | λ i : = − υ | ν i if Y λ \ Y ν is a box with color i and i < υ − | ν i if Y λ \ Y ν is a box with color i and i ≥ F [ i , i + [ | λ i : = − υ − | µ i if Y µ \ Y λ is a box with color i and i < υ | µ i if Y µ \ Y λ is a box with color i and i ≥ K [ i , i + [ | λ i : = υ n i ( λ ) | λ i for i ∈ Z . Here, we used the bijection between Z -pyramids and partitions (cf. Lemma 2.3) andthe identity n i ( λ ) = n [ i , i + [ ( p λ ) ,where p λ is the Z -pyramid associated with λ . Thus the U υ ( sl ( Z )) -action on F Z defined in Theo-rem 4.6 is identified with a suitable rescaling of the standard U υ ( sl ( ∞ )) -action on the Fock space.5. T HE CIRCLE QUANTUM GROUP AND THE FOLDING PROCEDURE
In this section, we recall the definition of the circle quantum group and we provide a real-ization of it in terms of Hall algebras of continuum quivers. Finally, we generalize the “foldingprocedure” of Hayashi-Misra-Miwa (see [Hay90, Section 6.2] and [MM90, Section 2]) to the circlequantum group. We will readapt a interpretation of the folding procedure due to Varagnolo-Vasserot (cf. [VV99, Section 6]) which uses the Hall algebras realization of the quantum group.This folding procedure will be a key tool to the proof of the action of the circle quantum groupon the Fock space in the next section.
Let K be either Q , R or α ( Z ) where α : Z → R is a strictly increasing map whose imagecontains 0 and is invariant under integer translations. Set S K : = K / Z and denote by π K : K → S K the projection map.5.1. The bialgebra U υ ( sl ( S K )) . The notion of (closed on the left, open on the right) intervalsgeneralizes in a straightforward way to S K . We say that an interval J of S K is strict if J = S K .We denote by F ( S K ) the algebra of piecewise constant, right–continuous, Z -valued functions f : S R → R , with finitely many points of discontinuity, whose points of discontinuity belong to S K . There is an obvious map π K : F ( K ) → F ( S K ) . Formula (4.1) defines bilinear forms h· , ·i and ( · , · ) on F ( S K ) . One defines F ( S K ) ± as before. There is an obvious map π K : F ( K ) ± → F ( S K ) ± . Remark . Note that ( S , f ) = f ∈ F ( S K ) . In fact it is easy to see that the kernel of ( · , · ) is equal to Z 1 S . △ Define, for a strict interval J ⊂ S K , Int ( J ) : = (cid:8) I ⊂ K | I = [ a , b [ is a K -interval , π K ( I ) = J (cid:9) .Thus Int ( J ) consists of all integer translates of some (any) K -interval ˜ J such that π K ( ˜ J ) = J . Definition 5.2.
Let J and J ′ be strict intervals of S K . We say that J is left adjacent to J ′ if J ∩ J ′ = ∅ , J ∩ J ′ = ∅ and J ∪ J ′ is an interval of S K . We denote it by J → J ′ . ⊘ We are now ready to give the definition of the circle quantum group.
Definition 5.3 ([SS17, Definition 1.1]) . Let U υ ( sl ( S K )) be the topological e Q -bialgebra generatedby elements E J , F J , K ± J ′ , where J (resp. J ′ ) runs over all strict intervals (resp. intervals) of S K ,modulo the following set of relations: • Drinfeld-Jimbo relations : – for any intervals I , I , I and strict interval J , [ K I , K I ] = K I E J K − I = υ ( I , J ) E J , (5.2) K I F J K − I = υ − ( I , J ) F J ; (5.3) – if J , J are strict intervals such that J ∩ J = ∅ , [ E J , F J ] = – for any strict interval J , [ E J , F J ] = K J − K − J υ − υ − ; (5.5) • join relations : – if J , J are strict intervals such that J is left adjacent to J , K J K J = K J ∪ J ; – if J , J are strict intervals such that J is left adjacent to J and J ∪ J is again a strictinterval, E J ∪ J = υ E J E J − υ − E J E J , F J ∪ J = υ − F J F J − υ F J F J ; • nest relations : OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 19 – if J , J are strict intervals such that J ∩ J = ∅ , [ E J , E J ] = [ F J , F J ] = – if J , J are strict intervals such that J ⊆ J , υ h J , J i E J E J = υ h J , J i E J E J , υ h J , J i F J F J = υ h J , J i F J F J .The coproduct is given by ∆ ( K J ) = K J ⊗ K J , ∆ ( E [ a , b [ ) = E [ a , b [ ⊗ + ∑ a < c < b υ − ( υ − υ − ) E [ a , c [ K [ c , b [ ⊗ E [ c , b [ + K [ a , b [ ⊗ E [ a , b [ , ∆ ( F [ a , b [ ) = ⊗ F [ a , b [ − ∑ a < c < b υ − ( υ − υ − ) F [ c , b [ ⊗ F [ a , c [ K − [ c , b [ + F [ a , b [ ⊗ K − [ a , b [ .Here the sums on the right-hand-side run over all possible K -values c ∈ [ a , b [ . ⊘ Definition 5.4.
We define a F ( S K ) -gradation on U υ ( sl ( S K )) by settingdeg ( E J ) = J , deg ( F J ) = − J and deg ( K J ′ ) = J and any interval J ′ . ⊘ As in Section 4.1, we define the subalgebras U ǫυ ( sl ( S K )) , with ǫ = ±
1, as well as the negativesubalgebra U ≤ υ ( sl ( S K )) and the positive subalgebra U ≥ υ ( sl ( S K )) .5.2. Representations of continuum quivers.
Let us begin with the line K , with K ∈ { R , Q } as before. We start by recalling the notion of persistence modules (see [BBCB18], [DEHH18,Section 2.3] for a recent report). Definition 5.5.
Let k be a field. A K -persistence module is a functor F : ( K , ≤ ) → Vect k fromthe poset ( K , ≤ ) to the category Vect k consisting of finite-dimensional vector spaces over k withlinear maps between them. Explicitly, F is determined by:i) a finite-dimensional k -vector space F ( t ) for every t ∈ K ,ii) a k -linear map F ( s ≤ t ) : V s → V t for every pair of real numbers s ≤ t such that – F ( t ≤ t ) is the identity map from V t to itself, – given real numbers s ≤ t ≤ u , one has F ( s ≤ u ) = F ( t ≤ u ) ◦ F ( s ≤ t ) . ⊘ Morphisms of K -persistence modules are defined in the obvious way. Remark . The poset ( K , ≤ ) can be interpreted as a category, whose objects are points of K andthe set of maps between two objects s , t ∈ K consists of exactly one map ι t , s if s ≤ t , otherwise isempty. Thanks to this interpretation, one can extend the definition above replacing ( K , ≤ ) withany (small) category, as done e.g. in [BBCB18, Section 2]. △ Definition 5.7.
A point t ∈ K is regular for a persistence module F if there exists an interval I ⊆ K where t ∈ I and F ( a ≤ b ) is an isomorphism for all pairs a , b ∈ I . Otherwise we say that t is critical . A persistence module F is tame if it has finitely many critical values. ⊘ The set of all regular points will be called the regular locus and its complement the critical locus . In the literature of persistent homology [Carl09, Oud15], this is called a pointwise finite-dimensional persistencemodule.
Example . Any K -interval J = [ a , b [ gives rise to a persistence module k J , for which k J ( t ) = k if t ∈ J , otherwise k J ( t ) = { } , and for any pair s , t ∈ K one has that k J ( s ≤ t ) is the identity mapif s , t ∈ J , otherwise k J ( s ≤ t ) is the zero map. By [BBL18, Proposition 2.2], k J is an example of anindecomposable module with local endomorphism ring. △ From now on, we shall restrict to a smaller class of persistence modules.
Definition 5.9.
We say that a persistence module F is coherent if it is tame and the map t dim F ( t ) is right–continuous and compactly supported. ⊘ The fundamental theorem of persistent homology says that any (pointwise finite-dimensional)persistence module is a direct sum of indecomposable modules with local endomorphism ring(cf. [BBCB18, Theorem 1.1]). If we apply it to our coherent persistence modules, we obtain:
Lemma 5.10.
For any coherent K -persistence module F there exists a partition { x ∈ K | F ( x ) = { }} = G i J i into finitely many K -intervals J i = [ a i , b i [ such that F ( v ≤ u ) is an isomorphism for any pair v ≤ u ofelements belonging to the same interval J i . From now on, we shall also call a coherent persistence module a representation of the continuumquiver K . We denote the data of a representation F of K in the following way: V t : = F ( t ) and x t , s : = F ( s ≤ t ) . Summarizing, a representation of K is a collection of data ( V , x ) : = ( V t , x t , s ) t , s with i) V t is a finite-dimensional k -vector space for every t ∈ K ,ii) x t , s : V s → V t is a k -linear map for every pair s , t ∈ K with s ≤ t ,such thata) the map t dim ( V t ) is right–continuous, compactly supported and with finitely manydiscontinuities,b) we have x t , r = x t , s ◦ x s , r for any triple r ≤ s ≤ t ,c) there exists a partition { t ∈ K | V t = { }} = F i J i into finitely many intervals J i = [ a i , b i [ such that x u , v is an isomorphism for any pair v ≤ u of elements belonging to the sameinterval J i .We denote by Rep k K the category of representations of K . Then Rep k K is abelian. It may berealized as a certain colimit of categories of representations of (locally) finite type A quivers. Moreprecisely, for any locally finite subset S ⊂ K , let us denote by Rep ( S ) k K the full subcategory of Rep k K consisting of representations ( V t , x t , s ) t , s for which x t , s is an isomorphism for s , t ∈ K \ S .Then Rep ( S ) k K is equivalent to the category Rep k Q S of representations of the quiver Q S , whosevertices are the maximal intervals in K \ S and whose arrows are given by the adjacency relation.It is clear that Q S is either a finite type A quiver (if | S | < ∞ ) or a quiver of type A ∞ (if S is infinite).The categories Rep ( S ) k K form a direct system with respect to the inclusion S ⊂ S ′ and we have Rep k K ≃ lim −→ Rep k Q S .It follows that Rep k K is hereditary. Remark . The above equivalence justifies our choice to call the objects of
Rep k K the represen-tations of K . △ Let us next consider the case of the circle S K . Let Γ be the oriented fundamental groupoid of S K : its objects are homotopy classes of orientation preserving paths [
0, 1 ] ∩ K → S K ; these maybe parametrized by triples ( s , t , n ) where s , t are (not necessarily distinct) elements S K and n ∈ N is the winding number of the path. For γ = ( s , t , n ) we will sometimes write γ ′ = s , γ ′′ = t . OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 21
By following Remark 5.6, we replace ( K , ≤ ) in Definition 5.5 by the category whose objectsare points t ∈ S K and morphisms are given by γ ∈ Γ : thus, the notion of coherent persistencemodules makes sense. We denote by Rep k S K the category of these coherent persistence modules,which we shall call representations of the continuum quiver S K . Note that Lemma 5.10 holds also inthis case, thanks to [BBL18, Theorem 1.1]. Summarizing, a representation of S K is a collection ofdata ( V , x ) : = ( V t , x γ ) t , γ with t ∈ S K and γ ∈ Γ withi) V t is a finite-dimensional k -vector space for every t ∈ S K ,ii) x γ : V γ ′ → V γ ′′ is a k -linear map for every γ ∈ Γ ,such thata’) the map t dim ( V t ) is right–continuous and with finitely many discontinuities,b’) we have x γ ◦ x γ = x γ · γ for any composable pair γ , γ of elements of Γ ,c’) there exists a partition S K = F i J i into finitely many intervals J i = [ a i , b i [ such that x γ isan isomorphism for any path γ contained in a single interval J i .Again, Rep k S K is a hereditary abelian category. Like Rep k K it may be realized as a colimit Rep k S K = lim −→ Rep ( S ) k S K , where S ranges over all finite subsets of S K . Note that Rep ( S ) k S K is nowequivalent to a category Rep k Q S of representations of an affine (cyclically oriented) quiver of type A . For any strict interval J = [ a , b [ of S K there is a indecomposable object k J of Rep k S K , and oneobtains in this way all rigid indecomposables.Taking the dimension function f ( V ) : t dim ( V t ) defines a map dim from the Grothendieckgroups K ( Rep k K ) and K ( Rep k S K ) to F ( K ) and F ( S K ) respectively. Remark . Let X be a scheme, L a line bundle on X and s a global section of L . Following [Tal18,Example 3.18], a parabolic sheaf with real weights on X is the assignment of an O X -module E r foreach r ∈ R , with maps E r → E r ′ when r ≤ r ′ , that are compatible with respect to composition,and such that E r → E r + ≃ L ⊗ O X E r is identified with multiplication by the section s .Given a real number r , we denote by ⌊ r ⌋ the largest integer which is less or equal to r and weset { r } : = r − ⌊ r ⌋ . A parabolic sheaf with real weights E is finitely presented if each E r is a finitelypresented sheaf on X , and moreover there exist finitely many real numbers 0 ≤ r < . . . < r k < r ∈ R then E r ≃ E r i ⊗ L ⌊ r ⌋ via the given map, where r i is the largest of thefixed numbers that is less or equal to { r } . In other words, the parabolic sheaf E is completelydetermined by the weights r i , the finitely presented sheaves E r i , and the maps between them.Then, our definition of representations of S R coincides with the definition of finitely presentedparabolic sheaves E for which each E r has zero-dimensional support for any r ∈ R . △ Hall algebras of continuum quivers.
When k = F q is a finite field, Rep k K and Rep k S K arefinitary categories, i.e., the sets Ext i ( V , V ′ ) are finite for any V , V ′ and i =
0, 1. In this context wemay consider the Hall algebras H K and H S K of Rep k K and Rep k S K respectively, as in Section 3.3.Recall that υ : = q . Let M K , M S denote the set of all isomorphism classes of objects of Rep k K and Rep k S K respectively. As vector spaces we have H K : = { h : M K → C | | supp ( f ) | < ∞ } and H S K : = { h : M S → C | | supp ( f ) | < ∞ } .As before, denote by [ V ] the characteristic function of an object V . The function dim induces anatural gradation H K = M f ∈ F ( K ) H K [ f ] and H S K = M f ∈ F ( S K ) H S K [ f ] . (5.6)The multiplications in H K and H S K are defined in the same fashion as in Section 3.3. Note thatthe multiplication is graded with respect to the decomposition (5.6). We let H sph S K be the subalgebra of H S K generated by the elements [ k J ] where J runs through theset of strict intervals in S . Theorem 5.13. (i) The assignment [ k J ] υ − E J for all intervals J defines an isomorphism of graded algebras H K ≃ U + υ ( sl ( K )) .(ii) The assignment [ k J ] υ − E J for all strict intervals J defines an isomorphism of graded alge-bras H sph S K ≃ U + υ ( sl ( S K )) .Proof. Notice that any order-preserving bijection K → ]
0, 1 [ ∩ K induces both a fully faithful em-bedding Rep k K ֒ → Rep k S K and an inclusion of algebras U + υ ( sl ( K )) ֒ → U + υ ( sl ( S K )) . Hence,by the compatibility between the Hall algebra construction and fully faithful embeddings (seee.g. [S12, Section 1.8]) it is enough to treat the case of Rep k S K and U + υ ( sl ( S K )) . By the samefunctorial properties of Hall algebras, H sph S K is isomorphic to an inductive limit of Hall alge-bras H sph ( Rep ( S ) k ( S K )) as S ranges over all finite subsets of S K . On the other hand, denote by U + υ ( sl ( S ) ( S K )) the subalgebra of U + υ ( sl ( S K )) generated by the elements E J for J = [ a , b [ satisfy-ing a , b ∈ S . Then U + υ ( sl ( S K )) is the inductive limit of U + υ ( sl ( S ) ( S K )) as S ranges over all finitesubsets of S K . Hence it suffices to prove that the assignment E J υ [ k J ] extends to an algebraisomorphism U + υ ( sl ( S ) ( S K )) ∼ −→ H sph ( Rep ( S ) k ( S K )) for any S . Since S is finite, this reduces to thewell-known identification between the spherical Hall algebra of a cyclic quiver with n verticesand U + υ ( b sl ( n )) (see e.g. [SS17, Section 4.2]). (cid:3) We will need a slight variant of the above result, involving U − υ ( sl ( S K )) instead of U + υ ( sl ( S K )) . Corollary 5.14. (i) The assignment [ k J ]
7→ − υ − F J for all intervals J defines an isomorphism of graded algebras H K ≃ U − υ ( sl ( K )) .(ii) The assignment [ k J ]
7→ − υ − F J for all strict intervals J defines an isomorphism of gradedalgebras H sph S K ≃ U − υ ( sl ( S K )) .Proof. This comes from the isomorphism U + υ ( sl ( S K )) ≃ U − υ ( sl ( S K )) given by E J
7→ − F J . (cid:3) Folding procedure.
To unburden the notation, let us simply denote by f f the projection π K : F ( K ) → F ( S K ) and likewise s s for the projection K → S K . We will denote by s , t , . . .elements of K and by a , b , . . . elements of S K .Following Section 3.3 we will now construct a family of maps γ f : H S K [ f ] → H K [ f ] for every f ∈ F ( K ) . Fix a function f ∈ F ( K ) and a collection of vector spaces V t , for t ∈ K , suchthat dim ( V t ) = f ( t ) for all t . Put: V = M t V t , V = M a V a , V a = M s = a V s .Thus, even if they may be canonically identified, V is a K -graded vector space while V is S K -graded. Note that since supp ( f ) is compact, V a is finite-dimensional for any a . For any s ∈ K wedefine V ≥ s : = M t ≥ s V t , OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 23 and we denote by V •≥ the associated filtration of V . There is an induced filtration V •≥ of V where V ≥ s = M a V ≥ s , a and V ≥ s , a = M t ≥ s , t = a V t .The filtrations V • > , V • > are defined in the same way, replacing ≥ by > . The associated gradedspace gr ( V ) : = M s ( V ≥ s / V > s ) is canonically isomorphic to V as a K -graded vector space.Let us denote by E V the groupoid of all representations of K in V : objects are collections ofmaps x t , s : V s → V t for s < t satisfying conditions (b) and (c) in Section 5.2 and morphisms aregiven by elements of ∏ t GL ( V t ) . We likewise denote by E V the groupoid of all representationsof S K in V : objects are collections of maps x γ : V γ ′ → V γ ′′ for γ ∈ Γ satisfying (b’) and (c’) andmorphisms are elements of ∏ a GL ( V a ) . Finally, let E V , V denote the groupoid of representationsin E V preserving the filtration V •≥ : objects are collections x γ as above such that for any γ ∈ Γ andany s ∈ K satisfying s = γ ′ we have x γ ( V ≥ s ) ⊆ V ≥ γ ( s ) – here γ ( s ) is the deck transformation of s associated to γ ; morphisms are given by elements of ∏ a P a where P a ⊂ GL ( V a ) is the parabolicsubgroup of elements preserving the induced filtration V •≥ ∩ V a .There is an obvious functor j : E V , V → E V . Passing to the associated graded we also get afunctor p : E V , V → E V defined as follows. For every s , t ∈ K with s < t , let γ correspond to theoriented path s t in K ; the map x γ : V s → V t sends V ≥ s to V ≥ t and V > s to V > t ; we let x t , s bethe induced map V s → V ≥ s / V > s → V ≥ t / V > t = V t . Lemma 5.15.
The functor p : E V , V → E V has essentially finite fibers.Proof. Let ρ = ( x t , s ) s , t ∈ E V . Denote by D ρ ⊂ K the set of critical points of ρ and let D ρ ⊂ S K beits reduction mod Z . Any object ˜ ρ = ( x γ ) γ in the fiber p − ( ρ ) has a critical set D ˜ ρ included in D ρ . Let S be a finite subset of S K containing D ρ . Let E ( S ) V , E ( S ) V , E ( S ) V , V be the (full) subgroupoids of E V , E V , E V , V whose objects are those representations whose critical sets are contained in π − K ( S ) and S respectively. The functor p : E V , V → E V restricts to a functor p ( S ) : E ( S ) V , V → E ( S ) V and thereis a cartesian square E ( S ) V , V E V , V E ( S ) V E Vp ( S ) p .Since ρ ∈ E ( S ) V , it suffices to show that p ( S ) has essentially finite fibers. This is obvious since E ( S ) V , V has finitely many objects up to isomorphism. (cid:3) For a groupoid X , we denote by C ( X ) the space of complex functions on Obj ( X ) which areinvariant under isomorphism and have essentially finite support. We will identify C ( E V ) and C ( E V ) with H K [ f ] and H S K [ f ] respectively. We set γ f : = υ − h ( f ) p ! ◦ j ∗ : H S K [ f ] → H K [ f ] ,where h ( f ) : = − ∑ ℓ< h f , τ ℓ ( f ) i , with τ ( f ) : t f ( t − ) being the translation operator in F ( K ) , and where p ! is taken in the senseof groupoids, i.e., p ! ( u )( ρ ) : = ∑ ˜ ρ ∈ Obj ( p − ( ρ )) / ∼ u ( ˜ ρ ) | Aut ( ρ ) || Aut ( ˜ ρ ) | . Proposition 5.16.
For any pair h , h ′ ∈ F ( S K ) and any f ∈ F ( K ) such that f = h + h ′ we have ∑ g + g ′ = fg = h , g ′ = h ′ υ ( g , ∑ ℓ> τ ℓ ( g ′ )) γ g ( u ) ⋆ γ g ′ ( u ′ ) = γ f ( u ⋆ u ′ ) for any u ∈ H S K [ h ] , u ′ ∈ H S K [ h ′ ] .Proof. Let f , h , h ′ be as above and fix some u ∈ H S K [ h ] , u ′ ∈ H S K [ h ′ ] . Let S be a finite subset of S K containing the critical sets of all representations occurring in u , u ′ as well as all the discontinuitiesof f , h , h ′ . Then S contains also the critical sets of any representation occurring in u ⋆ u ′ , as well asin γ g ( u ) , γ g ′ ( u ′ ) and hence also of γ g ( u ) ⋆ γ g ′ ( u ′ ) . Arguing as in the proof of Lemma 5.15, we seethat it is enough to prove the statement of the theorem when we replace everywhere E V , . . . by E ( S ) V , . . .. But in this case Rep ( S ) k S K and Rep ( S ) k K are equivalent to categories of representations of acyclic quiver and a A ∞ quiver respectively and we are in the precise setting of Proposition 3.3. (cid:3) Using the above Proposition, we may define an algebra morphism from H S K to a certain com-pletion of U ≤ υ ( sl ( K )) that we are going to introduce now. Let F ( K ) c be the space of piecewiseconstant, right–continuous functions f : R → Z whose points of discontinuity all belong to K and project onto a finite subset of S K . In other words, F ( K ) c is an analogue of F ( K ) but withoutthe bounded support condition. Let U υ ( sl ( K )) c be the algebra generated by elements K ± f for f ∈ F ( K ) c subject to the relations K f K g = K f + g , for f , g ∈ F ( K ) c . We set U ≤ υ ( sl ( K )) c : = M g ∈ F ( S K ) U ≤ υ ( sl ( K )) c [ g ] U ≤ υ ( sl ( K )) c [ g ] : = ∏ f ∈ F ( K ) , f = g n U υ ( sl ( K )) c ⋉ U − υ ( sl ( K ))[ f ] o .As before, one can show that U ≤ υ ( sl ( K )) c is an algebra (for this, we use that any g ∈ F ( K ) canbe written in finitely many ways as a sum g = g ′ + g ′′ with g , g ′ fixed). For x ∈ H S K [ g ] we set r ( x ) : = ∑ f : f = g K o ( f ) γ f ( u ) ∈ U ≤ υ ( sl ( K )) c where o ( f ) : = ∑ ℓ< τ ℓ ( f ) . Corollary 5.17.
The map r : H S K → U ≤ υ ( sl ( K )) c is an algebra homomorphism. One can show that the map r is injective by realizing H S K as a direct limit of Hall algebras ofcyclic quivers and using the injectivity of the map r in the finite case. Remark . Using the above Proposition, we can pull-back via r any weight representation of U ≤ υ ( sl ( K )) which extends to U ≤ υ ( sl ( K )) c . In the next section, we will define the Fock spacerepresentation of H S K or U − υ ( sl ( S K )) as the pullback by r of the Fock space representation F K of U ≤ υ ( sl ( K )) . △ Let us compute explicitly the image of the elements F J . OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 25
Lemma 5.19.
Let J be a strict interval of S K . Thenr ( F J ) = ∑ J ,..., J ℓ υ ℓ − ( υ − − υ ) ℓ − F J · · · F J ℓ ∏ i , ℓ> K Ji + ℓ (5.7) where the sum ranges over all tuples J , J , . . . , J ℓ such that J = π ( J + · · · + J ℓ ) , J < J < · · · and π K ( J ) → π K ( J ) → · · · .Proof. Let f ∈ F ( K ) be such that f = J . We claim that γ f ( F J ) = υ − ℓ ( υ − − υ ) ℓ − F J · · · F J ℓ if there exists J , J , . . . , J ℓ such that f = J + · · · + J ℓ , J < J < · · · and π K ( J ) → π K ( J ) → · · · and γ f ( F J ) = f = ∑ ℓ i = J i with J < J < · · · J ℓ . For γ f ( F J ) tobe nonzero, there must exist a filtration M ⊂ M ⊂ · · · ⊂ M ℓ = k J of the indecomposable S K -module k J such that dim ( M i / M i − ) = J i for i =
1, . . . , ℓ . This is possible if and only if π K ( J ) → π K ( J ) → · · · and moreover in that case we have L i M i / M i − ≃ L i k J i . The claimnow follows from the facts that | Aut ( k J ) | = q −
1, while | Aut ( L i I J i ) | = ( q − ) ℓ and h ( f ) = − ℓ ,and finally from the easily checked relation [ L i k J i ] = [ k J ] ⋆ · · · ⋆ [ k J ℓ ] . To obtain formula (5.7),observe that for i < k we have ( τ ℓ ( J k ) , J i ) = ℓ > i > k there exists a unique ℓ such that ( τ ℓ ( J k ) , J i ) = ( τ ℓ ( J k ) , J i ) = ℓ . (cid:3)
6. F
OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP
In this section, we define an action of the circle quantum group on the Fock space F K .6.1. The Fock space F K of U υ ( sl ( S K )) . We will now define the main object of the present paper,namely the Fock space representation of the circle quantum group. As a vector space, this Fockspace is again F K = M p ∈ Pyr ( K ) e Q | p i .For I a K -interval and p a K -pyramid, we set n > I ( p ) = ∑ m ≥ n τ m ( I ) ( p ) , n < I ( p ) = ∑ m ≥ n τ − m ( I ) ( p ) , n I ( p ) = ∑ m ∈ Z n τ m ( I ) ( p ) ,where n J ( p ) is defined as in Theorem 4.6 and where τ m ( I ) is the translation of I by m ≥ I = [ a , b ) then τ m ( I ) = [ a + m , b + m ) . If I , J are K -intervals then we write I < J if I = [ a , b ) , J = [ c , d ) and b < c . Theorem 6.1.
The following formulas define an action of the quantum group U υ ( sl ( S K )) on F K :E J | p i = ∑ J ′ ,..., J ′ ℓ υ − ℓ − ∑ i n < J ′ i ( p ) ( − υ ) − ∑ i h J ′ i , p i ( υ − υ − ) ℓ − | p − ∑ i J ′ i i , (6.1) F J | p i = ∑ J ′′ ,..., J ′′ ℓ υ ℓ − + ∑ i n > J ′′ i ( p ) ( − υ ) ∑ i h J ′′ i , p i ( υ − − υ ) ℓ − | p + ∑ i J ′′ i i , (6.2) K J | p i = υ n J ( p ) | p i , where the sums range over all tuples of removable K -intervals ( J ′ , . . . , J ′ l ) (resp. all tuples of addable K -intervals ( J ′′ , . . . , J ′′ l ) ) satisfying the conditionsa’) J ′ > J ′ > · · · > J ′ ℓ ,b’) π K ( J ′ ) → π K ( J ′ ) → · · · → π K ( J ′ ℓ ) , c’) π K ( J ′ ) ⊔ · · · ⊔ π K ( J ′ ℓ ) = J.(resp.a”) J ′′ < J ′′ < · · · < J ′′ ℓ ,b”) π K ( J ′′ ) → π K ( J ′′ ) → · · · → π K ( J ′′ ℓ ) ,c”) π K ( J ′′ ) ⊔ · · · ⊔ π K ( J ′′ ℓ ) = J.)Proof.
The proof of this theorem occupies the rest of the Section.6.1.1.
Well-definedness.
Let us first check that E J and F J are well-defined, i.e. that the sums in-volved in their definition are in fact finite. We treat the case of the operators F J , the case of E J being similar.Let us fix a K -pyramid p . By definition, a collection of K -intervals J ′′ , J ′′ , . . . , J ′′ ℓ satisfying theconditions (a”), (b”), (c”) in Theorem 6.1 induce a subdivision J → J → · · · → J ℓ of J , with J i = π K ( J ′′ i ) . For any given interval I ⊂ S K and any pyramid q , there are at most finitely many I ′′ ∈ Int ( I ) which are addable to q (because q is of compact support). Hence it is enough to provethat only finitely many subdivisions J → J → · · · → J ℓ of J may give rise to a collection ofaddable intervals ( J ′′ , . . . , J ′′ ℓ ) .Write J = [ a , b ) and J = [ c = a , c ) , J = [ c , c ) , . . . , J ℓ = [ c ℓ − , c ℓ = b ) . We claim that if thereexists addable K -intervals J ′′ , . . . , J ′′ ℓ such that J ′′ i ∈ Int ( J i ) for all i and J ′′ < J ′′ < · · · < J ′′ ℓ thennecessarily c , c , . . . , c ℓ − ∈ π K ( D K ( p )) . Indeed, writing J ′′ i = [ a ′′ i , b ′′ i ) for i =
1, . . . , ℓ (so that π K ( a ′′ i ) = c i − , π K ( b ′′ i ) = c i ) we have by Remark 2.11 b ′′ i ∈ D K ( p ) if b ′′ i < i = ℓ , a ′′ i ∈ D K ( p ) ∪ { } if a ′′ i ≥ i = D K ( p ) is finite, this implies the desired finiteness of possible tuples ( J , . . . , J ℓ ) . The argu-ment for E J is similar, using the reverse condition J ′ > J ′ > · · · > J ′ ℓ .6.1.2. Join and nest relations.
Using Formula (5.7) and Theorem 4.6 we deduce that the operators(6.2) indeed define an action of U + υ ( sl ( S K )) as wanted. To prove that the operators (6.1) definean action of U − υ ( sl ( S K )) , one may argue in a similar fashion — using Hall algebra of the oppositequivers with υ − = q — and show that the assignment E J ∑ J ,..., J ℓ υ − ℓ ( υ − υ − ) ℓ − E J · · · E J ℓ ∏ i , ℓ> K − Ji − ℓ where the sum ranges over all tuples J , . . . , J ℓ of intervals such that J = π ( J + · · · + J ℓ ) , J > J > · · · and π K ( J ) → π K ( J ) → · · · , defines an algebra morphism U − υ ( sl ( S K )) → U − υ ( sl ( K )) c .This argument proves the join and nest relations for the E J ’s and the F J ’s.6.1.3. The Drinfeld-Jimbo relations.
It remains to check that the actions of U ± υ ( sl ( S K )) and of U υ ( sl ( S K )) glue together to form an action of U υ ( sl ( S K )) . Each of the relations (5.1 – 5.5) involves finitelymany generators E J , F J , K J acting on a pyramid p .Let α : Z → R be a strictly increasing map, whose image is invariant under integer trans-lations, contains 0 as well as D K ( p ) and the endpoints of all involved intervals J . Put N = ( α − ([
0, 1 ))) . Arguing as in Section 4.3 we construct natural isomorphism U υ ( sl ( S N Z )) ≃ U υ ( sl ( S α ( Z ) )) and F N Z ≃ F α ( Z ) compatible with the formulas in Theorem 6.1. Thus it is enoughto prove the theorem in the case K = N Z , which we assume from now on.Relations (5.1), (5.2), (5.3) are obvious. Using the join relations, we may reduce relations (5.4),(5.5) to the cases where J , J , resp. J are of length N ; this is clear for (5.1) and results from a OCK SPACE REPRESENTATION OF THE CIRCLE QUANTUM GROUP 27 lengthy but straightforward computation for (5.2). In the case of length N -intervals (i.e. of sim-ple roots of b sl ( N ) ) these relations are well-known; we nevertheless derive them below for thereader’s comfort: let p be a N Z -pyramid, and let J = [ i , i + N [ for some i ∈ N Z / Z . Let p ′ be another N Z pyramid. By construction, we have h p ′ | E J F J p i = h p ′ | F J E J p i = p ′ = p − J ′ + J ′′ for a pair of intervals J , J ′ such that π ( J ′ ) = π ( J ′ ) = J . Let us first show that if p = p ′ then h p ′ | F J E J p i = h p ′ | E J F J p i . In that situation, J ′ and J ′′ are unique and we have h p ′ | E J F J p i = h p ′ | E J ( p + J ′′ ) ih p + J ′′ | F J p i = υ n > J ′′ ( p ) − n < J ′ ( p + J ′′ ) , h p ′ | F J E J p i = h p ′ | F J ( p − J ′ ) ih p − J ′ | E J p i = υ n > J ′′ ( p − J ′ ) − n < J ′ ( p ) .There are two possibilities: J ′′ > J ′ or J ′′ < J ′ . In the first one we have n > J ′′ ( p ) = n > J ′′ ( p − J ′ ) and n < J ′ ( p + J ′′ ) = n < J ′ ( p ) , while in the second one we have n > J ′′ ( p ) = n > J ′′ ( p − J ′ ) + n < J ′ ( p + J ′′ ) = n < J ′ ( p ) −
1. In both cases the desired equality follows.Thus only when p = p ′ may we have a contribution to the commutator [ E J , F J ] . Let I < I < · · · < I c be the different addable or removable intervals of p which are congruent to J . We maypartition [ c ] = A ⊔ R with A = { i | I i is addable } and R = { i | I i is removable } . Let us alsowrite a > h = | A ∩ [ h + n ] | , r > h = | R ∩ [ h + n ] | , a < h = | A ∩ [ h − ] | , r < h = | R ∩ [ h − ] | .The contribution of the interval I h to h p | [ E J , F J ] p i is equal to h p | E J ( p + I h ) ih p + I h | F J p i = υ ( a > h − r > h ) − ( a < h − r < h ) if h ∈ A , while it is −h p | F J ( p − I h ) ih p − I h | E J p i = − υ ( a > h − r > h ) − ( a < h − r < h ) if h ∈ R . All together we get h p | [ E J , F J ] p i = ∑ h ∈ A υ ( a > h − r > h ) − ( a < h − r < h ) − ∑ h ∈ R υ ( a > h − r > h ) − ( a < h − r < h ) . (6.3)It thus only remains to check that the r.h.s. of relation (6.3) coincides with h p | ( K J − K − J ) p i / ( υ − υ − ) = ( υ | A |−| R | − υ | R |−| H | ) / ( υ − υ − ) .This is a purely combinatorial statement, which may be proved as follows. We first check it when R = [ u ] , A = [ u + n ] , and then we prove that the r.h.s. of relation (6.3) remains unchangedwhen we exchange the position of a pair of adjacent elements, one which belongs to A and theother to R . We leave the details to the reader. (cid:3) Non-cyclicity.
In the finite setup (i.e., for Hayashi’s Fock space of U υ ( b sl ( n )) ) the Fock spaceis not cyclic. The same holds here: Proposition 6.2.
The subspace U υ ( sl ( S K )) · | i of F K is strict.Proof. We claim that the element | [ [ i does not belong to U υ ( sl ( S K )) · | i . Let us argue by con-tradiction. Let u : = P ( F J , . . . , F J s ) be a linear combination of monomials in generators F J , . . . , F J s such that u · | i = [ [ . Choose a finite subset α ⊂ S K containing all the endpoints of theintervals J i and let α : Z → K be such that π K ( α ( Z )) = α . There is a canonical embedding U υ ( sl ( S α ( Z ) )) → U υ ( sl ( S K )) whose image contains u . Moreover, U υ ( sl ( S α ( Z ) )) is isomorphic to U υ ( b sl ( N )) , where N = | α | , and the restriction of F K to U υ ( sl ( S α ( Z ) )) contains the Fock space F α ( Z ) as the subspace spanned by all pyramids p satisfying D K ( p ) ∈ α ( Z ) . Hence it is enoughto check that | [ [ i 6∈ U υ ( sl ( S α ( Z ) )) · | i or equivalently that the element | λ i with λ = ( N ) does not belong to the subspace U υ ( b sl ( N )) · | i in Hayashi’s Fock space. This last statement may bechecked by some simple direct calculations which we leave to the reader. (cid:3) Remark . The Fock space is a cyclic representation of the Hall algebra H S K . This follows byreduction to the case of cyclic quivers which is treated in [VV99]. △ Highest weight vectors.
The vacuum vector | i is an obvious highest weight vector. Some-what surprisingly, it turns out to be the only highest weight vector in F K when K = Q or K = R . Proposition 6.4.
Assume that K = Q or K = R . We have ( F K ) U + υ ( sl ( S K )) = e Q | i . Proof.
Let v : = ∑ p α p | p i be a highest weight vector. By homogeneity we may assume that all p for which α p = s . Assume that s = v e Q | i ). Because α p isnonzero for only finitely many pyramids p , the set D : = [ p , α p = D K ( p ) is nonempty and finite. Set x = max ( D ) . Choose some small ǫ > [ x − ǫ , x ] ∩ D = { x } .Let p be a pyramid for which α p = x ∈ D K ( p ) (hence the support of p is an interval of theform [ a , x ] ). Set J = [ x − ǫ , x ) . By construction, J is removable from p . Moreover p is the onlypyramid which can contribute to p − J in E J · v ; this follows from condition (a’) in the definitionof the action of E J (cf. Theorem 6.1) and from the fact that x − ǫ D K ( q ) for any pyramid q occuring in v . We deduce that h p − J | E J · v i = h p − J | E J · p i 6 =
0, which is in contradiction withthe assumption on v . (cid:3) Remark . Hayashi’s Fock space decomposes into a direct sum of highest weight representa-tions. In [S00], these highest weight vectors are obtained by the action of the center of the Hallalgebra on the vacuum vector. In the setting of the continuum quivers Q / Z or R / Z such a cen-ter only appears in a suitable completion (see [SS17]). This suggests that a better object to studywould be a similar completion of our Fock space. We hope to return to this in the future. △ R EFERENCES[AKSS19] A. Appel, T. Kuwagaki, F. Sala, and O. Schiffmann,
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