A basis theorem for the affine Kauffmann category and its cyclotomic quotients
aa r X i v : . [ m a t h . R T ] J un A BASIS THEOREM FOR THE AFFINE KAUFFMANN CATEGORY AND ITSCYCLOTOMIC QUOTIENTS
MENGMENG GAO, HEBING RUI, LINLIANG SONG
Abstract.
The affine Kauffmann category is a strict monoidal category and can be considered asa q -analogue of the affine Brauer category in (Rui et al. in Math. Zeit. 293, 503-550, 2019). In thispaper, we prove a basis theorem for the morphism spaces in the affine Kauffmann category. Thecyclotomic Kauffmann category is a quotient category of the affine Kauffmann category. We alsoprove that any morphism space in this category is free over an integral domain K with maximal rankif and only if the u -admissible condition holds in the sense of Definition 1.13. Introduction
In the last several decades, one of important developments in representation theory is a categorifi-cation of modules of Lie algebras or quantum groups. For example, Ariki’s categorification theorem [1]establishes relationships between various important stories (blocks, irreducible modules, projective in-decomposable modules, etc) in the representation theory of cyclotomic Hecke algebras and importantinvariants (weight spaces, dual canonical basis, canonical basis, etc) of certain irreducible modulesof Lie algebras sl ∞ or b sl p . In [8], Brundan et. al introduced the affine oriented Brauer category andthe cyclotomic oriented Brauer category. A special case of the cyclotomic oriented Brauer category isthe oriented Brauer category. Associated to it, there is a locally unital and locally finite dimensionalalgebra. Reynolds [30] constructed a categorical action of the Lie algebra sl ∞ or b sl p on the Grothen-dick group of certain module category for this locally unital and locally finite dimensional algebra.Since oriented Brauer category is the category version of the walled Brauer algebra [21], Reynolds’result can be considered as a categorification related to the walled Brauer algebra. See also [7] for theoriented skein category which is the category version of the quantized walled Brauer algebra [22].The paper is a part of our project for studying categorifications related to finite dimensional algebrasarising from Schur-Weyl dualities in types B, C and D . In [32], two of authors introduced the affineBrauer category and the cyclotomic Brauer category. The affine Nazarov-Wenzl algebras [27] and thecyclotomic Nazarov-Wenzl algebras [2] appear as endomorphism algebras of objects in these categories,respectively. Furthermore, they establish a higher Schur-Weyl duality between cyclotomic Brauercategories and BGG parabolic category O in types B, C and D . This enables them to computedecomposition matrices of the cyclotomic Nazarov-Wenzl algebra over the complex field C . A specialcase of the cyclotomic Brauer category is the Brauer category [23](whose additive Karoubi envelope isthe Deligne category Rep O ω [13, 14]). For the Brauer category, further results have been obtained.More explicitly, using the representation theory of Brauer category, they [34] gave a categorificationrelated to the Brauer algebra in [6]. In this picture, certain modules of coideal algebras in [3] comeinto picture. Thanks to certain exact truncation functors, representations of Brauer algebras can bereflected in the representations of the Brauer category. Mathematics Subject Classification.
Key words and phrases.
Affine Kauffmann category, cyclotomic Kauffmann category, quantum group of type
B/C/D ,basis theorem.M. Gao and H. Rui is supported partially by NSFC (grant No. 11971351). L. Song is supported partially by NSFC(grant No. 11501368).
Throughout this paper, K is an integral domain. We are going to introduce the affine Kauffmanncategory AK . It is a K -linear strict monoidal category generated by a single object . The category AK comes equipped with infinitely many algebraically independent parameters ∆ , ∆ , . . . . On evaluatingthese parameters at scalars in K , we obtain specialization AK ( ω ) and introduce the cyclotomic Kauff-mann category CK f associated to a monic polynomial f ( t ) ∈ K [ t ]. After we establish relationshipsbetween AK (resp., CK f ) and the category of endofunctors of the module category (resp., parabolicBGG category O ) associated to the quantum symplectic groups and quantum orthogonal groups, weare able to prove that any morphism space in AK is free over K and the affine Birman-Murakami-Wenzl algebra [18] appears as an endomorphism algebra End AK ( ω ) ( ⊗ r ) for some positive integer r .We also prove that any morphism space in CK f is free over K with maximal rank if and only if the u -admissible condition holds in the sense of Definition 1.13. In this case, the cyclotomic Birman-Murakami-Wenzl algebra [4, 28] appears as an endomorphism algebra End CK f ( ⊗ r ) for some positiveinteger r .In a subsequent work, we will investigate the representations of the cyclotomic Kauffmann categoryand the cyclotomic Brauer category. Based on previous observations, we believe that representationsof the cyclotomic Birman-Murakami-Wenzl algebra (resp., cyclotomic Nazarov-Wenzl algebra) can bereflected in the representation theory of cyclotomic Kauffmann category (resp., cyclotomic Brauercategory).In the remaining part of this section, we are going to introduce these categories and formulate ourmain results precisely.We start by recalling the definition of the category FT of framed tangles. It is the K -linear strictmonoidal category generated by a single object (see e.g., [35]). Thus, the set of objects in FT is { m | m ∈ N } , where m represents ⊗ m , and represents the unit object. For any objects m and s ,morphisms f : m → s are isotropy classes of framed tangles in [0 , × [0 , × R in 3-dimensional realspace R with boundary (cid:26) (1 − im + 1 , , | i = 1 , , . . . , m (cid:27) ∪ (cid:26) (1 − js + 1 , , | j = 1 , , . . . , s (cid:27) . Such tangles will be drawn by projecting them onto the xy -plane, and there are neither triple inter-sections nor tangencies. Further, any crossing of a tangle will be recorded as either over crossing orunder crossing. The resulting diagrams are called ( m, s )-tangle diagrams. Isotropy translates into theequivalence relation on diagrams generated by planar isotropy fixing the boundary together with theReidemeister Moves of types (RI)-(RIII) as follows:=(RI) , = (RII) = , =(RIII) . Tensor product of morphisms is given by horizontal concatenation and composition of morphisms isgiven by vertical stacking (in a strict monoidal category). For example, g ◦ h = gh , g ⊗ h = g h . Suppose K contains δ, δ − , z, ω . In this paper, we always assume δ − δ − = z ( ω − . (1.1) HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Let I be the tensor ideal of FT , which is uniquely determined by Kauffmann skein relation (S),twisting relation (T) and free loop relation (L) as follows: − =(S) z ( − ) , =(T) δ , =(L) ω . The Kauffmann category K is the quotient category FT /I [35]. In order to simplify notation, , , and will be denoted by U, A, T and T − , respectively. The following result givesthe presentation of the Kauffmann category K . Theorem 1.1. [35]
The Kauffmann category K is the strict K -linear monoidal category generated bya single object and four elementary morphisms U, A, T and T − subject to the relations as follows: (1) (1 ⊗ A ) ◦ ( T ⊗ ) ◦ (1 ⊗ U ) ◦ (1 ⊗ A ) ◦ ( T − ⊗ ) ◦ (1 ⊗ U ) = 1 , (2) T ◦ T − = T − ◦ T = 1 , (3) ( T ⊗ ) ◦ (1 ⊗ T ) ◦ (1 ⊗ T ) = (1 ⊗ T ) ◦ (1 ⊗ T ) ◦ ( T ⊗ ) , (4) T − T − = z (1 − U ◦ A ) , (5) (1 ⊗ A ) ◦ ( T ⊗ ) ◦ (1 ⊗ U ) = δ , (6) A ◦ U = ω , (7) (1 ⊗ A ) ◦ ( U ⊗ ) = 1 = ( A ⊗ ) ◦ (1 ⊗ U ) , (8) T = ( A ⊗ ) ◦ (1 ⊗ T − ⊗ ) ◦ (1 ⊗ U ) , (9) T − = ( A ⊗ ) ◦ (1 ⊗ T ⊗ ) ◦ (1 ⊗ U ) . Theorem 1.1(1)-(3) correspond to (RI)-(RIII), and Theorem 1.1(4)-(6) correspond to (S)-(L). Fur-ther, Theorem 1.1(7)-(9) are depicted as (1.2)-(1.4) as follows:= = , (1.2)= , (1.3)= . (1.4)We are going to state Turaev’s result [35] on bases of morphism spaces in K . Suppose m + s iseven. Following [18], endpoints at bottom (resp., top) row of an ( m, s )-tangle diagram d are labelledby 1 , , . . . , m (resp., 1 , , . . . , s ) from left to right. Then d decomposes { , , . . . , m, s, . . . , , } into m + s pairs conn( d ) = { ( i k , j k ) | ≤ k ≤ m + s } , called the ( m, s )-connector of d . Let conn ( m, s ) be theset of all ( m, s )-connectors. Later on, we always assume that i < i + 1 < j < j − i and j , and i k < j k and i k < i l whenever k < l . A strand connecting a pair on different rows (resp.,the same row) is called a vertical (resp., horizontal) strand. Moreover, a horizontal strand connectinga pair on the top (resp., bottom) row is also called a cup (resp., cap). For example, U is a cup and A is a cap.Motivated by [18, Definition 5.4], an ( m, s )-tangle diagram is said to be totally descending if it canbe traversed successively such that ( i k , j k ) passes over ( i l , j l ) whenever k < l and ( i k , j k ) crosses ( i l , j l )and if neither a strand crosses itself nor there is a loop. It is well-known (e.g. [18, Proposition 5.7])that d = e as morphisms in FT , if d, e are totally descending and conn ( d ) = conn ( e ) . (1.5)An ( m, s )-totally descending tangle diagram is said to be reduced, if two strands cross each otherat most once. Any ( m, s )-connector determines many reduced totally descending tangle diagrams. MENGMENG GAO, HEBING RUI, LINLIANG SONG
Thanks to (1.5), it is reasonable to denote one of such reduced totally descending tangle diagrams by D c for any connector c . Theorem 1.2. [35]
Suppose m, s ∈ N . (1) If m + s is odd, then Hom K ( m , s ) = 0 . (2) If m + s is even, then Hom K ( m , s ) has K -basis given by { D c | c ∈ conn ( m, s ) } . In particular,Hom K ( m , s ) is of rank ( m + s − . Motivated by Theorem 1.2 and our previous work on affine Brauer category in [32], we introduceaffine Kauffmann category whose objects are same as those of K . This is one of main objects in thispaper. Definition 1.3.
The affine Kauffmann category AK is the K -linear strict monoidal category gener-ated by a single object and six elementary morphisms U, A, T, T − , and X ± : → subject toTheorem 1.1(1)-(9) together with the following relations:(1) X ◦ X − = 1 = X − ◦ X ,(2) T ◦ ( X ⊗ ) ◦ T = 1 ⊗ X ,(3) A ◦ ( X ⊗ ) = A ◦ (1 ⊗ X − ) and ( X ⊗ ) ◦ U = (1 ⊗ X − ) ◦ U .In this paper, X and X − are drawn as • and ◦ , respectively. So, Definition 1.3(1)-(3) can bedepicted as (1.6)-(1.8) as follows: •◦ = = ◦• , (1.6) • = • , (1.7) • = ◦ , • = ◦ . (1.8) Lemma 1.4.
Suppose AK is the affine Kauffmann category. (1) There is a K -linear monoidal contravariant functor σ : AK → AK switching A and U andfixing
T, T − , X, X − . (2) There is a monoidal functor from K to AK sending the generators of K to the generators of AK with the same names.Proof. Easy exercise. (cid:3)
Thanks to Lemma 1.4(2), any tangle diagram can be interpreted as a morphism in AK . Lemma 1.5.
As morphisms in AK , we have: (1) = δ , = δ − , (2) = δ , = δ − , (3) ◦ = • , ◦ = • .Proof. We have = ◦ = ◦ = = δ − . One can prove the first equation in (1) similarly. Thanks to (1.2) and (1.8), we have • = ◦ = ◦ . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY So far, we have proved (1) and the first equation in (3). Applying the contravariant functor σ inLemma 1.4 on these equations yields (2) and the second equation in (3). (cid:3) A point on a strand of a tangle diagram is called a critical point if it is either an endpoint or apoint such that the tangent line at it is horizontal. A segment of a tangle diagram is defined to be aconnected component of the diagram obtained when all crossings and critical points are deleted. A dotted (m,s)-tangle diagram d is an ( m, s )-tangle diagram such that there are finitely many • ’s or ◦ ’son each segment of d . Both • and ◦ are called dots later on. If there are h • ’s (resp., ◦ ’s) on a segment,then such • ’s (resp., ◦ ’s) can be viewed as • ◦ . . . ◦ • (resp,. ◦ ◦ . . . ◦ ◦ ), and will be denoted by • h (resp., ◦ h ). In order to simplify notation, we say that h “ • ” is the same as − h “ ◦ ” if h <
0. In otherwords, • h = ◦− h and ◦ h = •− h if h < •• = ◦ • ◦ • = • = ◦− . Let T m,s be the set of all dotted ( m, s )-tangle diagram. By arguments similar to those in [32], any d in T m,s can be interpreted as a morphism in Hom AK ( m , s ). Conversely, if a diagram d is obtained bytensor product and composition of U, A, T, T − , X, X − together with such that there are m (resp., s ) endpoints on the bottom (resp., top) row, then d ∈ T m,s . So, Hom AK ( m , s ) is spanned by T m,s .For any i ∈ Z , let ∆ i = • i . (1.9)Then ∆ i ∈ End AK ( ) and ∆ = ω . By [16, Proposition 2.2.10], End AK ( ) is commutative and∆ i ◦ ∆ j = ∆ i ⊗ ∆ j for all admissible i, j . Lemma 1.6.
For any positive integer j , ∆ − j = δ − ∆ j + δ − z P j − i =1 (∆ i − j − ∆ i ∆ i − j ) .Proof. Thanks to Lemma 1.5(1), we have ◦ j = δ − ◦ j ( . ) = δ − • j = δ − •• j − ( . ) = δ − • • j − ( S ) = δ − • • j − + δ − z •• j − − δ − z • • j − ( . ) = δ − • • j − + δ − z ◦ j − − δ − z • ◦ j − = δ − ••• j − + δ − z ◦ j − − δ − z • ◦ j − = · · · = δ − • j + δ − z j − X i =1 ◦ j − i − δ − z j − X i =1 • ◦ i j − i = δ − • j + δ − z j − X i =1 ◦ j − i − δ − z j − X i =1 • ◦ i j − i , by Lemma 1.5(2). (cid:3) Suppose d ∈ T m,s . Throughout this paper, ˆ d is always the tangle diagram obtained from d byremoving all loops and all dots on it. MENGMENG GAO, HEBING RUI, LINLIANG SONG
Definition 1.7.
Suppose d ∈ T m,s . d is said to be normally ordered if(a) all of its loops are crossing-free, and there are no other strands shielding any of them from theleftmost edge of the picture,(b) there is no on d ,(c) there is no ◦ on each loop. All bullets on each loop will be at the leftmost boundary of it. Forexample, • is allowed but non of • , ◦ and • • are allowed,(d) ˆ d is a reduced totally descending tangle diagram,(e) whenever a dot ( • or ◦ ) appears on a vertical strand, it is on the boundary of the bottom row,(f) whenever a dot ( • or ◦ ) appears on a cap (resp., cup), it is on the leftmost boundary of thecap (resp., the rightmost boundary of a cup),(g) • and ◦ can not occur on the same strand.For any m, s ∈ N , let NT m,s = { d ∈ T m,s | d is normally ordered } . The following tangle diagramsrepresent two morphisms in Hom AK ( , ). The left one is in NT , whereas the right one is not. ◦• ◦•• , •◦ ••• . Definition 1.8.
For any d, d ′ ∈ NT m,s , write d ∼ d ′ if • they have the same number of • i , for any i ∈ N \ • conn( ˆ d ) = conn( ˆ d ′ ), • there are same number of • or ◦ on their corresponding strands.We have ˆ d = ˆ d ′ as morphisms in K and hence in AK if d, d ′ ∈ NT m,s and d ∼ d ′ . So d = d ′ as morphisms in AK . We will identify each equivalence class of NT m,s with any element in it. Thefollowing is the first main result of this paper. Theorem 1.9.
Suppose m, s ∈ N . (1) If m + s is odd, then Hom AK ( m , s ) = 0 . (2) If m + s is even, then Hom AK ( m , s ) has K -basis given by NT m,s / ∼ . In particular, Hom AK ( m , s ) is of infinite rank. Definition 1.10. [33, Definition 2.19] Suppose u , . . . , u a are units in K and ω i ∈ K , i ∈ Z . Let ω = { ω i | i ∈ Z } and u = { u , u , . . . , u a } . Then ω is called admissible if(1) ω − i = δ − ω i + δ − z P i − l =1 ( ω l − i − ω l ω l − i ), for any positive integer i ,(2) ω i = − P aj =1 b a − j ω i − j if i ≥ a , where b j = ( − a − j e a − j ( u ) and e i ( u ) is the i th elementarysymmetric function on u , u , . . . , u a . Definition 1.11.
Suppose f ( t ) = Q ai =1 ( t − u i ), where u , . . . , u a are units in K . For any admissible ω , define CK f = AK /I where I is the right tensor ideal generated by f ( • ) together with ∆ i − ω i for all i ∈ Z \ { } .We call CK f the cyclotomic Kauffmann category. It is available only if ω is admissible in the senseof Definition 1.10. In fact, Definition 1.10(1) follows from Lemma 1.6 and Definition 1.10(2) followsfrom the equation A ◦ ( X i − a ⊗ ◦ ( f ( X ) ⊗ ◦ U = 0in CK f for any i ≥ a . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Assumption 1.12.
Let K be an integral domain containing units q, q − q − , u , . . . , u a . We alwaysassume δ = α a Y i =1 u i , and ω = 1 + z − ( δ − δ − ) , where z = q − q − and α ∈ { , − } if a is odd and α ∈ {− q, q − } if a is even.We always keep the Assumption 1.12 when we talk about CK f over K . Later on, we denote q − q − by z q . Definition 1.13. [33, Lemma 2.28] Suppose u is an indeterminate. Then ω is called u -admissible if X i ≥ ω i u i = u u − − ( z q δ ) − + ( z q δ ) − a Y i =1 u i + ug a ( u ) u − ! a Y i =1 u i a Y i =1 u − u − i u − u i , X i ≥ ω − i u i = 1 u − z q δ ) − − ( z q δ ) − a Y i =1 u i − ug a ( u )( u − ! a Y i =1 u − i a Y i =1 u − u i u − u − i , where g a ( u ) = 1 (resp., − u ) if a is odd (resp., even).Recall that ⌊ ℓ ⌋ is the maximal integer such that ⌊ ℓ ⌋ ≤ ℓ , where ℓ is a real number. Definition 1.14.
For any m, s ∈ N , let NT m,s be the set of all d ∈ NT m,s such that there is no • i on d , ∀ i ∈ N . For any positive integer ℓ , let NT ℓm,s be the subset of all d ∈ NT m,s such that − i, j ∈ (cid:26) ⌊ ℓ − ⌋ , ⌊ ℓ − ⌋ − , . . . , −⌊ ℓ ⌋ (cid:27) if there are i (resp., j ) • ’s near each endpoint at top (resp., bottom) row of d .The equivalence relation ∼ in Definition 1.8 induces equivalence relations ∼ on both NT m,s and NT am,s . We have d = d ′ as morphisms in AK and hence in CK f if d, d ′ ∈ NT am,s and d ∼ d ′ . So, eachequivalence class in NT am,s can be identified with any element in it. Theorem 1.15 is the second mainresult of this paper. Theorem 1.15.
Keep the Assumption 1.12. Suppose m, s ∈ N . (1) If m + s is odd, then Hom CK f ( m , s ) = 0 . (2) If m + s is even, then Hom CK f ( m , s ) has K -basis given by NT am,s / ∼ if and only if ω is u -admissible. In this case, Hom CK f ( m , s ) is of rank a m + s ( m + s − . Connections to quantum symplectic and orthogonal groups
We follow the conventions in [5]. In this paper, g is always either a complex symplectic Lie algebra sp n or orthogonal Lie algebras so n , so n +1 . Let R be the root system associated to g with fixed simpleroots Π = { α , . . . , α n } such that α i = ε i − ε i +1 , 1 ≤ i ≤ n − α n = ε n (resp., 2 ε n , ε n − + ε n ) if g = so n +1 (resp., sp n , so n ). Then the set of positive roots R + = { ε i ± ε j | ≤ i < j ≤ n } ∪ Z g (2.1)where Z so n +1 = { ε i | ≤ i ≤ n } , Z sp n = { ε i | ≤ i ≤ n } , and Z so n = ∅ . From here onwards, wealways assume N = 2 n + 1 , (resp., 2 n ) if g = so n +1 (resp., g ∈ { so n , sp n } ), (2.2)and i ′ = N + 1 − i, ≤ i ≤ N. Then the half sum of positive roots ̺ = n X i =1 ̺ i ε i , (2.3) MENGMENG GAO, HEBING RUI, LINLIANG SONG where ̺ i = N − i + b g and b sp N = 1 and b so N = 0. Later on, we assume ̺ i ′ = − ̺ i ≤ i ≤ n and ̺ n +1 = 0 if g = so n +1 .On the real vector space spanned by R , there is an inner product ( | ) such that ( ε i | ε j ) = 2 δ i,j (resp., δ i,j ) if g = so n +1 (resp., otherwise). So,(2 ̺ | α i ) = ( α i | α i ) for any simple root α i . (2.4)The fundamental weights ̟ , . . . , ̟ n satisfy ( ̟ i | α j ) = δ ij d i , where d i = ( α i | α i ). The integralweight lattice P = ⊕ ni =1 Z ̟ i and the set of dominant integral weights P + = ⊕ ni =1 N ̟ i .If g = so n , let F = C ( q / ) where q / is an indeterminate. Otherwise, let F = C ( q / ). Later on, v = q / if g = so n +1 , and v = q if g = sp n , so n . (2.5)The quantum group U v ( g ) of simply connected type is the F -algebra generated by { x ± i , k λ | ≤ i ≤ n, λ ∈ P} subject to the relations in [25, § U v ( h ) be the subalgebragenerated by { k λ | λ ∈ P} . The quantum group U v ( g ) of adjoint type is the F -algebra generatedby { x ± i , k λ | λ ∈ Q} , where Q is the root lattice associated to g . Later on, we denote by Q + thepositive root lattice. It is known that U v ( g ) is a Hopf algebra and the comultiplication ∆, counit ε and antipode S satisfy∆( x + i ) = x + i ⊗ k α i ⊗ x + i , ε ( x + i ) = 0 , S ( x + i ) = − k − α i x + i , ∆( x − i ) = x − i ⊗ k − α i + 1 ⊗ x − i , ε ( x − i ) = 0 , S ( x − i ) = − x − i k α i , ∆( k µ ) = k µ ⊗ k µ , ε ( k µ ) = 1 , S ( k µ ) = k − µ . (2.6)For all u ∈ U v ( g ), let ∆( u ) = ( − ⊗ − ) ◦ ∆( u ) , (2.7)where is the C -linear automorphism of U v ( g ) such that x ± i = x i ± , k µ = k − µ and q / = q − / (resp., q / = q − / ) if g = so n (resp., otherwise). Then∆( x + i ) = x + i ⊗ k − α i ⊗ x + i , ∆( x − i ) = x − i ⊗ k α i + 1 ⊗ x − i , ∆( k µ ) = k µ ⊗ k µ , (2.8)for all admissible i and µ .In this paper, a U v ( g )-module M is always a left module. It is called a weight module if M = ⊕ λ ∈P M λ , where M λ = { m ∈ M | k µ m = v ( λ | µ ) m, for any µ ∈ P} , called the λ -weight space of M . Write wt ( m ) = λ if m ∈ M λ . It is easy to check that a weight U v ( g )-module is always a weight U v ( g )-module and vice versa. Further, a U v ( g )-homomorphism between two weight modules is alwaysa U v ( g )-homomorphism and vice versa. This enables us to use previous results on U v ( g ) in [31],directly.Let W be the Weyl group corresponding to R . It is a Coxeter group generated by { s i | ≤ i ≤ n } ,where s i is the simple reflection s α i . For any 1 ≤ i ≤ n , let T i = T ′′ i, +1 , the braid group generator defined by Lusztig in [25, § T w = T i · · · T i k if s i · · · s i k is a reduced expression of w .Throughout this paper, w is always the longest element in W . Let ℓ ( ) be the length functionon W . Fix a reduced expression ~ Q ℓ ( w ) j =1 s i j of w and define β = α i , β j = s i s i · · · s i j − ( α i j ) and d β j = ( β j | β j ), 1 ≤ j ≤ ℓ ( w ). Then R + = { β j | ≤ j ≤ ℓ ( w ) } HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY such that there is a convex ordering on R + [29] satisfying β j < β k whenever j < k and β i + β j = β k for some k, i < k < j if β i + β j ∈ R + . Motivated by [26], we consider root elements x ± β j = T i T i · · · T i j − ( x ± i j )for all admissible j . For any r ∈ N ℓ ( w ) , let x + r = ~ Y ℓ ( w ) i =1 ( x + β i ) r i and x − r = ~ Y i = ℓ ( w ) ( x − β i ) r i . (2.9)Then { x − r k µ x + s | r , s ∈ N ℓ ( w ) , µ ∈ P} forms a PBW-like basis of U v ( g ). Let U + v ( g ) (resp., U − v ( g )) bethe subalgebra of U v ( g ) with basis { x + r | r ∈ N ℓ ( w ) } (resp., { x − r | r ∈ N ℓ ( w ) } ). The Borel subalgebras U v ( b ) = U v ( h ) ⊗ F U + v ( g ) (resp., U v ( b − ) = U − v ( g ) ⊗ F U v ( h )).In [11, 24, 26] etc, root elements are defined via braid generators T i = T ′′ i, − in [25, § x ± β j ’s be the corresponding root elements. Checking their actions on generatorsyields T ′′ i, +1 = τ ′ T ′′ i, − τ ′ , ≤ i ≤ n, (2.10)where τ ′ is the F -linear automorphism of U v ( g ) such that τ ′ ( x ± i ) = − x ± i and τ ′ ( k µ ) = k µ for alladmissible µ and i . So, x ± β j = − τ ′ (ˆ x ± β j ) . (2.11)Thanks to [11, Theorem 9.5], ˆ x ± α i = x ± i for any simple root α i . So x ± α i = − τ ′ (ˆ x ± α i ) = x ± i . Let τ be the F -linear anti-automorphism of U v ( g ) such that τ ( x + i ) = x − i , τ ( x − i ) = x + i and τ ( k µ ) = k µ for all admissible i and µ . Let τ := − ◦ τ where is the C -linear automorphism of U v ( g ) in (2.7). Then τ is a C -linear anti-automorphism of U v ( g ). Checking the actions on generators yields τ T i = T i τ . So, τ ( x ± β i ) = x ∓ β i , for any β i ∈ R + . (2.12) Definition 2.1.
For any i ∈ Z , define A i = { x ∈ C ( q / ) | ev ( x ) ≥ i } where ev : C ( q / ) → Z suchthat ev (0) = ∞ and ev ( x ) = j if 0 = x = ( q / − j g/h and g, h ∈ C [ q / ] such that ( g, q / −
1) = 1,( h, q / −
1) = 1.Obviously, A i ⊂ A i − for any i ∈ Z and A i is a subring of C ( q / ) if i ∈ N . Lemma 2.2.
Suppose ≤ i < j ≤ ℓ ( w ) . Then (1) x − β i x − β j − v ( β i | β j ) x − β j x − β i = P r ∈ N ℓ ( w c − r x − r , (2) x + β j x + β i − v − ( β i | β j ) x + β i x + β j = P r ∈ N ℓ ( w c + r x + r ,such that c ± r ∈ A and P ℓ ( w ) l =1 r l β l = β i + β j . Moreover, c ± r = 0 unless r l = 0 for all l such that either l ≤ i or l ≥ j .Proof. By [24, Proposition 5.5.2],ˆ x + β j ˆ x + β i − v − ( β i | β j ) ˆ x + β i ˆ x + β j = X r ∈ N ℓ ( w ˆ c r ~ Y ℓ ( w ) i =1 (ˆ x + β i ) r i , (2.13)where ˆ c r = 0 unless r l = 0 for all l such that l ≤ i or l ≥ j , and P ℓ ( w ) l =1 r l β l = β i + β j . Recall τ ′ in(2.10). Acting τ ′ on both sides of (2.13) and using (2.11) yield the commuting relation about x + β j x + β i .In this case, c + r = ˆ c r up to a sign. In order to prove (2), we need to verify c + r ∈ A . Thank to [26, Theorem 6.7(ii)], ˆ c r = g ( v ) Q ℓ ( w ) j =1 ([ r j ] ! d βj ) − for some g ( v ) ∈ Z [ v, v − ], where[ i ] d = v di − v − di v d − v − d and [ k ] ! d = k Y i =1 [ i ] d .So, ˆ c r ∈ A , and hence c + r ∈ A . Note that τ ( A ) ⊆ A . Acting τ on both sides of the equation in(2) and using (2.12) yield (1). (cid:3) For any ordered sequence of positive roots I , we say that I is of length (resp., weight) j (resp., P jl =1 β i l ) and write ℓ ( I ) = j (resp., wt ( I ) = P jl =1 β i l ) if I = ( β i , β i , . . . , β i j ). In this case, define x ± I = ~ Q jl =1 x ± β il . If j = 0, write x ± I = 1. Corollary 2.3.
Suppose I = ( β i , β i , . . . , β i j ) . Then x − I can be written as an A -linear combinationof x − r , r ∈ N ℓ ( w ) .Proof. The result follows from Lemma 2.2, immediately. (cid:3)
Suppose ν, γ ∈ R + such that ν > γ with respect to a fixed convex order < . Following [9], ( ν, γ ) iscalled a minimal pair of β if β = ν + γ ∈ R + and if there is no ν ′ , γ ′ ∈ R + such that ν ′ + γ ′ = β and ν > ν ′ > β > γ ′ > γ . Let Υ β be the set of minimal pairs of β , β ∈ R + .Suppose ( ν, γ ) is a minimal pair of β . Following [9], define p ν,γ = 1 , (resp., 0) if ( d ν , d β , d γ ) = (1 , ,
1) (resp., otherwise). (2.14)Later on, we simply denote [ i ] by [ i ]. Lemma 2.4. ( [9, (4.4)] ) If ( ν, γ ) ∈ Υ β , β ∈ R + , then (1) x + γ x + ν − v ( ν | γ ) x + ν x + γ = [ p ν,γ + 1] x + β , (2) x − ν x − γ − v − ( ν | γ ) x − γ x − ν = [ p ν,γ + 1] x − β .Proof. (1) follows from [9, (4.4)] and the positive embedding in [9, p 1365]. Acting τ on (1) and using(2.12) yield (2). Finally, we remark that the current v is q − in [9]. (cid:3) So far, results in this section are available for any reduced expression of w . From here to the endof this section, we choose a reduced expression of w as follows. If g ∈ { sp n , so n +1 } , w = ~ Y ni =1 s i,n s n s n,i (2.15)where s i,j = s i s i +1 ,j if i < j and s i,i = 1 and s i,j = s i,j +1 s j if i > j . If g = so n , w = ~ Y n − i =1 s i,n s n s n − ,i , (resp., ~ Y n − i =1 s i,n s n s n − ,i · s n s n − ) if 2 | n (resp., 2 ∤ n ). (2.16)Then the corresponding convex orders on R + are given as follows: ε i − ε j < ε i − ε j +1 < ε i + ε k < ε i + ε k − < ε i +1 − ε i +2 , if g = so n , ε i − ε j < ε i − ε j +1 < ε i < ε i + ε k < ε i + ε k − < ε i +1 − ε i +2 , if g = so n +1 , ε i − ε j < ε i − ε j +1 < ε i < ε i + ε k < ε i + ε k − < ε i +1 − ε i +2 , if g = sp n , (2.17)for all admissible positive integers i, j, k such that i < j if ε i + ε j appears in (2.17). In this case, wehave root elements { x ± β j | ≤ j ≤ ℓ ( w ) } with respect to braid group generators T ′′ i, +1 .From here to the end of this paper, we keep using these β j ’s and x ± β j ’s. Unless otherwise specified,the convex order < is always the one in (2.17). Corollary 2.5.
Suppose i < j and ≤ l ≤ n − . Then (1) Υ ε l = { ( ε l + ε n , ε l − ε n ) } if g = sp n , (2) Υ ε l = { ( ε k , ε l − ε k ) | l < k ≤ n } if g = so n +1 , HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY (3) Υ ε i − ε j = { ( ε m − ε j , ε i − ε m ) | i < m < j } , (4) Υ ε i + ε j = { ( ε j − ε j +1 , ε i + ε j +1 ) } ∪ { ( ε l + ε j , ε i − ε l ) | i < l < j } , (5) Υ ε l + ε n = { ( ε k + ε n , ε l − ε k ) | ≤ l < k < n } ∪ B , where B = { ( ε n , ε l ) } (resp., { (2 ε n , ε l − ε n ) } , ∅ ) if g = so n +1 (resp., sp n , so n ).Proof. (1)-(5) can be verified directly via (2.17). (cid:3) Proposition 2.6.
For any r = ( r i ) ∈ N ℓ ( w ) , let c ± r be given in Lemma 2.2. Then c ± r ∈ A | r |− , where | r | = P i r i . Since we verify Proposition 2.6 case by case and could not find a conceptual proof, we will givedetails in Appendix A. We also give a proof of Proposition 2.7 in Appendix B. It involves tediouscomputation.
Proposition 2.7.
Suppose β ∈ R + . We have (1) ∆( x − β ) = x − β ⊗ k − β + 1 ⊗ x − β + P K,H h K,H x − K ⊗ k − wt ( K ) x − H , (2) ∆( x − β ) = x − β ⊗ k β + 1 ⊗ x − β + P K,H g K,H x − K ⊗ k wt ( K ) x − H ,where K, H range over non-empty sequences of positive roots such that wt ( K ) + wt ( H ) = β , and h K,H , g
K,H ∈ A max { ℓ ( K ) ,ℓ ( H ) } . Throughout this paper, V is always the natural U v ( g )-module and { v i | i ∈ N } is always a basis of V such that wt ( v i ) = − wt ( v i ′ ) = ε i , 1 ≤ i ≤ n and wt ( v n +1 ) = 0 if g = so n +1 , where N = { , , . . . , N } .The following result is well-known. Lemma 2.8.
As homomorphisms in
End( V ) , we have (1) x + i = E i,i +1 − E ( i +1) ′ ,i ′ , and x − i = E i +1 ,i − E i ′ , ( i +1) ′ if i = n , where E i,j ’s are matrix unitswith respect to { v i | i ∈ N } , (2) k µ = δ g , so n +1 E n +1 ,n +1 + P ≤ j ≤ n ( v ( µ | ε j ) E j,j + v − ( µ | ε j ) E j ′ ,j ′ ) for all admissible µ , (3) x + n = E n,n +1 − v − E n +1 ,n ′ , x − n = [2] E n +1 ,n − v − [2] E n ′ ,n +1 , if g = so n +1 , (4) x + n = E n,n ′ , x − n = E n ′ ,n , if g = sp n , (5) x + n = E n − ,n ′ − E n, ( n − ′ , x − n = E n ′ ,n − − E ( n − ′ ,n , if g = so n . We are going to give formulae on x ± β and τ ( x ± β ) as endomorphisms of V in Lemmas 2.9–2.12. Theidea is to use minimal pairs in Corollary 2.5 and Lemma 2.4 so as to use the corresponding formulaefor simple roots in Lemma 2.8. The computations are straightforward, hence we omit details. Lemma 2.9.
For all admissible i < j , (1) x + ε i − ε j = E i,j − q i − j +1 E j ′ ,i ′ , τ ( x + ε i − ε j ) = E j,i − q i − j +1 E i ′ ,j ′ , (2) x − ε i − ε j = E j,i − q j − i − E i ′ ,j ′ , τ ( x − ε i − ε j ) = E i,j − q j − i − E j ′ ,i ′ . Lemma 2.10.
Suppose g = so n +1 . For all admissible i < j and admissible k , we have (1) x + ε k = − q k − n − E n +1 ,k ′ + E k,n +1 , τ ( x + ε k ) = − q k − n + [2] E k ′ ,n +1 + [2] E n +1 ,k , (2) x − ε k = − q n − k + [2] E k ′ ,n +1 + [2] E n +1 ,k , τ ( x − ε k ) = − q n − k − E n +1 ,k ′ + E k,n +1 , (3) x + ε i + ε j = ( − n − j − q − [2] − E i,j ′ + ( − n − j q i + j − n − [2] − E j,i ′ , (4) τ ( x + ε i + ε j ) = ( − n − j − [2] q E j ′ ,i + ( − n − j [2] q i + j − n + E i ′ ,j , (5) x − ε i + ε j = ( − n − j − [2] q E j ′ ,i + ( − n − j [2] q n − i − j + E i ′ ,j , (6) τ ( x − ε i + ε j ) = ( − n − j − q − [2] − E i,j ′ + ( − n − j q n − i − j − [2] − E j,i ′ . Lemma 2.11.
Suppose g = sp n . For all admissible i < j and admissible k , we have (1) x +2 ε k = [2] − ( q k − n − + q k − n +1 ) E k,k ′ , τ ( x +2 ε k ) = [2] − ( q k − n − + q k − n +1 ) E k ′ ,k , (2) x − ε k = [2] − ( q n − k +1 + q n − k − ) E k ′ ,k , τ ( x − ε k ) = [2] − ( q n − k +1 + q n − k − ) E k,k ′ , (3) x + ε i + ε j = ( − n − j ( E i,j ′ + q i − j ′ E j,i ′ ) , τ ( x + ε i + ε j ) = ( − n − j ( E j ′ ,i + q i − j ′ E i ′ ,j ) , (4) x − ε i + ε j = ( − n − j ( E j ′ ,i + q j ′ − i E i ′ ,j ) , τ ( x − ε i + ε j ) = ( − n − j ( E i,j ′ + q j ′ − i E j,i ′ ) . Lemma 2.12.
Suppose g = so n . For all admissible i < j , we have (1) x + ε i + ε j = ( − n − j E i,j ′ + ( − n − j − q i − j ′ +2 E j,i ′ , (2) τ ( x + ε i + ε j ) = ( − n − j E j ′ ,i + ( − n − j +1 q i − j ′ +2 E i ′ ,j , (3) x − ε i + ε j = ( − n − j E j ′ ,i + ( − n − j +1 q j ′ − i − E i ′ ,j , (4) τ ( x − ε i + ε j ) = ( − n − j E i,j ′ + ( − n − j − q j ′ − i − E j,i ′ . The quasi-R-matrix Θ of U v ( g ) [25, § ~ Y j = ℓ ( w ) exp v − βj [(1 − v β j ) x − β j ⊗ x + β j ] , Θ = ~ Y ℓ ( w ) i =1 exp v βi [(1 − v − β i ) x − β i ⊗ x + β i ] , (2.18)where v β j = v d βj and exp v ± βj ( x ) = P ∞ k =0 v ± k ( k +1) β j x k [ k ] ! dβj . Thanks to [25, Theorem 4.1.2(a)], Θ isuniquely determined by (2.19) as follows:Θ = 1 ⊗ , and ∆( u )Θ = Θ∆( u ) (2.19)for all u ∈ U v ( g ). Write Θ = X β ∈Q + Θ β and Θ = X β ∈Q + Θ β , where both Θ β and Θ β are in U − v ( g ) − β ⊗ U + v ( g ) β and U ± v ( g ) ± β is the subspace of U ± v ( g ) spanned by { x ± I | wt ( I ) = β } . Thanks to [20, 7.1(2)–(3)],( ι ⊗ ι )(Θ β ) = Θ β , ( † ⊗ † )(Θ β ) = P (Θ β )for any β ∈ Q + , where P is the tensor flip and, ι (resp., † ) is the anti-automorphism (resp., automor-phism) of U v ( g ) fixing x ± i (resp., switching x + i and x − i ) and sending k µ to k − µ . Since τ = † ◦ ι ,( τ ⊗ τ )(Θ β ) = P (Θ β ) . Noting that ΘΘ = ΘΘ = 1 ⊗
1, we have ( τ ⊗ τ )(Θ β ) = P (Θ β ). So,Θ β = P ◦ ( τ ⊗ τ )(Θ β ) . (2.20) Proposition 2.13.
Let M be a U v ( g ) -module. For any = m ∈ M , Θ( m ⊗ v j ) = [1 ⊗ z q P i
If 1 ≤ j ≤ n , then wt ( v j ) = ε j . So Θ β acts on m ⊗ v j non-trivially only if either β = 0 or β = ε i − ε j for some i , 1 ≤ i < j . Thanks to (2.18), Θ = 1 ⊗ ε k − ε l − z q x − ε k − ε l ⊗ x + ε k − ε l = X r ≥ X I z rq x − I ⊗ x + I , for all 1 ≤ k < l ≤ n, (2.21)where I ranges over all sequences of positive roots ( ε i − ε i , . . . , ε i r − − ε i r ) such that i = k and i r = l . In particular, we have (2.21) for Θ ε i − ε j . Obviously, τ ( x − ε i − ε i ) v j = 0. Otherwise wt ( τ ( x − ε i − ε i ) v j ) = ε i − ε i + ε j , and i < i < j , a contradiction. So, P ◦ ( τ ⊗ τ )(RHS of (2.21))acts on m ⊗ v j as zero. By (2.20), we have the required formula on Θ( m ⊗ v j ) under the assumption1 ≤ j ≤ n .Now, we assume g = so n +1 and j = n + 1. So, wt ( v j ) = 0. In this case, Θ β acts on m ⊗ v n +1 non-trivially only if either β = 0 or β = ε l , 1 ≤ l ≤ n . Thanks to (2.18),Θ ε l − z q / x − ε l ⊗ x + ε l = X s>l X wt ( J )= ε s a s,J x − ε l − ε s x − J ⊗ x + ε l − ε s x + J , for some a s,J ∈ F . (2.22)Note that τ ( x − ε l − ε s ) v n +1 = 0. So, P ◦ ( τ ⊗ τ )(RHS of (2.22)) acts on m ⊗ v n +1 as zero. Using (2.20)again, one immediately has the required formula on Θ( m ⊗ v n +1 ).Suppose n ′ ≤ j ≤ ′ . Since wt ( v j ) = − ε j ′ , Θ β acts on m ⊗ v j non-trivially only if β and g satisfyone of conditions as follows: • β ∈ { , ε d + ε j ′ , ε j ′ − ε t | ≤ d ≤ n, j ′ < t ≤ n } , and g = so n +1 , • β ∈ { , ε d + ε j ′ , ε j ′ − ε t | ≤ d ≤ n, j ′ < t ≤ n } ∪ { ε j ′ } , and g = so n +1 .We have (2.21) for Θ ε j ′ − ε t . Since x + ε ir − − ε ir v j = 0, the RHS of (2.21) acts on m ⊗ v j as zero. So,Θ ε j ′ − ε t ( m ⊗ v j ) = z q x − ε j ′ − ε t ⊗ x + ε j ′ − ε t ( m ⊗ v j ) . Thanks to (2.18), we haveΘ ε c = z q X k>c x − ε c − ε k x − ε c + ε k ⊗ x + ε c − ε k x + ε c + ε k + δ g , so n +1 q [2] − z q / ( x − ε c ) ⊗ ( x + ε c ) + X c
1) = (1 ⊗ R M ,M ) ◦ ( R M ,M ⊗ ◦ (1 ⊗ R M ,M ) . (2.29)Similarly, (2.28)-(2.29) are still available if two of M , M , M are finite-dimensional weight modules.By Proposition 2.13 and (2.27), one can check that: π − ◦ Θ | V ⊗ = X i = i ′ ( qE i,i ⊗ E i,i + q − E i,i ⊗ E i ′ ,i ′ ) + X i = j,j ′ E j,j ⊗ E i,i + z q X i 1. The following resultfollows from (2.30) directly. Lemma 2.14. Let E : V ⊗ → V ⊗ be such that E ( v k ⊗ v l ) = δ k,l ′ P N i =1 q ̺ i ′ − ̺ k ς i ′ ς k v i ⊗ v i ′ for alladmissible k and l . (1) If either g = so n +1 or g = so n +1 and ( k, l ) = ( n + 1 , n + 1) , then R − V,V ( v k ⊗ v l ) = qv k ⊗ v k , if k = l , v l ⊗ v k , if k > l and k = l ′ , q − v l ⊗ v k − z q P i>k q ̺ i − ̺ k ς i ς k v i ′ ⊗ v i , if k > l and k = l ′ , v l ⊗ v k + z q v k ⊗ v l , if k < l and k = l ′ , q − v l ⊗ v k + z q ( v k ⊗ v l − P i>k q ̺ i − ̺ k ς i ς k v i ′ ⊗ v i ) , if k < l and k = l ′ . (2) If g = so n +1 , then R − V,V ( v n +1 ⊗ v n +1 ) = v n +1 ⊗ v n +1 − z q P i>n +1 q ̺ i v i ′ ⊗ v i .In any case, R − V,V − R V,V = z q (1 − E ) . In fact, the left action of R − V,V on V ⊗ is the same as the right action of ˇ R on V ⊗ in [31, Lemma 2.4].This makes us to use results in [31], freely. Lemma 2.15. Let α : V ⊗ → F and β : F → V ⊗ be two linear maps such that α ( v k ⊗ v l ) = δ k,l ′ q − ̺ k ς k for all admissible k and l , and β (1) = P N i =1 q ̺ i ′ ς i ′ v i ⊗ v i ′ . Then both α and β are U v ( g ) -homomorphisms, where F is considered as the trivial U v ( g ) -module given by the counit.Proof. We have explained that a weight U v ( g )-module is a weight U v ( g )-module and vice versa. Sinceboth V ⊗ and F are weight U v ( g )-module, we can use corresponding results in [31]. HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Thanks to [31, Coro. 2.3], β is a U v ( g )-homomorphism and Im β ∼ = F as U v ( g )-modules. Thecorresponding isomorphism is denoted by β ′ . Let be the embedding Im β ֒ → V ⊗ . Then β = ◦ β ′ .Recall E in Lemma 2.14. It is proved in [31] that E is a U v ( g )-homomorphism and hence a U v ( g )-homomorphism. Further, E = β ◦ α = ◦ β ′ ◦ α . So β ′ ◦ α is a U v ( g )-homomorphism, forcing α to bea U v ( g )-homomorphism, too. (cid:3) In Propositions 2.16-2.17, we assume K and AK are defined over F with defining parameters δ, z and ω such that z = z q and δ = ε g q N − ε g (2.31)where ε g = − g = sp n and 1, otherwise. Since z q is invertible in F , ω is uniquely determined by(1.1). Proposition 2.16. Let U v ( g ) -mod be the category of all U v ( g ) -modules. There is a monoidal functor Φ : K → U v ( g ) -mod such that (1) Φ( ) = F and Φ( ) = V , (2) Φ( A ) = α , Φ( U ) = β , Φ( T ) = R − V,V and Φ( T − ) = R V,V ,where α and β are given in Lemma 2.15.Proof. When g = sp n , this result was given in [37]. We need to deal with so n and so n +1 soas to study the dualities between cyclotomic Birman-Murakami-Wenzl algebras and parabolic BGGcategory O in types B, C and D in the future. For this reason, we give a sketch of proof.Thanks to results on Schur-Weyl duality between U v ( g ) and Birman-Murakami-Wenzl algebrasin [31], we need only to verify relations given in Theorem 1.1(1),(5)-(9). In [31], two of us verified( Id V ⊗ E ) ◦ ( R ∓ V,V ⊗ Id V ) ◦ ( Id V ⊗ E ) = δ ± ( Id V ⊗ E ) and ( Id V ⊗ E ) = ω ( Id V ⊗ E ) (2.32)as U v ( g )-homomorphism. So, for any 1 ≤ j ≤ N, we have( Id V ⊗ E ) ◦ ( R ∓ V,V ⊗ Id V ) ◦ ( Id V ⊗ E )( v j ⊗ v ⊗ v ′ ) = δ ± ( Id V ⊗ E )( v j ⊗ v ⊗ v ′ ) . Since E = β ◦ α and β is injective, we have( Id V ⊗ α ) ◦ ( R ∓ V,V ⊗ Id V ) ◦ ( Id V ⊗ β )( v j ⊗ 1) = δ ± ( v j ⊗ , ≤ j ≤ N.This verified Theorem 1.1(1),(5). Similarly, Theorem 1.1(6) can be verified via the second equationin (2.32). It is not difficult to verify( Id V ⊗ α )( β ⊗ Id V )( v j ) = v j = ( α ⊗ Id V )( Id V ⊗ β )( v j ) , ≤ j ≤ N.This implies Theorem 1.1(7). Thanks to Lemma 2.14, we can check that( α ⊗ Id V ) ◦ ( Id V ⊗ R V,V ) = ( Id V ⊗ α ) ◦ ( R − V,V ⊗ Id V ) (2.33)as operators acting on V ⊗ . For example, when k = l ′ = j and k = l , we have( α ⊗ Id V ) ◦ ( Id V ⊗ R V,V )( v j ⊗ v k ⊗ v l ) = ( Id V ⊗ α ) ◦ ( R − V,V ⊗ Id V )( v j ⊗ v k ⊗ v l ) = q − ̺ j ς j v k . Now, Theorem 1.1(8) follows from Theorem 1.1(7) and (2.33). Finally, one can verify Theorem 1.1(9)in a similar way. (cid:3) Proposition 2.17. Let E nd ( U v ( g ) -mod ) be the category of endofunctors of U v ( g ) -mod. Then thereis a strict monoidal functor Ψ : AK → E nd ( U v ( g ) -mod ) such that (1) Ψ( ) = Id and Ψ( ) = − ⊗ V , (2) Ψ( Y ) M = Id M ⊗ Φ( Y ) , Ψ( X ) M = δR − M,V R − V,M , Ψ( X − ) M = δ − R V,M R M,V for all Y ∈ { A, U, T, T − } and any U v ( g ) -module M , where Φ is given in Proposition 2.16 and δ is given in (2.31) . Proof. Thanks to Proposition 2.16, it is enough to verify Definition 1.3(1)-(3). First of all, Defini-tion 1.3(1) follows since Ψ( X ) M Ψ( X − ) M = Ψ( X − ) M Ψ( X ) M = Id M . By (2.28), and R M,N R − M,N = Id N ⊗ M , we have R − M ⊗ V,V R − V,M ⊗ V = ( Id M ⊗ R − V,V ) ◦ ( R − M,V ⊗ Id V ) ◦ ( R − V,M ⊗ Id V ) ◦ ( Id M ⊗ R − V,V ) . Therefore, Ψ(1 ⊗ X ) M = Ψ( T ◦ ( X ⊗ ) ◦ T ) M , (2.34)proving Definition 1.3(2).Let L ( µ ) be the irreducible highest weight U v ( g )-module with highest weight µ . Then V ∼ = L ( ε ), F ∼ = L (0) and V ⊗ V ∼ = L (0) ⊕ L (2 ε ) ⊕ L ( ε + ε ). Note that α : V ⊗ ։ L (0) is the projection. So,Ψ( A ◦ ( X ⊗ ) ◦ (1 ⊗ X )) M = (Ψ( A ) ◦ Ψ( X ⊗ ) ◦ Ψ(1 ⊗ X )) M =(Ψ( A ) ◦ Ψ( X ⊗ ) ◦ Ψ( T ) ◦ Ψ( X ⊗ ) ◦ Ψ( T )) M , by (2.34)=(Ψ( A ) ◦ Ψ( T ) ◦ Ψ( X ⊗ ) ◦ Ψ( T ) ◦ Ψ( X ⊗ )) M , by (2.28)–(2.29)= δ − (Ψ( A ) ◦ Ψ( X ⊗ ) ◦ Ψ( T ) ◦ Ψ( X ⊗ )) M , by Lemma 1.5(1) and Proposition 2.16= δ ( Id M ⊗ α ) ◦ ( R − M,V R − V,M ⊗ Id V ) ◦ ( Id M ⊗ R − V,V ) ◦ ( R − M,V R − V,M ⊗ Id V ) M ⊗ V ⊗ = δ ( Id M ⊗ α ) ◦ ( R − V,V ⊗ Id M ) ◦ ( R − M,V ⊗ Id V ) ◦ ( Id V ⊗ R − M,V ) ◦ ( Id V ⊗ R − V,M ) ◦ ( R − V,M ⊗ Id V ) M ⊗ V ⊗ , by (2.29)= δ (( Id M ⊗ α ) ◦ ( R − V,V ⊗ Id M ) ◦ R − M,V ⊗ R − V ⊗ ,M ) M ⊗ V ⊗ , by (2.28)=(( Id M ⊗ α ) ◦ R − M,V ⊗ R − V ⊗ ,M ) M ⊗ V ⊗ , by Lemma 1.5(1) and Proposition 2.16=( Id M ⊗ α ) M ⊗ V ⊗ , since φ stabilizes M ⊗ L ( η ) and acts on M ⊗ L (0) as scalar 1,=Ψ( A ) M , where φ = R − M,V ⊗ R − V ⊗ ,M and η ∈ { , ε , ε + ε } . This proves the first equation in Definition 1.3(3).Finally, one can check the second equation in Definition 1.3(3) in a similar way. (cid:3) Connections to category O We consider O , the subcategory of U v ( g )-mod whose objects M satisfy the following conditions:(1) M = ⊕ µ ∈P M µ and dim M µ < ∞ ,(2) there are finitely many weights λ , λ , . . . , λ t ∈ P such that µ ∈ λ i − Q + for some i if µ is aweight of M .Obviously, O is closed under tensor product.Recall that Φ = P β ∈Q + Φ β , where Φ ∈ { Θ , Θ } and Φ β ∈ U − v ( g ) − β ⊗ U + v ( g ) β . Thanks toΘΘ = ΘΘ = 1 ⊗ u ) = ∆( u )Θ (3.1)for all u ∈ U v ( g ). By (2.18),Θ = X β ∈Q + X wt ( J )= β z ℓ ( J ) v g J x − J ⊗ x + J , Θ = X β ∈Q + X wt ( J ′ )= β z ℓ ( J ′ ) v h J ′ x − J ′ ⊗ x + J ′ , (3.2)where J (resp., J ′ ) ranges over weakly increasing (resp., decreasing) sequences of positive roots and g J , h J ′ ∈ A \ A (see Definition 2.1). In particular, g J = h J ′ = 1 when wt ( J ) = wt ( J ′ ) = 0.Let m be the multiplication map of U v ( g ). For any J in (3.2), let b J = z ℓ ( J ) v g J . We keep using b J ’s in Lemmas 3.1-3.3 as follows. Lemma 3.1. Suppose M ∈ O . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY (1) For any m ∈ M λ , define σ M ( m ) = v − ( λ | λ +2 ̺ ) m ( Id ⊗ S )(Θ) m , where S is the antipode of U v ( g ) in (2.6) and ̺ is given in (2.3) . Then σ M : M → M is a U v ( g ) -homomorphism. (2) σ M = v − ( λ | λ +2 ̺ ) Id M if M is a highest weight U v ( g ) -module with the highest weight λ .Proof. Since Θ∆( x ± i ) = ∆( x ± i )Θ, we have m ( Id ⊗ S )(Θ∆( x ± i )) = m ( Id ⊗ S )(∆( x ± i )Θ). Thanks to(2.6) and (3.2), it is routine to check m ( Id ⊗ S )(Θ∆( x ± i )) = 0. Using this fact and (2.8) yields X ν ∈Q + X wt ( J )= ν b J x + i x − J S ( x + J ) = X ν ∈Q + X wt ( J )= ν b J x − J S ( x + J ) k − α i x + i , X ν ∈Q + X wt ( J )= ν b J x − i x − J S ( x + J ) = X ν ∈Q + X wt ( J )= ν b J x − J S ( x + J ) k α i x − i . (3.3)Suppose m ∈ M λ . By (2.4) and (3.3), we have x + i σ M ( m ) = v − ( λ | λ +2 ̺ ) x + i m ( Id ⊗ S )(Θ) m = v − ( λ | λ +2 ̺ ) X ν ∈Q + X wt ( J )= ν b J x − J S ( x + J ) k − α i x + i m = v − ( λ | λ +2 ̺ ) v − (2 α i | λ + α i ) m ( Id ⊗ S )(Θ) x + i m = v − ( λ + α i | λ + α i +2 ̺ ) m ( Id ⊗ S )(Θ) x + i m = σ M ( x + i m ) . Similarly, one can verify x − i σ M ( m ) = σ M ( x − i m ). Finally, k η σ M ( m ) = σ M ( k η m ) for any η ∈ P since k η m ( Id ⊗ S )(Θ) = m ( Id ⊗ S )(Θ) k η . This completes the proof of (1). Under the assumption in (2), σ M is a scalar map. Since σ M ( m λ ) = v − ( λ | λ +2 ̺ ) X ν ∈Q + X wt ( J )= ν b J x − J S ( x + J ) m λ = v − ( λ | λ +2 ̺ ) m λ , where m λ is the maximal vector of M , we have (2). (cid:3) Suppose u = P x ⊗ y ∈ U v ( g ) ⊗ . For all 1 ≤ i < j ≤ r , define u j,ir = P ( u ) i,jr where u i,jr = P ⊗ i − ⊗ x ⊗ j − i − ⊗ y ⊗ r − j . Later on, we use u i,j (resp., u j,i ) to replace u i,jr (resp., u j,ir ) if we know r from the context. Lemma 3.2. Suppose ν ∈ Q + . (1) (∆ ⊗ Id )(Θ ν ) = P ν ′ + ν ′′ = ν Θ , ν ′ (1 ⊗ k ν ′′ ⊗ , ν ′′ , where ν ′ , ν ′′ ∈ Q + . (2) ( Id ⊗ ∆)(Θ ν ) = P ν ′ + ν ′′ = ν Θ , ν ′ (1 ⊗ k − ν ′′ ⊗ , ν ′′ , where ν ′ , ν ′′ ∈ Q + . (3) ( Id ⊗ ∆)(Θ ν ) = P ν ′ + ν ′′ = ν Θ , ν ′ (1 ⊗ k ν ′ ⊗ , ν ′′ , where ν ′ , ν ′′ ∈ Q + . (4) P ν ′ + ν ′′ = ν (∆ ⊗ Id )(Θ ν ′ )Θ , ν ′′ = P ν ′ + ν ′′ = ν ( Id ⊗ ∆)(Θ ν ′ )Θ , ν ′′ , where ν ′ , ν ′′ ∈ Q + . (5) P µ,η ∈Q + P J,H b J b H x − J x − H ⊗ k µ x + H S ( x + J ) = 1 ⊗ for all possible J, H such that wt ( J ) = µ and wt ( H ) = η . (6) ∆( u ) = P µ ∈Q + P wt ( J )= µ b J ∆( x − J )( S ⊗ S )∆ op ( x + J ) , where u = m ( Id ⊗ S )(Θ) and ∆ op ( x ) = (∆( x )) , for all x ∈ U v ( g ) .Proof. Acting ( − ⊗ − ⊗ − ) on both sides of equations in [25, Proposition 4.2.2-4.2.4] yields (1)-(4).By (3), LHS of (5) = ( Id ⊗ m )( Id ⊗ ⊗ S )( Id ⊗ ∆)(Θ) = ( Id ⊗ ε )(Θ) = 1 ⊗ , where ε is the counit of U v ( g ). Finally,RHS of (6) = X µ ∈Q + X wt ( J )= µ b J ∆( x − J )∆( S ( x + J )) = ∆( u ) , where the first equality follows from the well-known equality ∆ ◦ S = ( S ⊗ S ) ◦ ∆ op . (cid:3) The following is the counterpart of [15, (3.4)]. The proof is motivated by Drinfeld’s arguments. Lemma 3.3. Suppose M, N are two objects in O . As morphisms in End( M ⊗ N ) , P π Θ π − P Θ∆( m ( Id ⊗ S )(Θ)) = m ( Id ⊗ S )(Θ) ⊗ m ( Id ⊗ S )(Θ) . Proof. Suppose y , . . . , y ∈ U v ( g ). Motivated by Drinfeld’s formula in [15, line -4, Page 34], we define( y ⊗ y ⊗ y ⊗ y ) × ( y ⊗ y ) = y y S ( y ) ⊗ y y S ( y ) . It is routine to check that( z ⊗ z ⊗ z ⊗ z ) × [( y ⊗ y ⊗ y ⊗ y ) × ( x ⊗ x )] = ( z y ⊗ z y ⊗ z y ⊗ z y ) × ( x ⊗ x ) , (3.4)for any x i , y j , z k ∈ U v ( g ) for all admissible i, j, k .Suppose u = m ( Id ⊗ S )(Θ) and Θ = P ν ∈Q + (1 ⊗ k ν )Θ ν ( k − ν ⊗ P Θ P = Θ , in End( M ⊗ N ). By (2.27), Θ = π Θ π − in End( M ⊗ N ). So, P π Θ π − P Θ∆( u ) =Θ , X ν ∈Q + X wt ( J )= ν b J (Θ∆( x − J ))( S ⊗ S )∆ op ( x + J ) , by Lemma 3.2(6)=Θ , X ν ∈Q + X wt ( J )= ν b J ∆( x − J )Θ( S ⊗ S )∆ op ( x + J ) , by (3.1)=[Θ , (∆ ⊗ ∆ op )(Θ)] × Θ . So, we need to verify [Θ , (∆ ⊗ ∆ op )(Θ)] × Θ = u ⊗ u in End( M ⊗ N ). This is the case since[Θ , (∆ ⊗ ∆ op )(Θ)] × Θ=[Θ , X λ,λ ′ ,η,η ′ Θ , η (1 ⊗ k η ′ ⊗ ⊗ )Θ , η ′ (1 ⊗ ⊗ k η + η ′ )Θ , λ (1 ⊗ k λ ′ ⊗ ⊗ )Θ , λ ′ ] × Θ=[Θ , X λ,η,η ′ Θ , η (1 ⊗ k η ′ ⊗ ⊗ )Θ , η ′ (1 ⊗ ⊗ k η + η ′ )Θ , λ ] × (1 ⊗ , by (3.4), Lemma 3.2(5)=[Θ , X η,η ′ Θ , η (1 ⊗ k η ′ ⊗ ⊗ )Θ , η ′ (1 ⊗ ⊗ k η + η ′ )] × (1 ⊗ u ) , by (3.4)= X β,η,η ′ X wt ( J )= β X wt ( J )= η X wt ( J )= η ′ b J b J b J k β x + J x − J S ( x + J x + J ) ⊗ x − J k − β x − J k η ′ uS ( k η + η ′ )= X β,η,η ′ X wt ( J )= β X wt ( J )= η X wt ( J )= η ′ b J b J b J k β x + J x − J S ( x + J ) S ( x + J ) ⊗ x − J k − β x − J k − η u = π X β,η,η ′ X wt ( J )= β X wt ( J )= η X wt ( J )= η ′ b J b J b J x + J k − η x − J S ( x + J ) S ( x + J ) ⊗ x − J x − J u π − = π [ X β,η,η ′ Θ , β ( k − η ⊗ ⊗ , η Θ , η ′ ] × (1 ⊗ u ) π − = π [ X β,η,η ′ Θ , η ′ ( k β ⊗ ⊗ , β Θ , η ] × (1 ⊗ u ) π − , by Lemma 3.2(1)–(2),(4)= π X η ′ Θ , η ′ × ( X β,η X wt ( J )= β X wt ( J )= η b J b J k β x + J S ( x + J ) ⊗ x − J x − J u ) π − = π X η ′ Θ , η ′ × (1 ⊗ u ) π − , by Lemma 3.2(5)= π ( u ⊗ u ) π − = u ⊗ u, since u stabilizes any weight space . We remark that the first equality follows from Lemma 3.2(1),(3), and the fifth equality follows fromthe equation k ν u = uk ν for any ν ∈ P , and the sixth equality can be checked by direct calculation. (cid:3) HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Proposition 3.4. For any two objects M and N in O , σ M ⊗ N = (Θ πP ) σ M ⊗ σ N .Proof. Suppose ( m, n ) ∈ ( M λ , N µ ) and u = m ( Id ⊗ S )(Θ). Note that P π = πP in End( M ⊗ N ).Using Lemma 3.1 and (2.27), we have(Θ πP ) σ M ( m ) ⊗ σ N ( n ) = Θ πP Θ P v − ( λ | µ ) v − ( λ | λ +2 ̺ ) v − ( µ | µ +2 ̺ ) um ⊗ un = v − ( λ + µ | λ + µ +2 ̺ ) Θ πP Θ P π − um ⊗ un = v − ( λ + µ | λ + µ +2 ̺ ) ∆( u )( m ⊗ n )= σ M ⊗ N ( m ⊗ n ) , where the third equality follows from Lemma 3.3 and ΘΘ = 1 ⊗ (cid:3) Recall that Π is the set of simple roots and R is the root system. Fix positive integers q , . . . , q k such that P ki =1 q i = n . Define I = Π \ { α p , α p , . . . , α p k } , and I = I ∪ { α n } (3.5)where p i = P ij =1 q j for all admissible i .For each I ∈ { I , I } , there is a subroot system R I = R ∩ Z I . The corresponding positive roots R + I = R + ∩ Z I . Let w ,I be the longest element in the standard parabolic subgroup W I of W withrespect to I . Then w = w ,I y where y is a distinguished right coset representative in W I \ W .As in [19], we fix reduced expressions of w ,I and y so as to get a corresponding reduced expressionof w . We have the corresponding convex order of R + . To distinguish from the convex order in(2.17), we denote it by ˇ β < ˇ β < ... < ˇ β ℓ ( w ) . In this case, we also denote the root elements whichare defined via braid generators T i = T ′′ i, − in [25, § x ˇ β j ’s. The parabolic subalgebra U v ( p I )is generated by { x − i | α i ∈ I } ∪ { x + i , k µ | α i ∈ Π , µ ∈ P} , and the corresponding Levi subalgebra U v ( l I ) is generated by { x ± i | α i ∈ I } ∪ { k µ | µ ∈ P} . Let U v ( u + I ) be the subalgebra generated by { ˇ x +ˇ β j | j > ℓ ( w ,I ) } . Similarly, let U v ( u − I ) be the subalgebra generated by { ˇ x − ˇ β j | j > ℓ ( w ,I ) } . It isknown that U v ( g ) = U v ( u − I ) ⊗ F U v ( p I ) and U v ( p I ) = U v ( l I ) ⊗ F U v ( u + I ).Let Λ p I = { λ ∈ P | λ | α j )( α j | α j ) ∈ N , ∀ α j ∈ I } . Then Λ p I is the set of all l I -dominant integral weights.We have v ( λ | µ ) = q ( λ,µ ) , where ( , ) is the symmetric bilinear form such that ( ε i , ε j ) = δ i,j . We will keep using q ( λ,µ ) later on.It is known that the irreducible U v ( l I )-module L I ( λ ) with highest weight λ is of finite dimensional ifand only if λ ∈ Λ p I . By inflation, L I ( λ ) can be considered as a U v ( p I )-module. For any λ ∈ Λ p I , theparabolic Verma module with highest weight λ is M p I ( λ ) := U v ( g ) ⊗ U v ( p I ) L I ( λ ) , λ ∈ Λ p I .As vector spaces, M p I ( λ ) ∼ = U v ( u − I ) ⊗ L I ( λ ). Given a c = ( c , c , . . . , c k ) ∈ Z k such that c k = 0 if I = I , we define λ I, c = k X j =1 c j ( ε p j − +1 + ε p j − +2 + . . . + ε p j ) , (3.6)where p = 0, and p j ’s are given in (3.5). One can check λ I, c ∈ Λ p I and dim F L I ( λ I, c ) = 1.In the remaining part of this paper, we denote by m I the highest weight vector of L I ( λ I, c ) for any I ∈ { I , I } . Then L I ( λ I, c ) = F m I . Suppose R + \ R + I = { β t j | ≤ j ≤ ℓ ( w − ,I w ) } (3.7)and β t j < β t l in the sense of (2.17) whenever j < l . So, t j < t l if and only if j < l . Lemma 3.5. Suppose r ∈ N . Let S I,r = (cid:26) ~ Y j = ℓ ( w − ,I w ) ( x − β tj ) i j m I ⊗ v j | i j ∈ N , j ∈ N r (cid:27) , (3.8) where v j = v j ⊗ v j ⊗ . . . ⊗ v j r . Then S I,r is a basis of M p I ( λ I, c ) ⊗ V ⊗ r .Proof. It is enough to prove that S I, is a basis of M p I ( λ I, c ). Suppose M = M p I ( λ I, c ). We have M ∼ = U v ( u − I ) ⊗ F L I ( λ I, c ), and S is a basis of M , where S = ( ~ Y ℓ ( w ,I )+1 j = ℓ ( w ) (ˇ x − ˇ β j ) r j ⊗ m I | r j ∈ N , ℓ ( w ,I ) + 1 ≤ j ≤ ℓ ( w ) ) . Note that M = ⊕ η M η where η ∈ λ I, c − Q + . For all admissible η , define ( S I, ) η = S I, ∩ M η and S η = S ∩ M η . Then ♯S η = ♯ r ∈ N ℓ ( w − ,I w ) | ℓ ( w ) X j = ℓ ( w ,I )+1 r j − ℓ ( w ,I ) ˇ β j = λ I, c − η and dim M η = ♯S η < ∞ . Since { ˇ β j | ℓ ( w ,I ) + 1 ≤ j ≤ ℓ ( w ) } = R + \ R + I and ♯ ( S I, ) η = ♯ r ∈ N ℓ ( w − ,I w ) | ℓ ( w − ,I w ) X j =1 r j β t j = λ I, c − η = ♯S η , it is enough to verity that S I, spans M .If the result were false, we can find a non-zero element x − r ⊗ m I which is not a linear combina-tion of elements in S I, . Take the maximal integer j such that r j = 0 and β j ∈ R + I . If j = 1,then x − r ⊗ m I = ~ Q j = ℓ ( w ) ( x − β j ) r j ⊗ ( x − β ) r m I = 0, a contradiction. The general case follows fromLemma 2.2(1) together with arguments for induction on j . (cid:3) In the remaining part of this section, c is always the one in (3.6). If g = so n +1 , we define b j = c j − p j − ) + N − ε g , if 1 ≤ j ≤ k ,0 , if j = k + 1, − c k − j +2 + 2 p k − j +2 − N + ε g , if k + 2 ≤ j ≤ k + 1. (3.9)Otherwise, define b j = ( c j − p j − ) + N − ε g , if 1 ≤ j ≤ k , − c k − j +1 + 2 p k − j +1 − N + ε g , if k + 1 ≤ j ≤ k , (3.10)where ε g is given in (2.31). Definition 3.6. Define f I ( t ) = Q j ∈ J I ( t − u j ) and f I ( t ) = Q j ∈ J I ( t − u − j ), where • J I = { , , . . . , k } and J I = J I \ { k + 1 } if g = so n +1 , • J I = { , , . . . , k + 1 } and J I = J I \ { k + 1 , k + 2 } if g = so n +1 , • u j = ε g q b j for any j ∈ J I and b j ’s are in (3.9)-(3.10). Definition 3.7. Define V I,j = { v l | p j − + 1 ≤ l ≤ p j } if either I = I and 1 ≤ j ≤ k or I = I and1 ≤ j ≤ k − 1, and V I,j = { v l ′ | p k − j +1 ≤ l ≤ p k − j + 1 } , k + 1 ≤ j ≤ k and I = I ; { v l ′ | p k − j − + 1 ≤ l ≤ p k − j } , k + 1 ≤ j ≤ k − I = I ; { v l , v l ′ | p k − + 1 ≤ l ≤ p k } ∪ { δ g , so n +1 v n +1 } , j = k and I = I . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Recall the functor Ψ in Proposition 2.17. Let Ψ M : AK → U v ( g )-mod be the functor obtained bythe composition of Ψ followed by evaluation at M ∈ U v ( g )-mod. Thanks to Proposition 2.17, for any d ∈ Hom AK ( m , s ) and d ∈ Hom AK ( h , t ), m, s, h, t ∈ N , we haveΨ M ( d ⊗ d ) = Ψ( d ) M ⊗ V ⊗ s ◦ (Ψ( d ) M ⊗ Id V ⊗ h ) . (3.11)Obviously both M p I ( λ I, c ) and finite dimensional weight U v ( g )-modules are in O . Since O is closedunder tensor product, the image of Ψ M p I ( λ I, c ) is in O . Thus, we have a functor Ψ M p I ( λ I, c ) : AK → O .In the following, we denote Ψ M p I ( λ I, c ) ( d ) by d for any morphism d in AK . Lemma 3.8. If g ∈ { so n , sp n } , then M p I ( λ I, c ) ⊗ V has a parabolic Verma flag N ⊆ N ⊆ . . . ⊆ N k = M p I ( λ I, c ) ⊗ V such that N j /N j − ∼ = M p I ( λ I, c + ε p j − +1 ) , if ≤ j ≤ k , M p I ( λ I, c − ε p k ) , if I = I and j = k + 1 , , if I = I and j = k + 1 , M p I ( λ I, c − ε p k +1 − j ) , if k + 2 ≤ j ≤ k . (3.12) Moreover, X preserves previous flag and f I ( X ) acts on M p I ( λ I, c ) ⊗ V trivially, where f I ( t ) is givenin Definition 3.6.Proof. Keep the notation in Definition 3.7. By arguments similar to those in [32, Lemma 4.11], wehave the required flag where N j is the left U v ( g )-module generated by m I ⊗ u ’s and(1) u ∈ ∪ jl =1 V I,l if I = I or I = I and 1 ≤ j ≤ k ,(2) u ∈ ∪ kl =1 V I,l if I = I and j = k + 1,(3) u ∈ ∪ j − l =1 V I,l if I = I and k + 2 ≤ j ≤ k .Note that Θ = P β ∈Q + Θ β and Θ β ∈ U − v ( g ) − β ⊗ U + v ( g ) β . Since U + v ( g ) β m I = 0 for any β = 0, wehave X − ( m I ⊗ v j ) = δ − Θ πP Θ πP ( m I ⊗ v j ) = δ − q − λ I, c ,wt ( v j )) Θ( m I ⊗ v j ) , where δ is given in (2.31). So, X − preserves the required flag. Let M I = M p I ( λ I, c ). Thanks toLemma 3.1(2), σ M I ⊗ σ V = q − [( λ I, c ,λ I, c +2 ̺ )+( ε ,ε +2 ̺ )] Id M I ⊗ V , σ N j /N j − = q − ( ν,ν +2 ̺ ) Id N j /N j − if N j /N j − ∼ = M p I ( ν ). By Proposition 3.4, σ M I ⊗ V = (Θ πP ) σ M I ⊗ σ V . Since X − acts on M I ⊗ V as δ − (Θ πP ) , X − | N j /N j − = q − ( ν,ν +2 ̺ ) q ( λ I, c ,λ I, c +2 ̺ ) q ( ε ,ε +2 ̺ ) ε g q − N+ ε g Id N j /N j − = ε g q − b j Id N j /N j − , where b j ’s are given in (3.10). So, f I ( X − ) acts on M I ⊗ V trivially, where f I ( t ) is given in Definition3.6. Now, the corresponding result on X follows immediately. (cid:3) Lemma 3.9. If g = so n +1 , then M p I ( λ I, c ) ⊗ V has a parabolic Verma flag N ⊆ N ⊆ . . . ⊆ N k +1 = M p I ( λ I, c ) ⊗ V such that N j /N j − ∼ = M p I ( λ I, c + ε p j − +1 ) , if ≤ j ≤ k , M p I ( λ I, c ) , if I = I and j = k + 1 , M p I ( λ I, c − ε p k ) , if I = I and j = k + 2 , , if I = I and j = k + 1 , k + 2 , M p I ( λ I, c − ε p k +1 − j ) , if k + 3 ≤ j ≤ k + 1 .Moreover, X preserves previous flag and f I ( X ) acts on M p I ( λ I, c ) ⊗ V trivially, where f I ( t ) is givenin Definition 3.6. Proof. The result can be verified by arguments similar to those in the proof of Lemma 3.8. We givethe explicit construction of N j ’s and leave others to the reader. Recall V I,l ’s in Definition 3.7. Then N j is the left U v ( g )-module generated by m I ⊗ u , where(1) u ∈ ∪ jl =1 V I,l if 1 ≤ j ≤ k ,(2) u ∈ ∪ kl =1 V I,l if I = I and k + 1 ≤ j ≤ k + 2,(3) u ∈ ∪ j − l =1 V I,l if I = I and k + 3 ≤ j ≤ k + 1,(4) u ∈ ∪ kl =1 V I,l ∪ { v n +1 } if I = I and j = k + 1,(5) u ∈ ∪ j − l =1 V I,l ∪ { v n +1 } if I = I and k + 2 ≤ j ≤ k + 1. (cid:3) The following definition is motivated by [12, (2.24)]. Definition 3.10. Suppose M is a U v ( g )-module and N is a finite dimensional weight U v ( g )-module.For any ψ ∈ End U v ( g ) ( M ⊗ N ), define Id ⊗ qtr N ( ψ ) : M → M such that Id ⊗ qtr N ( ψ ) = ( Id ⊗ tr N )((1 ⊗ e K ) ◦ ψ ) , where 1 ⊗ e K ( m ⊗ n λ ) = q − ( λ, ̺ ) m ⊗ n λ for any ( m, n λ ) ∈ ( M, N λ ).It is well known that the natural transformations between the identity functor and itself in U v ( g )-mod form an algebra which can be identified with the center Z ( U v ( g )) of U v ( g ). Let z j = Ψ(∆ j ) U v ( g ) (1) , (3.13)where ∆ j is given in (1.9). Then z j ∈ Z ( U v ( g )), for any j ∈ Z . Let γ = P π − Θ P π − Θ. Thanks toProposition 2.17, z j = ( Id U v ( g ) ⊗ α ) ◦ (( δγ ) j ⊗ Id V )( Id U v ( g ) ⊗ X i ∈ N q ̺ i ′ ς i ′ ( v i ⊗ v i ′ )) = ε g ( Id ⊗ qtr V )(( δγ ) j ) . Write z + ( u ) = P l ≥ z l u − l and z − ( u ) = P l ≥ z − l u − l , where u is an indeterminate. By argumentssimilar to those in the proof of [12, Theorem 3.5], we have π ( z + ( u ) + δ − q − q − − u u − σ ̺ ( δ ( u − q ε g )( q − q − )( u − Y j ∈ N − k wt ( v j ) q − ( ε g u ) − − k wt ( v j ) q ( ε g u ) − ) ,π ( z − ( u ) − δ − q − q − − u − σ ̺ ( − δ − ( u − q − ε g )( q − q − )( u − Y j ∈ N − k wt ( v j ) q ( ε g u ) − − k wt ( v j ) q − ( ε g u ) − ) , (3.14)where π is Harish-Chandra homomorphism and σ ̺ ( k µ ) = q ( ̺,µ ) k µ , and ̺ is given in (2.3). Our z + ( u )(resp., z − ( u )) is Z + V ( u )(resp., Z − V ( u ) − z ) in [12, Theorem 3.5]. The difference is that we dealwith the quantum group U v ( g ) whereas they deal with U h ( g ). So, we have to use P π − Θ to replacetheir R . Lemma 3.11. Suppose j ∈ Z . Then Ψ(∆ j ) M p I ( λ I, c ) = ω j Id M p I ( λ I, c ) for some ω j ∈ F , where ∆ j isgiven in (1.9) . Further, ω is u -admissible in the sense of Definition 1.13, where u = { u j | j ∈ J I } ,and u j ’s and J I are given in Definition 3.6.Proof. Recall z j in (3.13). Suppose ω j = z j | M p I ( λ I, c ) , ∀ j ∈ Z . Obviously, there is a U v ( g )-homomorphism φ : U v ( g ) → M p I ( λ I, c ) such that φ (1) = m I . So,Ψ(∆ j ) M p I ( λ I, c ) m I = Ψ(∆ j ) M p I ( λ I, c ) φ (1)= φ ( z j ) = z j φ (1)= ω j m I , and Ψ(∆ j ) M p I ( λ I, c ) = ω j Id M p I ( λ I, c ) . This proves the first assertion. HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Let a = ♯J I . It is routine to check that δ, u , . . . , u a satisfy Assumption 1.12 where δ is in (2.31)and u k +1 = 1 (resp., q ε g ) if g = so n +1 (resp., I = I and g ∈ { so n , sp n } ) and u k +2 = 1 if I = I and g = so n +1 . In fact, when a is odd, δ = α Q j ∈ J I u j , where α = 1 (resp., − 1) if g ∈ { so n , so n +1 } (resp., g = sp n ). When a is even, we still have δ = α Q j ∈ J I u j , where α = − q (resp., q − ) if g = sp n (resp., so n ). We omit details since one can verify them by straightforward computation.Define u g = u − q − u − q (resp., 1) if g = so n +1 (resp., otherwise). Let z = z q . By (3.14), we have thefollowing equalities in End( M p I ( λ I, c )): X j ≥ ω j u − j + δ − q − q − − u u − σ ̺ δz u − q ε g u − Y j ∈ N − ε g q − u − k wt ( v j ) − ε g qu − k wt ( v j ) = δz u − q ε g u − k Y j =1 (1 − ε g u − q (2 ̺,ε pj ) − k wt ( v pj ) )(1 − ε g u − q − (2 ̺,ε pj − ) − k − wt ( v pj − ) )(1 − ε g u − q − (2 ̺,ε pj )+1 k − wt ( v pj ) )(1 − ε g u − q (2 ̺,ε pj − )+1 k wt ( v pj − ) ) u g = δz u − q ε g u − k Y j =1 (1 − ε g u − q N − ε g − p j +2 c j )(1 − ε g u − q ε g − N+2 p j − − c j )(1 − ε g u − q − (N − ε g − p j +2 c j ) )(1 − ε g u − q − ( ε g − N+2 p j − − c j ) ) u g = δz u − q ε g u − Y j ∈ J I − u − j u − − u ′ j u − u g = δz u − q ε g u − Y j ∈ J I u − u j − u − u j u g = z − δ − Y j ∈ J I u j + ug a ( u ) u − Y j ∈ J I u j Y j ∈ J I u − u j − u − u j , where g a ( u ) is defined in Definition 1.13, and X j ≥ ω − j u − j − δ − q − q − − u − σ ̺ − δ − z u − q − ε g u − Y j ∈ N − ε g qu − k wt ( v j ) − ε g q − u − k wt ( v j ) = − δ − zu g u − q − ε g u − k Y j =1 (1 − ε g u − q − (2 ̺,ε pj )+1 k − wt ( v pj ) )(1 − ε g u − q (2 ̺,ε pj − )+1 k wt ( v pj − ) )(1 − ε g u − q (2 ̺,ε pj ) − k wt ( v pj ) )(1 − ε g u − q − (2 ̺,ε pj − ) − k − wt ( v pj − ) )= − δ − zu g u − q − ε g u − k Y j =1 (1 − ε g u − q − (N − ε g − p j +2 c j ) )(1 − ε g u − q − ( ε g − N+2 p j − − c j ) )(1 − ε g u − q N − ε g − p j +2 c j )(1 − ε g u − q ε g − N+2 p j − − c j )= − δ − zu g u − q − ε g u − Y j ∈ J I − u j u − − u − j u − = − δ − zu g u − q − ε g u − Y j ∈ J I u − u j u − u − j = − z − δ − Y j ∈ J I u j − ug a ( u )( u − Y j ∈ J I u − j Y j ∈ J I u − u j u − u − j . So, ω is u -admissible in the sense of Definition 1.13. (cid:3) It is proved in [33, Corollary 2.29] that ω is admissible if it is u -admissible. So, we have CK f I (seeDefinition 1.11), where f I ( t ) is given in Definition 3.6. Theorem 3.12. Ψ M p I ( λ I, c ) factors through CK f I , where f I ( t ) is given in Definition 3.6.Proof. Thanks to Lemmas 3.8-3.9 and 3.11,Ψ M p I ( λ I, c ) ( f I ( X )) = 0 and Ψ M p I ( λ I, c ) (∆ j − ω j ) = 0 , ∀ j ∈ Z . Recall that a right tensor ideal B of a K -linear strict monoidal category C is the collection of K -submodules { B ( a , b ) | B ( a , b ) ⊂ Hom C ( a , b ) , ∀ a , b ∈ C} , such that h ◦ h ◦ h ∈ B ( a , d ) , and h ⊗ d ∈ B ( b ⊗ d , c ⊗ d ) (3.15)whenever ( h , h , h ) ∈ Hom C ( a , b ) × B ( b , c ) × Hom C ( c , d ) for all objects a , b , c , d in C . By (3.11),Ψ M p I ( λ I, c ) ( B ( m , s )) = 0, for all m, s ∈ N , where B is the right ideal of AK generated by f I ( X ) and∆ j − ω j , j ∈ Z . So Ψ M p I ( λ I, c ) factors through CK f I . (cid:3) A basis theorem for affine Kauffmann category The aim of this section is to prove Theorem 1.9, which says that any morphism space in AK is freeover K with infinite rank. Lemma 4.1. For any positive integer m , define η m = · · · · · · m m and γ m = · · · · · · m m . (4.1) Then (1 m ⊗ γ m ) ◦ ( η m ⊗ m ) = ( γ m ⊗ m ) ◦ (1 m ⊗ η m ) = 1 m in AK and CK f .Proof. We have explained that d = d ′ as morphisms in AK if d, d ′ ∈ NT m,s such that d ∼ d ′ . So, d = d ′ as morphisms in CK f . The required equalities follow from this observation. (cid:3) Lemma 4.2. Suppose m, s ∈ N such that m = 0 and | m + s . Let η m : Hom AK ( m , s ) → Hom AK ( , m + s ) , γ m : Hom AK ( , m + s ) → Hom AK ( m , s ) be two linear maps such that η m ( d ) = ( d ⊗ m ) ◦ η m and γ m ( d ) = γ m ◦ ( d ⊗ m ) . Then (1) Hom C ( m , s ) ∼ = Hom C ( , m + s ) where C ∈ {AK , CK f } , (2) η m gives a bijection between NT m,s / ∼ and NT ,m + s / ∼ , and its inverse is γ m , (3) η m induces a bijection between NT am,s / ∼ and NT a ,m + s / ∼ and its inverse is γ m .Proof. Thanks to Lemma 4.1, η m and γ m are mutually inverse to each other. So, we have the requiredisomorphism when C = AK .By (3.15), it is easy to see that η m ( I ( m , s )) ⊂ I ( , m + s ) and γ m ( I ( , m + s )) ⊂ I ( m , s ) for any righttensor ideal I of AK . So, both η m and γ m induce required K -isomorphisms in CK f .Thanks to (1.8) and Lemma 1.5(3), for any b ∈ NT m,s / ∼ , there is a unique b ′ ∈ NT , r / ∼ such that η m ( b ) = b ′ as morphisms in AK . So, η m ( NT m,s / ∼ ) ⊆ NT , r / ∼ . Similarly, we have γ m ( NT , r / ∼ ) ⊆ NT m,s / ∼ . Since γ m is the inverse of η m , (2) holds. (3) can be proved similarly. (cid:3) Lemma 4.3. In AK , we have (1) • = • − z • + z ◦ , • = • − z • + z ◦ , (2) ◦ = ◦ + z ◦ − z • , ◦ = ◦ + z ◦ − z • , (3) • = • + z • − z • , • = • + z • − z • , (4) ◦ = ◦ − z ◦ + z ◦ , ◦ = ◦ − z ◦ + z ◦ .Proof. (1)-(4) can be checked via (1.6)-(1.8), Lemma 1.5(3), (S) and (RII), directly. (cid:3) Definition 4.4. Let ˆ T m,s be the subset of T m,s such that each dotted tangle diagram in ˆ T m,s satisfiesthe following conditions: HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY (1) neither a strand crosses itself nor two strands cross each other more than once,(2) Definition 1.7(a)-(c),(3) whenever a dot appears on a stand, it is near the endpoints of the strand,(4) Definition 1.7(d). Lemma 4.5. Any d in T m,s can be written as a linear combination of elements in ˆ T m,s .Proof. For any d ∈ T m,s which does not satisfies Definition 4.4(1), either there is a strand crossingitself or there are two strands crossing each other more than once. Thanks to Lemma 4.3(1)-(4), onecan slide dots ( • ’s or ◦ ’s) along each crossing in d modulo diagrams with fewer crossings. By (1.8)and Lemma 1.5(3), one can also slide dots ( • ’s or ◦ ’s) along each cup and each cap. By induction onthe number of crossings, we can assume that there is no dots in the local area on which either thereis a self-crossing strand or there are two strands crossing each other more than once. Using Theorem1.2 and the monoidal functor in Lemma 1.4(2), we see that d can be written as a linear combinationof dotted tangle diagrams which satisfy Definition 4.4(1).Now, we assume that d ∈ T m,s satisfying Definition 4.4(1). Then all loops of d are crossing free.Otherwise, there is a crossing which appears on a loop such that either there is a strand crossing itselfor there are two strands crossing each other more than once. Thanks to (1.8), (L) and Lemma 1.6,we can assume all loops of d satisfy Definition 1.7(b)-(c). In order to write d as a linear combinationof dotted tangle diagrams satisfying Definition 4.4(1)-(2), it is enough to move a loop containing k • to the left of a vertical line, where k is a positive integer. In fact, applying Lemma 4.3 repeatedlytogether with (1.8) and Lemma 1.5(3) yields the following equations: • k = • k = • k + z k X i =1 • i • k − i − ◦ k − i • i = • k + z k X i =1 • i • k − i + z k − i X j =1 • j • k − j − • k − j − z ◦ k − i • i + z i X j =1 • i − j ◦ k + j − i − • i − j − k = • k + z k X i =1 δ • k + z k − i X j =1 • j • k − j − • k − j − z k X i =1 δ − • i − k + z i X j =1 • i − j ◦ k + j − i − • i − j − k , (4.2)where the last equation follows from Lemma 1.5(1), (T), (RII) and= , = . (4.3)We remark that (4.3) follows from (1.2)-(1.4). Now, any d can be written as a linear combination ofdotted tangle diagrams satisfying Definition 4.4(1)–(2) since no new crossing appears when we movea loop with k • to the left of a vertical line. Now, we assume that d ∈ T m,s satisfying Definition 4.4(1)-(2). Using Lemmas 4.3,1.5(3), (1.8)together with induction on the number of crossings, we see that d can be written as a linear combinationof dotted tangle diagrams satisfying Definition 4.4(1)-(3) since neither new loops nor crossing occurswhen we apply the previous results to move dots along a strand.Finally, we can assume that d satisfies Definition 4.4(1)-(3). Thanks to Theorem 1.2 and the functorin Lemma 1.4(2), d can be written as a linear combination of dotted tangle diagrams in ˆ T m,s . (cid:3) Proposition 4.6. The K -module Hom AK ( m , s ) is spanned by NT m,s / ∼ .Proof. Thanks to Lemma 4.5, it is enough to verify that any d ∈ ˆ T m,s is a linear combination ofelements in NT m,s . If d has no crossing, then the result follows from (1.6),(1.8) and Lemma 1.5(3).Suppose that there are some crossings on d . Thanks to Lemmas 4.3, 1.5(3), (1.8), and (1.6), thereis a d ∈ NT m,s such that ˆ d = ˆ d and d = d up to some dotted tangle diagrams in ˆ T m,s with fewercrossings than that of d . Now the result follows from induction on the number of crossings. (cid:3) Lemma 4.7. Suppose r is a positive integer. For all admissible i, j , define X i r = 1 i − ⊗ X ⊗ r − i , r U j = 1 j − ⊗ U ⊗ r − j − and Z j r = 1 j − ⊗ Z ⊗ r − j − ,where Z ∈ { A, T, T − } . Then X i r = T i − r X i − r T i − r , ≤ i ≤ r . Further, X i r X j r = X j r X i r for all admissible i and j .Proof. The first assertion follows from (1.7) and the last follows from the interchange law. (cid:3) Suppose d ∈ NT m,s and m + s is even. In section 1, we have labelled the endpoints of ˆ d at the bottom(resp., top) row as 1 , , . . . , m (resp., 1 , , . . . , s ) from the left to the right and i < i + 1 < j < j − i and j . We also have conn ( ˆ d ) = { ( i l , j l ) | ≤ l ≤ m + s } such that i l < j l and i k < i k +1 and for all admissible k, l . Definition 4.8. For any d ∈ NT m,s such that conn ( ˆ d ) = { ( i l , j l ) | ≤ l ≤ m + s } and any i ∈ { , , . . . , m, , , . . . , s } , define b d,i = j if there are j “ • ” near the i th endpoint.Note that b d,i ∈ Z . Thanks to Definition 1.7, b d,j l = 0 for all admissible l . Later on, we simplydenote X i r by X i etc if we know r from the context. Then any d ∈ NT m,s is of form d = ∞ Y j =1 ∆ ji j · Y l = s Y lb d,l · ˆ d · Y j = m X jb d,j (4.4)where Y l is X l ∈ End AK ( s , s ) and i j ’s ∈ N such that only finite number of i j ’s are non-zero.For any positive integer b ≥ 2, let S b be the symmetric group on b letters 1 , , . . . , b . Then S b isgenerated by basic transpositions r i = ( i, i +1), 1 ≤ i ≤ b − 1. Now, S b acts on the right of N b via placepermutation. More explicitly, i σ = ( i σ (1) , i σ (2) , . . . , i σ ( b ) ) for all σ ∈ S b and i = ( i , i , . . . , i b ) ∈ N b .Define r k,l = r k r k +1 · · · r l − if k < l and r k,k = 1 and r k,l = r k − r k − . . . r l if k > l . If r i r i · · · r i k isa reduced expression of w ∈ S b , define T w = T i · · · T i k , and T invw = T − i · · · T − i k (4.5)in End AK ( b ), where T j is given in Lemma 4.7. It is well-known that both T w and T invw are independentof a reduced expression of w . We denote T r k,l (resp., T invr k,l ) by T k,l (resp., T invk,l ). Let B b be thesubgroup of S b generated by { r b − i +2 r b − i +1 r b − i +3 r b − i +2 | < i < b } and { r b − } . Define D b, b = (cid:26) r ,i r ,j · · · r b − ,i b r b,j b (cid:12)(cid:12)(cid:12)(cid:12) ≤ i < . . . < i r ≤ b ≤ i k < j k ≤ k ; 1 ≤ k ≤ b (cid:27) . (4.6)Thanks to [33, Lemma 4.3], D b, b is a complete set of right coset representatives for B b in S b and ♯ D b, b = (2 b − HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Proposition 4.9. Suppose d ∈ NT s, / ∼ and conn ( d ) = { ( i l , j l ) | ≤ l ≤ s } . As morphisms in AK , d = ⊗ s T invw , where w = ~ Q st =1 r t − ,i t r t,k t ∈ D s, s for some admissible k t ’s.Proof. Suppose d = d . Define d = d T j s , s T i s , s − and write conn ( d ) = { ( i l , j l ) | ≤ l ≤ s } . Obvi-ously, i t = i t if t < s and ( i s , j s ) = (2 s − , s ). In general, write conn ( d t − ) = { ( i t − l , j t − l ) | ≤ l ≤ s } and define d t = d t − T j t − s − t +1 , s − t +1) T i t − s − t +1 , s − t )+1 , ≤ t ≤ s . Then conn ( d t ) = { ( i k , j tk ) , (2 l + 1 , l + 2) | ≤ k ≤ s − t, s − t ≤ l < s } . In particular i t < j s − tt ≤ t for any 1 ≤ t ≤ s . So, both d s and ⊗ s are totally descending tangle diagrams withthe same connector. By (1.5), d s = ⊗ s as morphisms in FT and hence in K . Using the functor inLemma 1.4(2), we see that d s = ⊗ s as morphisms in AK . For example: d = d = , d = , d = (1.5) = ⊗ . Let w = ~ Q st =1 r t − ,i t r t,j s − tt . Then w ∈ D s, s and ⊗ s T invw = d s ~ Q st =1 T inv t − ,i t T inv t,j s − tt = d asmorphisms in AK . (cid:3) Consider U v ( h ) as a U v ( b )-module such that x + i acts as zero for all admissible i , and define M gen := U v ( g ) ⊗ U v ( b ) U v ( h ) . (4.7)Later on, M gen is called the generic Verma module . Then M gen is a right U v ( h )-module with basis { x − s ⊗ | s ∈ N ℓ ( w ) } , where x − s is defined in (2.9). Recall that V is the natural U v ( g )-moduleand dim V = N. Then M gen ⊗ V ⊗ r has F -basis { x − s ⊗ k µ ⊗ v i | s ∈ N ℓ ( w ) , i ∈ N r , µ ∈ P} , where v i = v i ⊗ . . . ⊗ v i r .From here to the end of this section, we assume g = so n . So, N = 2 n , and v = q . Definition 4.10. For any r ∈ N and j ∈ { , } , let M jr be the free A -module with basis { z jv x − s ⊗ k µ ⊗ v i | s ∈ N ℓ ( w ) , i ∈ N r , µ ∈ P} , where A is given in Definition 2.1.Obviously, M ⊂ M and z v M = M . Lemma 4.11. Suppose H, K are sequences of positive roots and j ∈ { , } . Then (1) x − K ⊗ x − H stabilizes M j , (2) x − K ⊗ x + H stabilizes M j .Proof. For any basis element v l ∈ V , by Lemmas 2.9, 2.12, x + H v l = g v k (resp., x − H v l = g v k ) forsome g , g ∈ A \ A and some v k , v k ∈ V if x + H v l = 0 (resp., x − H v l = 0). Then (1)-(2) follow fromCorollary 2.3. (cid:3) Lemma 4.12. Suppose s ∈ N ℓ ( w ) and µ ∈ P . For any i ∈ N , and any Ψ ∈ { Θ , Θ } , (1) Ψ(( x − s ⊗ k µ ) ⊗ v i ) ≡ x − s ⊗ k µ ⊗ v i (mod M ) , (2) P Ψ P (( x − s ⊗ k µ ) ⊗ v i ) ∈ M , (3) Ψ( M ) ⊆ M and P Ψ P ( M ) ⊆ M . Proof. Recall the terms z ℓ ( J ) v g J x − J ⊗ x + J ’s in (3.2). If ℓ ( J ) = 0, then z ℓ ( J ) v g J x − J ⊗ x + J = 1 ⊗ z ℓ ( J ) v g J x − J ⊗ x + J (( x − s ⊗ k µ ) ⊗ v i ) = x − s ⊗ k µ ⊗ v i . If ℓ ( J ) > 0, by Lemma 4.11(2), z ℓ ( J ) v g J x − J ⊗ x + J (( x − s ⊗ k µ ) ⊗ v i ) ∈ M . Thanks to (3.2),Θ(( x − s ⊗ k µ ) ⊗ v i ) ≡ x − s ⊗ k µ ⊗ v i (mod M ) . The corresponding result for Θ can be proved similarly. This proves (1). Note that x + β (1 ⊗ k µ ) = 1 ⊗ x + β k µ = 0 for any β ∈ R + . By (3.2), P Θ P ((1 ⊗ k µ ) ⊗ v i ) = 1 ⊗ k µ ⊗ v i . (4.8)This proves (2) when P i s i β i = 0 and Ψ = Θ. In general, let j = max { t | s t = 0 } . Suppose y Ψ = P Ψ P (( x − s ⊗ k µ ) ⊗ v i )and c ∈ N ℓ ( w ) such that c k = s k for any k = j and c j = s j − 1. Then y Θ − P Θ∆( x − β j )( v i ⊗ ( x − c ⊗ k µ ))= − P Θ( X K,H h K,H x − K v i ⊗ k − wt ( K ) x − H ( x − c ⊗ k µ )) , by Proposition 2.7(1)= − P Θ P ( X K,H h K,H k − wt ( K ) x − H ⊗ x − K (( x − c ⊗ k µ ) ⊗ v i )) ∈ M , by Lemma 4.11(1) and induction assumption on X i s i β i − wt ( K ) , (4.9)where K, H are sequences of positive roots such that wt ( K ) + wt ( H ) = β j , K = ∅ and h K,H ∈ A .In particular, h K,H = 1 when ( K, H ) = ( β j , ∅ ). For any U v ( g )-modules M, N and any a, b ∈ U v ( g ),it is easy to see P ( a ⊗ b ) P = b ⊗ a in End( M ⊗ N ). So, P Θ∆( x − β j )( v i ⊗ ( x − c ⊗ k µ )) = P ∆( x − β j )Θ( v i ⊗ ( x − c ⊗ k µ )) , by (3.1)= P ∆( x − β j ) P P Θ P (( x − c ⊗ k µ ) ⊗ v i ) ∈ P ∆( x − β j ) P ( M ) , by induction assumption on X i s i β i − β j ∈ M , by Proposition 2.7(2) and Lemma 4.11(1) . (4.10)Combining (4.9)–(4.10) yields y Θ ∈ M . The corresponding result for y Θ can be checked by argumentssimilar to those above. The only difference is that one has to replace (3.1) by (2.19). This proves (2).Finally, (3) follows from (1)-(2). (cid:3) To simplify the notation, we use d to replace Ψ M gen ( d ). Let ˆ ε i = wt ( v i ), 1 ≤ i ≤ N. So, ˆ ε i = ε i if1 ≤ i ≤ n , and ˆ ε i = − ε i ′ if n ′ ≤ i ≤ ′ . Lemma 4.13. For any ≤ i ≤ N and any µ ∈ P , X ± ((1 ⊗ k µ ) ⊗ v i ) ≡ v ± (2 n − (1 ⊗ k µ ± ε i ⊗ v i ) (mod M ) . Further, X ± M ⊆ M .Proof. Thanks to (2.27), P π = πP and π − ( M ) ⊆ M . Then X ((1 ⊗ k µ ) ⊗ v i ) = v n − P π − Θ P π − Θ((1 ⊗ k µ ) ⊗ v i ) , by Proposition 2.17 ≡ v n − P π − Θ P ((1 ⊗ k ˆ ε i + µ ) ⊗ v i ) (mod M ) , by Lemma 4.12 and (2.27) ≡ v n − π − (1 ⊗ k ˆ ε i + µ ) ⊗ v i ) (mod M ) , by (4.8) ≡ v n − (1 ⊗ k ε i + µ ⊗ v i ) (mod M ) , by (2.27) . So, XM ⊆ M . Finally, one can verify the result for X − by arguments similar to those above. Theonly difference is that one has to use the result on Θ to replace that for Θ in Lemma 4.12. (cid:3) HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY From here to the end of this section, we assume n > r . Definition 4.14. For any d ∈ NT r, , define η ( d ) ∈ N r such that η ( d ) i l = l and η ( d ) j l = l ′ , 1 ≤ l ≤ r ,where conn ( ˆ d ) = { ( i , j ) , . . . , ( i r , j r ) } .Since we are assuming that n > r , η ( d ) = η ( d ) for any d , d ∈ NT r, if and only if conn ( ˆ d ) = conn ( ˆ d ). Lemma 4.15. Suppose d, e ∈ NT r, such that conn ( ˆ d ) = { ( i , j ) , . . . , ( i r , j r ) } . For any µ ∈ P , thereis a c ∈ Z such that d ((1 ⊗ k µ ) ⊗ v η ( e ) ) ≡ δ η ( d ) ,η ( e ) v c (1 ⊗ k β ) (mod M ) , where β = P rt =1 b d,i t ε t + µ and b d,i t ’s are given in Definition 4.8.Proof. By Proposition 4.9, ˆ d = d r T invw as morphisms, where w = ~ Q rt =1 r t − ,i t r t,c t ∈ S r for some c t ’s and d r = ⊗ r . Then η ( ˆ d ) = η ( d r ) w . For any k ∈ N r , define δ k = 1 if k l − = k ′ l for all1 ≤ l ≤ r and δ k = 0 otherwise. Since A acts on V ⊗ via α (see Lemma 2.15), we have d r ( M r ) ⊆ M and d r ((1 ⊗ k η ) ⊗ v k ) = v b δ k (1 ⊗ k η ) , for ∀ η ∈ P , (4.11)where b ∈ Z . Note that T − j acts on M gen ⊗ V ⊗ r via Id M gen ⊗ Id ⊗ j − V ⊗ R V,V ⊗ Id ⊗ r − j − V . ByLemma 2.14, T − j stabilizes M r and T − j ((1 ⊗ k η ) ⊗ v k ) ≡ v ( wt ( v kj ) | wt ( v kj +1 )) (1 ⊗ k η ) ⊗ v k r j (mod M r ) (4.12)for any η ∈ P .Suppose s = η ( e ) and β s = µ + 2 P rj =1 b d,j ˆ ε s j . Then there exist c, a , a ∈ Z and x, x ∈ M r suchthat d ((1 ⊗ k µ ) ⊗ v s ) = d r T invw X b d, r r ◦ . . . ◦ X b d, ((1 ⊗ k µ ) ⊗ v s ) , by (4.4),= v a d r T invw ((1 ⊗ k β s ) ⊗ v s + x ) , by Lemma 4.13 and (4.12)= v a d r ((1 ⊗ k β s ) ⊗ v s w − + x ) , by (4.12) ≡ v c δ s w − (1 ⊗ k β s ) (mod M ) , by (4.11) . (4.13)Let c = s w − = ( s w − (1) , . . . , s w − (2 r ) ). Then c is an arrangement of 1 , ′ , . . . , r, r ′ . Note that δ c = 1if and only if c l − = c ′ l if and only if s w − (2 l − = s ′ w − (2 l ) for all 1 ≤ l ≤ r . So, δ c = 1 if and onlyif {{ w − (2 l − , w − (2 l ) } | ≤ l ≤ r } is the set of all caps in e . On the other hand, η ( d ) = η ( d r ) w .So {{ w − (2 l − , w − (2 l ) } | ≤ l ≤ r } is the set of all caps in d . Thus, δ c = 1 if and only if conn (ˆ e ) = conn ( ˆ d ), proving δ c = δ η ( d ) ,η ( e ) . (cid:3) Lemma 4.16. For any positive integer i and any µ ∈ P , we have (1) ∆ i M ⊆ M , (2) ∆ i (1 ⊗ k µ ) ≡ P nj =1 v ̺ j ′ (1 ⊗ k i ˆ ε j + µ ) (mod M ) .Proof. Thanks to Lemma 2.15, U ( M ) ⊆ M and A ( M ) ⊆ M . By Lemma 4.13, we have∆ i M = A ◦ ( X i ⊗ Id V ) ◦ U ( M ) ⊆ M and∆ i (1 ⊗ k µ ) = ( Id M gen ⊗ α ) ◦ ( X i ⊗ Id V )((1 ⊗ k µ ) ⊗ n X j =1 v ̺ j ′ v j ⊗ v j ′ ) ≡ ( Id M gen ⊗ α )( n X j =1 v ̺ j ′ ((1 ⊗ k i ˆ ε j + µ ) ⊗ v j ⊗ v j ′ )) (mod M ) ≡ n X j =1 v ̺ j ′ (1 ⊗ k i ˆ ε j + µ ) (mod M ) . (cid:3) The proof of Theorem 4.17 depends on quantum group U v ( so n ) over C ( v / ). Since we considergeneric Verma module and natural module of U v ( so n ), it is enough use C ( v / ) instead of C ( v / ). Theorem 4.17. Suppose K = C ( v / ) . If ( δ, z ) = ( v n − , z v ) and ω is uniquely determined by (1.1) ,then Hom AK ( r, ) has K -basis given by NT r, / ∼ .Proof. Recall that any e ∈ NT r, / ∼ is of form in (4.4). By Proposition 4.6, it is enough to prove p d = 0 for all d if X d ∈ B p d (∆ , ∆ , . . . ) d = 0 , (4.14)where B is a finite subset of NT r, and p d ∈ K [ t , t , . . . ]. If it were false, we assume all previous p d ’sare non-zero and choose a j such that all previous p d ∈ K [ t , t , . . . , t j ]. In this case, write p d (∆ , ∆ , . . . ) = X s ∈ N j f d s ( v / )∆ s , (4.15)where f d s ( v / ) ∈ K and ∆ s = ∆ s ∆ s · · · ∆ s j j . Without loss of any generality, we can assumethat f d s ( v / ) ∈ A for all previous pairs ( d, s ) and moreover, there is a pair ( d , ˜ s ) such that f d ˜ s ( v / ) ∈ A \ A . Suppose conn ( ˆ d ) = { ( i l , j l ) | ≤ l ≤ r } and B = { d ∈ B | η ( d ) = η ( d ) } .Since we are assuming that n > r , η ( d ) = η ( d ) if and only if conn ( ˆ d ) = conn ( ˆ d ). Recall b d,i ’s inDefinition 4.8. Thanks to Lemmas 4.15–4.16, there are some c ( d ) ∈ Z depending on d such that X d ∈ B p d (∆ , ∆ , . . . ) d ((1 ⊗ ⊗ v η ( d ) ) ≡ X d ∈ B p d (∆ , ∆ , . . . ) d ((1 ⊗ ⊗ v η ( d ) ) (mod M ) ≡ X d ∈ B v c ( d ) p d ( n X l =1 v ̺ l ′ (1 ⊗ k ε l ) , n X l =1 v ̺ l ′ (1 ⊗ k ε l ) , . . . )(1 ⊗ k P rt =1 b d,it ε t ) (mod M ) . Thanks to (4.14)-(4.15), X d ∈ B v c ( d ) X s ∈ N j f d s ( v / ) j Y i =1 ( n X l =1 ⊗ k i ˆ ε l ) s i (1 ⊗ k P rt =1 b d,it ε t ) ≡ M ) . (4.16)Now, define deg( k ± ε i ) = ± i . Considering the terms in the LHS of (4.16) with thehighest degree yields X d ∈ B X s ∈ N j v c ( d ) f d s ( v / ) j Y i =1 ( n X l =1 ⊗ k iε l ) s i (1 ⊗ k P rt =1 b d,it ε t ) ≡ M ) (4.17)Note that (4.17) is something like [32, (3.27)]. Using arguments on the leading monomials at the endof the proof of [32, Proposition 3.12], we have v c ( d ) f d s ( v / ) ∈ A for all pairs ( d, s ) such that d ∈ B .In particular, v c ( d ) f d ˜ s ( v / ) ∈ A , a contradiction since f d ˜ s ( v / ) ∈ A \ A . (cid:3) Proof of Theorem 1.9 . If m + s is odd, then Hom AK ( m , s ) = 0. So, we assume m + s = 2 r for some r ∈ N . Thanks to Lemma 4.2(2), both η m and γ m give bijections between NT m,s / ∼ and NT , r / ∼ (as morphisms in AK ). Thanks to Proposition 4.6, it is enough to verify that NT r, / ∼ is linearindependent over K .We consider AK over the quotient ring C [ δ, δ − , z, ω ] /I and I is the ideal generated by δ − δ − − z ( ω − δ − δ − − z ( ω − 1) is irreducible, C [ δ, δ − , z, ω ] /I is a domain. Suppose X d ∈ B f d d = 0 , where B is a finite set of NT r, / ∼ and f d ∈ C [ δ, δ − , z, ω ] /I . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY We claim that f d = 0 for all d ∈ B . Since C [ δ, δ − , z, ω ] /I is a domain, f d = 0 if and only if z a δ b f d = 0 for any a ∈ N , b ∈ Z . So, we can assume each monomial of f d has neither δ − nor ω as itsfactor when we prove the claim. Thanks to Theorem 4.17, we have f d = 0 for all δ = v n − , z = v − v − whenever n > r . Now the claim follows from the fundamental theorem of algebra. So, NT r, / ∼ islinear independent over C [ δ, δ − , z, ω ] /I and hence over Z = Z [ δ, δ − , z, ω ] /J , where J is the idealgenerated by δ − δ − − z ( ω − AK Z (resp., AK K ) over Z (resp., K ). By base changeproperty, Hom K ⊗ Z AK Z ( r, ) has basis given by NT r, / ∼ . Then Theorem 1.9 follows from the factthat there is an obvious functor from AK K to K ⊗ Z AK Z , which sends the required basis element ofHom AK K ( r, ) to the corresponding basis element in Hom K ⊗ Z AK Z ( r, ). (cid:3) Affine Birman-Murakami-Wenzl algebras Thanks to Theorem 1.9, AK can be considered as K [∆ , ∆ , . . . ]-linear category. Suppose ω satisfiesDefinition 1.10(1). Consider K as the right K [∆ , ∆ , . . . ]-module on which ∆ i acts on K via ω i forany positive integer i . Define AK ( ω ) = K ⊗ K [∆ , ∆ ,... ] AK . Let K ′ be the subcategory of AK ( ω ) generated by and four elementary morphisms A, U, T and T − .Thanks to Theorem 1.9, we have the following result, immediately. This shows that K is a subcategoryof AK ( ω ). Corollary 5.1. Suppose m, s ∈ N . If ω satisfies Definition 1.10(1), then (1) Hom AK ( ω ) ( m , s ) has K -basis given by NT m,s / ∼ , (2) K ′ ≃ K . Definition 5.2. [18] The affine Birman-Murakami-Wenzl algebra W aff r, K is the K -algebra generatedby x ± , g ± i and e i (1 ≤ i ≤ r − 1) subject to the following relations:(1) g i g − i = g − i g i = 1,(2) x x − = x − x = 1,(3) e i = ω e i ,(4) g i g i +1 g i = g i +1 g i g i +1 ,(5) g i − g − i = z (1 − e i ),(6) x g x g = g x g x ,(7) y i z j = z j y i , if | i − j | ≥ 2, (8) x y j = y j x if j ≥ e i e i ± e i = e i ,(10) g i g i ± e i = e i ± e i and e i g i ± g i = e i e i ± ,(11) e x s e = ω s e , for any s ≥ g i e i = e i g i = δ − e i ,(13) e x g x = δe = x g x e ,where y i , z i ∈ { e i , g i } , 1 ≤ i ≤ r − z in Definition 5.2 is − z in [18]. Theorem 5.3. As K -algebras, End AK ( ω ) ( r ) ∼ = W aff r, K .Proof. The required algebra homomorphism γ : W aff r, K → End AK ( ω ) ( r ) satisfies γ ( e i ) = E i , γ ( x ± ) = X ± and γ ( g ± i ) = T ± i for all admissible i , where E i = U i ◦ A i . (5.1)In order to verify that γ is an algebra homomorphism, we need to check that the images of x ± , g ± i and e i satisfy Definition 5.2(1)-(12). Later on, we say Definition 5.2(1) holds if the images of x ± , g ± i and e i satisfy Definition 5.2(1).By (RII)-(RIII),(1.6), (L) and (S), we see that Definition 5.2(1)–(5) hold. Thanks to the definitionof AK ( ω ), Definition 5.2(11) holds. For 1 ≤ i ≤ r , let x i = g i − ...g x g ...g i − . Then γ ( x ± i ) = X ± i . Now, Definition 5.2(6) follows from Lemma 4.7. Definition 5.2(7)-(8) follows from the interchange law.Note that both sides of each equation in Definition 5.2(9)-(10) are totally descending tangle diagramswith the same connector. Thanks to (1.5) and the monoidal functor in Lemma 1.4(2), Definition 5.2(9)-(10) hold. Definition 5.2 (12) follows from Lemma 1.5(1)-(2). Thanks to Lemma 1.5(1)(3), we have E X T X = δE T X T X = δE X X = δE X − X = δE . One can verify X T X E = δE similarly. So, Definition 5.2(13) holds. This verifies that γ is analgebra homomorphism.Let W r, K be the subalgebra of W aff r, K generated by g ± i and e i , ≤ i ≤ r − 1. Thanks to [18, Theorem5.41] and Corollary 5.1(2), γ | W r, K : W r, K → End K ′ ( r ) is a K -algebra isomorphism. Let γ = γ | W r, K .Thanks to [18, Propotion 6.12(1)], W aff r, K have basis { x t γ − ( d ) x s } where d ∈ NT r,r / ∼ , t , s ∈ Z r ,and x t = x t · · · x t r r and x s = x s · · · x s r r such that γ ( x t γ − ( d ) x s ) ∈ NT r,r / ∼ . By Corollary 5.1(1), NT r,r / ∼ is a basis of End AK ( ω ) ( r ). Via (4.4), one can check that γ gives a bijective map between { x t γ − ( d ) x s } and NT r,r / ∼ . So, γ has to be a K -linear isomorphism, and hence an algebra isomor-phism. (cid:3) A basis theorem of cyclotomic Kauffmann category The aim of this section is to prove Theorem 1.15 under the Assumption 1.12. We always assumethat ω is admissible. So, both AK ( ω ) and CK f are available. Thanks to Definition 1.11, there is afunctor from AK ( ω ) to CK f . It results in an epimorphism ξ : End AK ( ω ) ( r ) ։ End CK f ( r ) . Let ˜ γ = ξ ◦ γ , where γ is the K -algebra isomorphism in Theorem 5.3. Then˜ γ : W aff r, K ։ End CK f ( r )is an epimorphism such that ˜ γ ( f ( x )) = f ( X ) ⊗ r − = 0. So, ˜ γ factors through the cyclotomicBirman-Wenzl-Murakami algebra W a,r = W aff r, K /J , where J is the two sided ideal of W aff r, K generatedby f ( x ) = Q ai =1 ( x − u i ). The induced epimorphism is denoted by γ : W a,r ։ End CK f ( r ). Thecurrent W a,r is the same as B ar ( q, δ − , ω i , − b i ) in [38, Definition 1.1], where b i is the coefficient of x i in f ( x ).For all p ∈ Z and all admissible positive integers i, j, l such that i ≤ j ≤ l , Yu [38] defined α pi,j,l = x pi g i,j e j,l +1 ∈ W a,r where g i,j = g i g i +1 · · · g j − and e j,l +1 = e j e j +1 · · · e l . Note that g i,j = 1 if i = j . For any r ∈ N \ 0, let B r, r = K -span (cid:26) ~ Y l = r α s l i l ,j l , l − | i < i < ... < i r , −⌊ a − ⌋ ≤ s l ≤ ⌊ a ⌋ (cid:27) . (6.1)Then B r, r ⊆ W a, r . Recall E i in (5.1). Mimicking Yu’s construction, we define γ pi,j,ℓ = X pi T invi,j E j,ℓ U ℓ ∈ Hom CK f ( ℓ − , ℓ + )for any p ∈ Z and all positive integers i, j, ℓ such that i ≤ j ≤ ℓ , where T invi,j = T − i T − i +1 · · · T − j − and E j,ℓ = E j E j +1 · · · E ℓ − . For any positive integer r , let B , r = K -span (cid:26) ~ Y l = r γ s l i l ,j l , l − | i < i < ... < i r , −⌊ a ⌋ ≤ s l ≤ ⌊ a − ⌋ (cid:27) . (6.2)Then B , r ∈ Hom CK f ( , r ). Recall the contravariant functor σ in Lemma 1.4(1). Since σ stabilizesthe right tensor ideal of AK generated by f ( • ) together with ∆ k − ω k for all k ∈ Z , it induces acontravariant functor σ : CK f → CK f . Lemma 6.1. For any positive integer r , Hom CK f ( , ) = σ ( B , r ) . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Proof. Obviously, σ ( B , r ) ⊆ Hom CK f ( , ). Suppose d ∈ NT r, / ∼ . Thanks to (4.4) and Proposi-tion 4.9, d ∈ d r End CK f ( ), where d r = ⊗ r = σ ( ~ Y k = r γ k − , k − , k − ) ∈ σ ( B , r ) . So, d ∈ σ ( B , r ) End CK f ( ).In [38, Lemmas 2.6,2.7], Yu proved that B r, r is a left W a, r -module, where B r, r is given in (6.1).Mimicking arguments there, one can verify End C ( ) B , r ⊆ B , r without any difficulty. Note that σ = Id and σ (End C ( )) = End C ( ). So, σ ( B , r ) End C ( ) ⊆ σ ( B , r ), forcing d ∈ σ ( B , r ). ByProposition 4.6, we have σ ( B , r ) ⊇ Hom CK f ( , ). (cid:3) Proposition 6.2. As K -module, Hom CK f ( , ) is spanned by NT a r, / ∼ .Proof. Suppose ~ Q rk =1 γ s k i k ,j k , k − ∈ B , r (see (6.2)). We claim that there is a d ∈ NT a r, such that d = ~ Q rk =1 σ ( γ s k i k ,j k , k − ) as morphisms in CK f . If so, by Lemma 6.1, we immediately have the result.In order to simplify notation, let E i,i = 1 and E i,j = E i − E i − · · · E j if i > j . Thanks to (1.5)and the functor in Lemma 1.4(2), A k = A k E k − T k T k − for all k > AK and hence in CK f . So, A k T invk − ,k +1 = A k E k − . Using it together with braid relations yields A l E l,j T invj,i = A l T invl,i T invl +1 ,j +1 , for all possible i ≤ j ≤ l .Obviously, i = j = 1. So, ~ Y rℓ =1 σ ( γ s ℓ i ℓ ,j ℓ , ℓ − ) = ~ Y rℓ =1 A ℓ − E ℓ − ,j ℓ T invj ℓ ,i ℓ X s ℓ i ℓ = ~ Y rℓ =1 A ℓ − T inv ℓ − ,i ℓ T inv ℓ,j ℓ +1 r Y k =1 X s k i k = A A · · · A r − ~ Y rℓ =1 T inv ℓ − ,i ℓ T inv ℓ,j ℓ +1 r Y k =1 X s k i k = ⊗ r ~ Y rℓ =1 T inv ℓ − ,i ℓ T inv ℓ,j ℓ +1 r Y k =1 X s k i k . Define B = { ⊗ r T invw | w ∈ D r, r } , where D r, r is given in (4.6). By Proposition 4.9, as setsof morphisms in AK , B ⊇ NT r, / ∼ . Thanks to Theorem 1.9, the number of morphisms in NT r, / ∼ is ♯ D r, r = (2 r − ♯B ≤ ♯ D r, r , B = NT r, / ∼ in AK . So, Proposition 4.9gives an explicit bijection between NT r, / ∼ and B . Moreover, there is a d ∈ NT r, such that d = ⊗ r ~ Q rℓ =1 T inv l − ,i l T inv l,j l +1 and any i ℓ is a left endpoint of a cap in d . Then the claim follows ifwe assume d = d ◦ Q rk =1 X s k i k . (cid:3) Assumption 6.3. Until the end of Proposition 6.20, we assume 2 r ≤ min { q , q , . . . , q k } . We alsokeep the setting for parabolic quantum groups in section 3. Fix I ∈ { I , I } in (3.5) such that g = so n +1 when I = I . Moreover, a is always deg f I ( t ) and k = ⌊ ( a − / ⌋ + 1, where f I ( t ) is givenin Definition 3.6.Suppose d ∈ NT a r, / ∼ such that conn ( ˆ d ) = { ( i l , j l ) | ≤ l ≤ r } . For any c ∈ Z and 0 ≤ c ≤ a − a i l ,c = p c + l, if 0 ≤ c ≤ k − p k − c − l + 1 , if I = I and k ≤ c ≤ a − p k − c − − l + 1 , if I = I and k ≤ c ≤ a − p = 0 and p j = P ji =1 q i . Since we are assuming q j ≥ r for all admissible j , we have thefollowing Lemma, immediately. Lemma 6.4. Suppose b, c ∈ { , , . . . .a − } and ≤ l ≤ r . We have (1) p c < a i l ,c ≤ p c + r ≤ p c +1 − r if ≤ c ≤ k − , (2) p k − c − + r ≤ p k − c − r < a i l ,c ≤ p k − c if I = I and k ≤ c ≤ a − , (3) p k − c − + r ≤ p k − c − − r < a i l ,c ≤ p k − c − if I = I and k ≤ c ≤ a − , (4) a i j ,b = a i l ,c if and only if ( j, b ) = ( l, c ) , (5) a i j ,b = l if and only if ( j, b ) = ( l, . Lemma 6.5. Suppose ≤ l ≤ r . Then β l,c ∈ R + \R + I , and β l,c < β l,c +1 if ≤ c < k and β l,c +1 < β l,c if k ≤ c < a − , where β l,c = ε a il,c − − ε a il,c , if ≤ c ≤ k − , ε a il,k + ε a il,k − , if c = k , ε a il,c − ε a il,c − , if k < c ≤ a − . (6.3) Proof. The result follows from Lemma 6.4 and (2.17). (cid:3) Example 6.6. Suppose ( k, q , q ) = (2 , , 9) in (3.5). Let ( r, d ) = (2 , γ ), where γ is given in (4.1).Then conn ( d ) = { (1 , , (2 , } . If ( I, a ) = ( I , • a , = 1, a , = 5, a , = 13, a , = 4, • a , = 2, a , = 6, a , = 12, a , = 3, • β , = ε − ε , β , = ε + ε , β , = ε − ε , and β , < β , < β , , where < is the convexorder in (2.17). • β , = ε − ε , β , = ε + ε , β , = ε − ε , and β , < β , < β , . Definition 6.7. Suppose d ∈ NT a r, / ∼ and conn ( ˆ d ) = { ( i l , j l ) | ≤ l ≤ r } . For 1 ≤ l ≤ r and1 ≤ c ≤ a − 1, define x ± ( d ) i l ,c = ~ Q cj =1 x ± β l,j and x ± ( d ) i l , = 1, where β l,j is given in (6.3).Given two sequences of positive roots K, H , we write x − K ∼ x − H and x + K ∼ x + H if K can be obtainedfrom H by place permutation. Recall V is the natural U v ( g )-module with basis { v , v , ..., v N } and j ′ = N + 1 − j for all 1 ≤ j ≤ N. For any positive root β , we have already described the action of x + β on V in Lemmas 2.9-2.12. We will freely use those results in the proof of Lemma 6.8. Lemma 6.8. Keep the notations in Definition 6.7. Suppose x + H ∼ x + ( d ) i l ,c . Assume h = a i l ,c if ≤ c ≤ k − and h = a ′ i l ,c otherwise. Then x + ( d ) i l ,c v h = yv l for some y ∈ A \ A , and x + H v h = 0 if and only if x + H = x + ( d ) i l ,c .Proof. If c = 0 there is nothing to prove. Via Lemma 6.4 and (6.3), x + β l,j v h = 0 if and only if j = c .Further, x + β l,c v h = y v a il,c − , if 1 ≤ c ≤ k − y v a il,k − , if c = k , y v a ′ il,c − , if k < c ≤ a − y , y , y ∈ A \ A . Note that a i l , = l . The result follows from induction on c . (cid:3) Recall S I,l in (3.8) for any l ∈ N and R + = { β j | ≤ j ≤ ℓ ( w ) } such that β i < β j in the sense of(2.17) if i < j . To simplify the notation, we write x − i = ~ Y j = ℓ ( w − ,I w ) ( x − β tj ) i j for any i ∈ N ℓ ( w − ,I w ) , then S I,l = n x − i m I ⊗ v j | i ∈ N ℓ ( w − ,I w ) , j ∈ N l o . We say x − i is of degree | i | = P ℓ ( w − ,I w ) j =1 i j . In this case, we also say x − i m I ⊗ v j is of degree | i | . Definition 6.9. For any l, j ∈ N , let M jI,l be the free A -module with basis given by { z deg( y )+ jv y | y ∈ S I,l } , where A is given in Definition 2.1.Obviously, M I,l ⊂ M I,l and z v M I,l = M I,l . Lemma 6.10. Suppose i ∈ N ℓ ( w − ,I w ) . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY (1) If β ∈ R + I , then z | i | +1 v x − β x − i m I ≡ M I, ) . (2) If β ∈ R + \ R + I (see (3.7) ), then z | i | +1 v x − β x − i m I ≡ z | i | +1 v v c x − s m I (mod M I, ) for some c ∈ Z and s ∈ N ℓ ( w − ,I w ) such that x − s ∼ x − β x − i .Proof. Obviously, (1)-(2) hold if x − i = 1. We assume x − i = 1 and hence | i | ≥ 1. Suppose x − c ∼ x − β x − i and R + I = { β j l | ≤ l ≤ ℓ ( w ,I ) } such that j a < j b for all admissible a < b . Thanks to Lemma 2.2and Proposition 2.6, for any x − H such that x − H ∼ x − β x − i , we have z | i | +1 v ( x − H − v b x − c ) = X r ∈ N ℓ ( w a r z | r | +1 v x − r (6.4)for some b ∈ Z and some a r ∈ A . First of all, we assume x − β x − i = x − c . By Lemma 2.2, a r = 0 if r t = 0 for some t such that β t ≥ β .If β 6∈ R + I , then x − c = x − ˜ c , where ˜ c ∈ N ℓ ( w − ,I w ) such that ˜ c j = c t j for all admissible j . In thiscase, (2) automatically holds. Otherwise, β ∈ R + I and β = β j l for some j l > t ℓ ( w − ,I w ) . Suppose x − H = x − i x − β jl in (6.4). Note that x − i x − β jl m I = 0. We claim the RHS of (6.4) acts on m I is in M I, . Ifso, we have (1), immediately.We prove our claim by induction on l . If l = 1 and β = β j , any term in the RHS of (6.4) actson m I is in M I, , and the claim follows. Suppose l > 1. Any monomial x − s in RHS of (6.4) satisfies s j = 0 unless β j < β j l . If P ℓ ( w ,I ) l =1 s j l = 0, then z | s | +1 v x − s m I ∈ M I, , and there is nothing to prove.Otherwise, let t be the minimal number such that s j t = 0, then j t < j l (i.e. t < l ). By inductionassumption on t and Lemma 2.2, z P jt − ℓ =1 s ℓ +1 v x − β jt ~ Y i = j t − ( x − β i ) s i m I = X r ∈ N ℓ ( w − ,Iw b r z | r | +1 v x r m I where b r ∈ A and b r = 0 unless r l = 0 for all 1 ≤ l ≤ ℓ ( w − ,I w ) such that t l > j t . So, z | s | v x − s m I = z P ℓ ( w ℓ = jt s ℓ − v ~ Y j t +1 i = ℓ ( w ) x − β i ( x − β jt ) s jt − X r ∈ N ℓ ( w − ,Iw b r z | r | +1 v x r m I . By induction assumption on P ℓ ( w ,I ) l =1 s j l − 1, we have z | s | +1 v x − s m I ∈ M I, , proving the claim.We have proved (1)-(2) when x − β x − i = x − c . In general, we assume x − H = x − β x − i . Using (6.4), we seethat the general case follows from those when x − β x − i = x − c . (cid:3) Corollary 6.11. Suppose H, K are sequences of positive roots and j ∈ { , } . Then (1) z ℓ ( K ) v x − K ⊗ x − H stabilizes M jI, , (2) z ℓ ( K ) v x − K ⊗ x + H stabilizes M jI, .Proof. For any basis element v l ∈ V , by Lemmas 2.9–2.12, x + H v l = g v k (resp., x − H v l = g v k ) forsome g , g ∈ A \ A and some v k , v k ∈ V if x + H v l = 0 (resp., x − H v l = 0). Then the (1)-(2) followfrom Lemma 6.10. (cid:3) Lemma 6.12. Suppose Φ ∈ { Θ , Θ } . For any s ∈ N ℓ ( w − ,I w ) and any basis element v l of V , we have (1) z | s | v P Φ P ( x − s m I ⊗ v l ) ≡ z | s | v x − s m I ⊗ v l (mod M I, ) , (2) P Φ P ( M jI, ) ⊆ M jI, , j = 0 , .Proof. We prove (1) by induction on P ℓ ( w − ,I w ) j =1 s j β t j . Since x + β m I = 0 for any positive root β , by(3.2), P Φ P ( m I ⊗ v l ) = m I ⊗ v l . This proves (1) when P ℓ ( w − ,I w ) j =1 s j β t j = 0. Otherwise, we pick ℓ , the maximal number such that s ℓ = 0. Let y Φ = z | s | v P Φ P ( x − s m I ⊗ v l ) and c ∈ N ℓ ( w − ,I w ) such that c k = s k for any k = ℓ and c ℓ = s ℓ − 1. Then, y Θ − z | s | v P Θ(∆( x − β tℓ )( v l ⊗ x − c m I ))= − z | s | v P Θ( X K,H g K,H z ℓ ( H ) v x − K v l ⊗ k wt ( K ) x − H x − c m I ) , by Proposition 2.7(2)= − z v P Θ P ( X K,H g K,H z ℓ ( H )+ | c | v k wt ( K ) x − H ⊗ x − K )( x − c m I ⊗ v l ) ∈ M I, , by Corollary 6.11(1) and induction assumption on ℓ ( w − ,I w ) X j =1 s j β t j − wt ( K ) , where K, H are sequences of positive roots such that wt ( K ) + wt ( H ) = β t ℓ , K = ∅ and g K,H ∈ A .In particular, g K,H = 1 when ( K, H ) = ( β t ℓ , ∅ ). We have z | s | v P Θ(∆( x − β tℓ )( v l ⊗ x − c m I )) = z | s | v P ∆( x − β tℓ )Θ( v l ⊗ x − c m I ) , by (2.19)= z v P ∆( x − β tℓ ) P P Θ P ( z | c | v x − c m I ⊗ v l )= z v P ∆( x − β tℓ ) P ( z | c | v x − c m I ⊗ v l + x ) , by induction assumption on ℓ ( w − ,I w ) X j =1 c j β t j ,where x ∈ M I, . By Proposition 2.7(1), as operators in End( M p I ( λ I, c ) ⊗ V ), we have z v P ∆( x − β tℓ ) P − z v x − β tℓ ⊗ z v k − β tℓ ⊗ x − β tℓ + X K ,H z ℓ ( H )+1 v h K ,H k − wt ( K ) x − H ⊗ x − K , (6.5)where K , H are non-empty sequences of positive roots such that wt ( K ) + wt ( H ) = β t ℓ and h K ,H ∈ A . Thanks to Corollary 6.11(1), the RHS of (6.5) sends M I, to M I, and z v x − β tℓ ⊗ M I, . Therefore, y Θ ≡ ( z v x − β tℓ ⊗ z | c | v x − c m I ⊗ v l ) (mod M I, ) ≡ z | s | v x − s m I ⊗ v l (mod M I, ) . This proves (1) for Θ. Using (3.1) instead of (2.19), one can check the result on y Θ similarly. Finally,the last assertion follows from previous result on y Φ . (cid:3) Lemma 6.13. Keep the setting above. (a) z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) ( X t ) ∈ A if one of conditions holds: (1) ≤ − t < c < k , (2) k ≤ c ≤ a − ,and − k < t ≤ c − k , (b) z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) ( X t ) ∈ A \ A if and only if one of conditions holds: (1) h = a i l ,c providedthat t = − c and < c < k , (2) h = a ′ i l ,c provided that either t = c − k + 1 and k ≤ c ≤ a − or a is even and ( t, c ) = ( − k, a − .Proof. First, we discuss the actions of Θ t and Θ t on M p I ( λ I, c ) ⊗ V . By Lemmas 2.9-2.12, ( x ± β ) = 0in End( V ) for any β ∈ R + . Thanks to (3.2), for any positive j ≥ 1, we haveΘ j = X H g H,j z ℓ ( H ) v x − H ⊗ x + H , Θ j = X H ′ h H ′ ,j z ℓ ( H ′ ) v x − H ′ ⊗ x + H ′ , (6.6)where H (resp., H ′ ) ranges over all sequences of positive roots such that ℓ ( H ) < ∞ (resp., ℓ ( H ′ ) < ∞ ) and g H,j , h H ′ ,j ∈ A . In order to divide such H, H ′ into different classes, we define ψ ( H ) = { s | β ℓ s > β ℓ s +1 , ≤ s ≤ b − } and φ ( H ) = { s | β ℓ s < β ℓ s +1 , ≤ s ≤ b − } , where H = ( β ℓ , ..., β ℓ b ). Thanks to (3.2), ψ ( H ) ≤ j − φ ( H ′ ) ≤ j − H and H ′ are those in (6.6)and g H,j ∈ A \ A (resp., h H ′ ,j ∈ A \ A ) if ψ ( H ) = j − φ ( H ′ ) = j − HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Thanks to Corollary 6.11(2) and (2.27),Φ( M jI, ) ⊆ M jI, , if j ∈ { , } and Φ ∈ { Θ , Θ , π, π − } . (6.7)It is easy to see P π = πP . Now, we use Lemma 6.12 to obtain:( P π − Θ P π − Θ) ± b ( z | s | v x − s m I ⊗ v h ) ≡ z | s | v ( π − Θ) ± b ( x − s m I ⊗ v h ) (mod M I, ) (6.8)for b ≥ x − s and v h are given before this lemma. If c ( s ,h ) , ( r ,l ) ( X t ) = 0, then wt ( x − s m I ⊗ v h ) = wt ( x − r m I ⊗ v l ), and hence wt ( v h ) = ( ε a il,c , if 0 ≤ c ≤ k − − ε a il,c , if k ≤ c ≤ a − . (6.9)In other words, h = ( a i l ,c , if 0 ≤ c ≤ k − a ′ i l ,c , if k ≤ c ≤ a − 1. (6.10)In the following, we always assume that (6.9) holds. Otherwise, c ( s ,h ) , ( r ,l ) ( X t ) = 0, and there isnothing to prove.For any sequence J of positive roots with ℓ ( J ) < ∞ , let c ( s ,h ) , ( r ,l ) ( J ) be the coefficient of x − r m I ⊗ v l in the expression of z | s | + ℓ ( J ) v x − J ⊗ x + J ( x − s m I ⊗ v h ). Let J be the sequence of positive roots such that x ± J = x ± ( d ) i l ,c . Then ℓ ( J ) = c . Thanks to Lemma 6.10, z − ( c + | s | ) v c ( s ,h ) , ( r ,l ) ( J ) ∈ A , if x − J ≁ x − J . (6.11)By Lemma 6.8, we have x + J v h = 0 , for any J such that x + J ∼ x + J and J = J , (6.12)and x + J v h = f v l for some f ∈ A \ A . By Lemma 6.10, z ( c + | s | ) v x − J x − s m I ≡ z ( c + | s | ) v v p x − r m I (mod M I, )for some p ∈ Z . Therefore, z − ( c + | s | ) v c ( s ,h ) , ( r ,l ) ( J ) ∈ A \ A . (6.13)For any t ∈ {⌊ a − ⌋ , ⌊ a − ⌋ − , . . . , −⌊ a ⌋} , let c ( s ,h ) , ( r ,l ) (Θ t ) be the coefficient of x − r m I ⊗ v l in theexpression of z | s | v Θ t ( x − s m I ⊗ v h ). Thanks to Lemma 6.5, ψ ( J ) = ( , if 1 ≤ c ≤ kc − k, if k < c ≤ a − φ ( J ) = ( c − , if 1 ≤ c ≤ kk − , if k < c ≤ a − . (6.14)So, • ψ ( J ) ≥ t if k ≤ c ≤ a − 1, 0 ≤ t ≤ c − k , • φ ( J ) ≥ − t if 0 < c < k , − c < t ≤ k ≤ c ≤ a − − c < t ≤ • ψ ( J ) = t − k ≤ c ≤ a − t = c − k + 1, • φ ( J ) = − t − < c < k , t = − c or 2 | a and ( c, t ) = ( a − , − k ).Suppose t is given in (a). If t ≤ 0, then φ ( J ) ≥ − t . Thanks to (6.6), x − J ⊗ x + J can not appear in theexpression of Θ − t . By (6.11)-(6.12), z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) (Θ − t ) ∈ A . If t > 0. Then ψ ( J ) ≥ t . Thanksto (6.6), x − J ⊗ x + J can not appear in the expression of Θ t . By (6.11)-(6.12), z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) (Θ t ) ∈ A .In any case, z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) (Θ t ) ∈ A . (6.15)If t is given in (b)(1) or 2 | a and ( c, t ) = ( a − , − k ) in (b)(2), then t < φ ( J ) = − t − 1. Thanksto (6.6), x − J ⊗ x + J do appear in Θ − t with coefficient z cv h J ,t , where h J ,t ∈ A \ A . By (6.11)-(6.13), z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) (Θ − t ) ∈ A \ A . If t is given in (b)(2) and t > 0, then ψ ( J ) = t − 1. Thanks to (6.6), x − J ⊗ x + J appears in Θ t with coefficient z cv g J ,t , where g J ,t ∈ A \ A . By (6.11)-(6.13), z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) (Θ t ) ∈ A \ A . In any case, z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) (Θ t ) ∈ A \ A . (6.16)We have Ψ M p I ( λ I, c ) ( X t ) = ( δP π − Θ P π − Θ) t , where δ is given in (2.31). So both (a) and (b)follow from (6.8) and (2.27), immediately. (cid:3) Assume N a = { , , , ..., a − } . Let φ : {⌊ a − ⌋ , ⌊ a − ⌋ − , . . . , −⌊ a ⌋} → N a be the map such that φ ( j ) = ( − j, if − k < j ≤ k + j − , if 0 < j ≤ k − 1, (6.17)and φ ( − k ) = a − 1, which is available only if a = 2 k . Corollary 6.14. Suppose Y ∈ { X, Θ } and c ∈ N a . (1) If φ ( t ) < c , then z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) ( Y t ) ∈ A . (2) Suppose φ ( t ) = c . Then z − ( | s | + c ) v c ( s ,h ) , ( r ,l ) ( Y t ) ∈ A \ A if and only if h is given in (6.10) .Proof. Note that conditions in (1) (resp., (2)) is equivalent to the conditions in Lemma 6.13(a) (resp.,(b)). So, the current results follow from Lemma 6.13 when Y = X . If Y = Θ, the results follows from(6.15) and (6.16). (cid:3) For any 1 ≤ i ≤ r − 1, define˜ R i = Id M p I ( λ I, c ) ⊗ Id ⊗ i − V ⊗ ˜ R ⊗ Id ⊗ r − i − V , where ˜ R ∈ End( V ⊗ ) such that ˜ R ( v j ⊗ v l ) = v ( wt ( v j ) | wt ( v l )) v j ⊗ v l for any v j , v l ∈ V . Define˜ X ± j = ˜ R j − ˜ X ± j − ˜ R j − , ≤ j ≤ r, (6.18)where ˜ X ± = ( δπ − Θ) ± ⊗ Id ⊗ r − V , and δ is given in (2.31). For any ψ, ψ ∈ End( M I, r ), we write ψ ≈ ψ if ψ ( y ) ≡ ψ ( y ) (mod M I, r ) for any y ∈ M I, r .As mentioned before, we use d to denote Ψ M p I ( λ I, c ) ( d ) for any admissible d . The following resultsfollow immediately from Lemma 2.14 and (6.8). Lemma 6.15. As morphisms in End( M I, r ) , we have (1) T ± j ≈ ˜ R j for all ≤ j ≤ r − , (2) X ± ≈ ˜ X ± . Lemma 6.16. If d ∈ { T , . . . , T r − } ∪ { X , . . . , X r } , and j ∈ { , } , then d ± M jI, r ⊆ M jI, r .Proof. Since T ± ℓ acts on M I, r via Id M p I ( λ I, c ) ⊗ Id ⊗ ℓ − V ⊗ R ± V,V ⊗ Id ⊗ r − ℓ − V , by Lemma 2.14, wehave the result when d ∈ { T , . . . , T r − } . Let Φ ∈ { Θ , Θ } . Thanks to Lemma 6.12, M jI, r is fixed by P Φ P . By (6.7), M jI, r is fixed by Θ , Θ and π ± . Note that Ψ M p I ( λ I, c ) ( X ± ) = ( δP π − Θ P π − Θ) ± .So we have the result when d = X . The general case follows from the equation X j = T j − X j − T j − . (cid:3) Definition 6.17. Suppose d ∈ NT a r, / ∼ and conn ( ˆ d ) = { ( i l , j l ) | ≤ i ≤ r } . For any e ∈ N ra , define(1) ˜ x − ( d ) e = x − s such that x − s ∼ ~ Π rl =1 x − ( d ) i l ,e l , where x − ( d ) i l ,e l is given in Definition 6.7,(2) v ( d ) e = v b ⊗ . . . ⊗ v b r ∈ V ⊗ r such that b j l = l ′ and b i l = a i l ,e l (resp., a ′ i l ,e l ) if 0 ≤ e l ≤ k − k ≤ e l ≤ a − Example 6.18. Keep the notations in Example 6.6. We have • β , < β , < β , < β , < β , < β , HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY • v ( d ) e = v ⊗ v ⊗ v ′ ⊗ v ′ and ˜ x − ( d ) e = 1 if e = (0 , • v ( d ) e = v ′ ⊗ v ′ ⊗ v ′ ⊗ v ′ and ˜ x − ( d ) e = x − β , x − β , x − β , x − β , x − β , x − β , , if e = (3 , d, d ′ ∈ NT a r, / ∼ and e ∈ N ra . Let c d,d ′ be the coefficient of ˜ x − ( d ′ ) e m I in d ( m I ⊗ v ( d ′ ) e ).It is reasonable since d ( m I ⊗ v ( d ′ ) e ) can be expressed as a linear combination of elements in S I, and ˜ x − ( d ′ ) e m I ∈ S I, (see Lemma 3.5). Thanks to (4.4), d = ˆ d Q rl =1 ◦ X b d,l l . Suppose conn ( ˆ d ) = { ( i l , j l ) | ≤ l ≤ r } . Then, b d,j l = 0, 1 ≤ l ≤ r . Define e ( d ) = ( φ ( b d,i ) , . . . , φ ( b d,i r )),where φ is given in (6.17). For any t , s ∈ N ra , we say t < s if t l ≥ s l , 1 ≤ l ≤ r . Lemma 6.19. Suppose e ∈ N ra and d, d ′ ∈ NT a r, / ∼ such that d = ˆ d Q rl =1 X b d,l l . We have (1) z −| e | v c d,d ′ ∈ A if either e ( d ) < e or e ( d ) = e and η ( d ) = η ( d ′ ) , where | e | = P ri =1 e i and η ( d ) is given in Definition 4.14. (2) z −| e | v c d,d ′ ∈ A \ A if e ( d ) = e and η ( d ) = η ( d ′ ) .Proof. For all 0 ≤ b < c ≤ r , let φ b,c : U v ( g ) ⊗ → U v ( g ) ⊗ (2 r +1) be the linear map such that φ b,c ( x ⊗ y ) = 1 ⊗ b ⊗ b +1th x ⊗ ⊗ ( c − b − ⊗ c +1th y ⊗ ⊗ (2 r − c ) , for all x ⊗ y ∈ U ( g ) ⊗ . Given an m I ⊗ v ( d ′ ) e ∈ S I, r , where v ( d ′ ) e is defined in Definition 6.17, weclaim ( d − v s ˆ d ~ Y rl =1 φ ,l (Θ) b d,l )( m I ⊗ v ( d ′ ) e ) ∈ M I, for some s ∈ Z depending on m I ⊗ v ( d ′ ) e .In fact, by Lemmas 6.15-6.16, we have Q rl =1 X b d,l l ≈ ~ Q rl =1 ˜ X b d,l l . Thanks to (2.27) and (6.18) ~ Y rl =1 ˜ X b d,j l ( m I ⊗ v ( d ′ ) e ) = v t ~ Y rl =1 φ ,l (Θ) b d,l ( m I ⊗ v ( d ′ ) e ) (6.19)for some t ∈ Z depending on m I ⊗ v ( d ′ ) e . By Proposition 4.9, ˆ d = d r T invw for some w ∈ D r, r , where d r = ⊗ r ∈ NT r, . By Lemmas 2.15, 6.16, d r M jI, r ⊆ M jI, and ˆ dM jI, r ⊆ M jI, if j ∈ { , } .Combining these results yields our claim.We write the RHS of (6.19) as a linear combination of S I, r in Lemma 3.5. For any v i = v i ⊗ v i ⊗ ... ⊗ v i r ∈ V ⊗ r , let a d,d ′ , i be the coefficient of ˜ x − ( d ′ ) e m I ⊗ v i in this expres-sion. Thanks to Corollary 6.14, z −| e | v a d,d ′ , i ∈ A if e ( d ) < e . Since ˆ dM I, r ⊆ M I, , we immediatelyhave z −| e | v c d,d ′ ∈ A in this case. This is the first assertion in (1).Now, we assume e = e ( d ). By Corollary 6.14 again, z −| e | v a d,d ′ , i ∈ A \ A if and only if i = η ( d ′ ),where η ( d ′ ) is given in Definition 4.14.Let b d,d ′ , i be the coefficient of ˜ x − ( d ′ ) e m I in ˆ d ( z | e | v ˜ x − ( d ′ ) e m I ⊗ v i ). We have z −| e | v b d,d ′ , i ∈ A since z | e | v ˜ x − ( d ′ ) e m I ⊗ v i ∈ M I, r and ˆ dM I, r ⊆ M I, .Thanks to (4.13), we have z −| e | v b d,d ′ , i ∈ A if i = η ( d ) and z −| e | v b d,d ′ , i ∈ A \ A if i = η ( d ). So, z −| e | v c d,d ′ ∈ A if η ( d ) = η ( d ′ ) and z −| e | v c d,d ′ ∈ A \ A if η ( d ) = η ( d ′ ). This proves (2) and the secondassertion in (1). (cid:3) By (2.5), v = q if g ∈ { sp n , so n } and v = q / if g = so n +1 . In Proposition 6.20, we assume (1) K = C ( q / ), (2) z = z q , (3) δ is given in (2.31), (4) ω is determined by (1.1), (5) ω is u -admissibleand (6) f I ( t ) is given in Definition 3.6 where I ∈ { I , I } . Proposition 6.20. Hom CK fI ( , ) has K -basis given by NT a r, / ∼ .Proof. By Proposition 6.2, it is enough to prove f d ( q / ) = 0 for all d if X d ∈ B f d ( q / ) d = 0 , where B is a finite subset of NT a r, / ∼ and f d ( q / ) ∈ K .If it were false, we assume all previous f d ( q / )’s are non-zero. Without loss of any generality, wecan assume that f d ( q / ) ∈ A for all previous d and moreover, there is at least one d ′ such that f d ′ ( q / ) ∈ A \ A . Let B = { d ∈ B | f d ( q / ) 6∈ A } and B = { d ∈ B | | e ( d ) | is maximal } , where e ( d ) is given before Lemma 6.19. Obviously B = ∅ .Keep the notation in Definition 6.17. Thanks to Lemma 6.19, the coefficient of ˜ x − ( d ) e ( d ) m I in P d ∈ B f d ( q / ) d ( m I ⊗ v e ( d ) ) is f d ( q / ) z | e ( d ) | v ( g ( q / ) + h ( q / )) for any d ∈ B , where g ( q / ) ∈ A \ A and h ( q / ) ∈ A . Then f d ( q / ) = 0 ∈ A , a contradiction since d ∈ B . (cid:3) In Theorem 6.21, we keep the following assumptions: • Z = Z [ˆ q ± , (ˆ q − ˆ q − ) − , ˆ u ± , . . . , ˆ u ± a ], where ˆ q and ˆ q − ˆ q − , ˆ u , . . . , ˆ u a are indeterminates. • ˆ f ( t ) = Q aj =1 ( t − ˆ u j ) ∈ Z [ t ], • z = z ˆ q , δ = α Q ai =1 ˆ u i where α ∈ { , − } if a is odd and α ∈ {− ˆ q, ˆ q − } if a is even, • δ, ω , z satisfy (1.1), • { ˜ ω j | j ∈ Z } is ˆ u -admissible in the sense of Definition 1.13 and ˜ ω = ω . Theorem 6.21. The Z -module Hom CK ˆ f ( , ) has basis given by NT a r, / ∼ .Proof. We assume g = so n (resp., sp n ) if α ∈ { ˆ q − , } (resp., α ∈ {− ˆ q, − } ), where n = P kj =1 q j such that 2 r ≤ min { q , q , . . . , q k } and k = ⌊ a − ⌋ + 1. We also assume I = I if a = 2 k and I = I if a = 2 k − 1, where I , I is given in (3.5). This enables us to use freely previous results in section 3and those in this section. In any case, v = q , where v is the defining parameter for U v ( g ) over F . Forany s ≥ i = ( i , i , . . . , i s ) ∈ Z s , let q i = ( q i , q i , . . . , q i s ) . For any a ∈ { k, k − } , we specialize (ˆ q, ˆ u , . . . , ˆ u a ) at ( q, ε g q b a ) where b k = ( b , . . . , b k ) and b k − is obtained from b k by removing b k +1 . In any case, b j ’s are given in (3.10) and ˆ f ( t ) is specializedto f I ( t ) in Definition 3.6, respectively. Since { ˜ ω j | j ∈ Z } is ˆ u -admissible, such ˜ ω j ’s are uniquelydetermined by ˆ u j ’s. So, ˜ ω is specialized to ω in Lemma 3.11 for g ∈ { sp n , so n } with respect to α .Since C ( q / ) is a Z -module on which ˆ q, ˆ u , . . . , ˆ u a act via q and ε g q b a in an obvious way, thereis a Z -linear monoidal functor F : AK Z → AK C ( q / ) sending generators to generators with thesame names. Let J I be the right tensor ideal of AK C ( q / ) generated by f I ( X ), ∆ j − ω j , j ∈ Z .Since F (∆ j − ˜ ω j ) = ∆ j − ω j for all j ∈ Z , and F ( ˆ f ( X )) = f I ( X ), F induces a Z -linear functor˜ F : CK ˆ f Z → CK f I C ( q / ) , where CK f I C ( q / ) = AK C ( q / ) /J I .Now, we prove P d ∈ NT a r, / ∼ g d d = 0 only if g d = 0 for all d , where g d ∈ Z . In fact, we have P d ∈ NT a r, / ∼ g d ( q, ε g q b a ) d = 0 where g d ( q, ε g q b a ) = ˜ F ( g d ). By Proposition 6.20, g d ( q, ε g q b a ) = 0 forall d ∈ NT a r, / ∼ . Thanks to (3.10), b j = ( c j + P j ≤ l ≤ k q l ) − ε g , if 1 ≤ j ≤ k − c k − j +1 − P k +2 ≤ l ≤ j q k − l +2 ) + ε g , if k + 1 ≤ j ≤ k . (6.20)So, we can view b j ’s as polynomials of q j ’s and c l ’s, where 1 ≤ j ≤ k and 1 ≤ l ≤ k (resp., 1 ≤ l ≤ k − a = 2 k (resp., a = 2 k − v a = ( ( b , . . . , b k , b k , . . . , b a − k +1 ) , if a = 2 k ,( b , . . . , b k , b k , . . . , b a − k +3 ) , if a = 2 k − 1. (6.21) HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Let φ a : C a → C a be the morphism such that φ a ( q , . . . , q k , c , . . . , c a − k ) = v a . Thanks to (6.20), theJacobi matrix J φ a of φ a satisfies J φ k = (cid:18) H k E − H k + 2 E − E (cid:19) , where E is the k × k identity matrix and H k is the upper-triangular matrix such that the ( l, j )th entryis 2 for all admissible j ≥ l . The Jacobi matrix J φ k − is obtained from J φ k by deleting its 2 k th rowand 2 k th column. It is routine to check thatdet J φ k = ( − k k and det J φ k − = ( − k − k − .So, φ a is always dominant. Define O a = { ( q , . . . , q k , c , . . . , c a − k ) | c t , q j ∈ Z and q j ≥ r, ≤ j ≤ k, ≤ t ≤ a − k } . Then O a is Zariski dense in C a . Since φ a is dominant, φ a ( O a ) is Zariski dense in C a . For any a ∈ { k, k − } define φ ( O a ) q = { ε g q v a | v a ∈ φ a ( O a ) } . Suppose 0 = q ∗ ∈ C such that q ∗ is nota root of 1. Specializing q at q ∗ yields a bijection between φ a ( O a ) q ∗ and φ a ( O a ). This shows that φ a ( O a ) q ∗ is Zariski dense in C a . This observation together with g d ( q, ε g q b a ) = 0 yield g d = 0. So, NT a r, / ∼ is Z -linear independent. Now, the result follows immediately from Proposition 6.2. (cid:3) Proof of Theorem 1.15 . If m + s is odd , then Hom CK f ( m , s ) = 0. So, we assume m + s = 2 r forsome r ∈ N .“ ⇐ =”: By Theorem 6.21, Hom CK ˆ f ( r, ) has Z -basis given by NT r, . By arguments on base change(see, e.g in [32]), we see that NT a r, / ∼ is linear independent over K . Thanks to Proposition 6.2,Hom CK ˆ f ( r, ) has K -basis given by NT a r, / ∼ . In general, the result follows from Lemma 4.2(3).“ = ⇒ ” : At the beginning of this section, we have explained that there is an algebra epimor-phism γ : W a,r → End CK f ( r ), where W a,r is the cyclotomic Birman-Wenzl-Murakami algebra. Good-man [17] has proved that W a,r is always free over K with rank c r ((2 r − − r !) + a r r !, where c isthe minimal positive integer such that e f ( x ) = 0 and c = deg f ( x ) ≤ a . Since γ is an epimor-phism and rank(End CK f ( r )) = a r (2 r − c = a . This shows that e , e x , . . . , e x a − is K -linear independent. Otherwise, we can find a f ( x ) such that c = deg f ( x ) < a , andrank( W a,r ) = c r ((2 r − − r !) + a r r ! < a r (2 r − W a,r is admissible inthe sense of [36, Corollary 4.5]. Goodman [17] proved that W a,r is admissible if and only if ω is u -admissible in our sense. This completes the proof. (cid:3) The following result follows immediately from arguments above. Corollary 6.22. Keep the Assumption 1.12. Suppose ω is u -admissible in the sense of Definition 1.13.Then W a,r ∼ = End CK f ( r ) as K -algebras. Appendix A. Proof of Proposition 2.6 The aim of this section is to prove Proposition 2.6. More explicitly, we need to prove c ± r ∈ A | r |− ,where c ± r are given in Lemma 2.2. Via the C -linear anti-automorphism τ of U v ( g ) and (2.12), it isenough for us to deal with c − r . So, we compute [ x − ν , x − α ] v = x − ν x − α − v ( ν | α ) x − α x − ν for any positive roots α, ν ∈ R + such that ν < α . Our arguments depend on minimal pairs in Corollary 2.5 with respect tothe convex order in (2.17). For such a pair α and ν , there are four cases we have to consider:(1) α + ν ∈ R + and ( α, ν ) is a minimal pair.(2) α + ν ∈ R + and { α, ν } is not a minimal pair.(3) α + ν / ∈ R + and there are β, γ ∈ R + such that α > β ≥ γ > ν and α + ν = β + γ .(4) α + ν / ∈ R + and there are no β, γ ∈ R + such that α > β ≥ γ > ν and α + ν = β + γ . In case (1), ( α, ν ) is one of minimal pairs in Corollary 2.5. In Lemma A.1-Lemma A.3, We givedetailed information on { α, ν } which appears in (2)-(4). Lemma A.1. Suppose α, β, γ, ν ∈ R + such that α > β > γ > ν and α + ν = β + γ ∈ R + . Then α + ν ∈ { ε i + ε j , ε k | i < j, k < n − } . Further, (1) If α + ν = ε i + ε j , then α ∈ { ε j − ε k , ε j + ε l , ε j , ε j | k > j + 1 , l > j } . In this case,a ) β = ε j − ε t , j < t < k , if α = ε j − ε k ,b ) β = ε j − ε f , j < f ≤ n , if α = ε j , ε j ,c ) β ∈ { ε j + ε s , ε j − ε t , ε j | s > l, t > j } , if α = ε j + ε l . (2) If α + ν = 2 ε i , then α ∈ { ε i + ε j | i < j < n } . In this case, β = ε i + ε k , where j < k ≤ n .Proof. We decompose a positive root µ into a summation of two positive roots as follows. If µ ∈ { ε i − ε j , ε i + ε j } and i < j , we have • ε i − ε j = ( ε m − ε j ) + ( ε i − ε m ) , i < m < j , • ε i + ε j = ( ε j − ε k ) + ( ε i + ε k ) = ( ε j + ε l ) + ( ε i − ε l ) = ( ε i ) + ( ε j ) = (2 ε j ) + ( ε i − ε j ) , j < k , i < l = j .Suppose µ ∈ { ε i , ε i } . Then ε i = ( ε k ) + ( ε i − ε k ) and 2 ε i = ( ε i + ε k ) + ( ε i − ε k ). Thanks to (2.17), α + ν / ∈ { ε i − ε j , ε k } for all admissible i, j, k . Now, (1)-(2) follow from (2.17). (cid:3) Lemma A.2. Suppose α, β, γ, ν ∈ R + such that α > β ≥ γ > ν and α + ν = β + γ 6∈ R + . Then oneof (a)-(j) holds: (a) ( α, ν, β, γ ) = ( ε k − ε l , ε i − ε j , ε k − ε j , ε i − ε l ) , i < k < j < l , (b) ( α, ν, β, γ ) = ( ε k + ε j , ε i + ε l , ε k + ε l , ε i + ε j ) , i < k < j < l , (c) ( α, ν, β, γ ) = ( ε k + ε j , ε i − ε l , ε k − ε l , ε i + ε j ) , i < k < j , i < k < l and j = l , (d) ( α, ν, β, γ ) = ( ε k , ε i − ε j , ε k − ε j , ε i ) , i < k < j , (e) ( α, ν, β, γ ) = ( ε k + ε j , ε i , ε k , ε i + ε j ) , i < k < j , (f) ( α, ν, β, γ ) = (2 ε k , ε i − ε j , ε k − ε j , ε i + ε k ) , i < k < j , (g) ( α, ν, β, γ ) = ( ε k + ε j , ε i , ε i + ε k , ε i + ε j ) , i < k < j , (h) ( α, ν, β, γ ) = (2 ε k , ε i , ε i + ε k , ε i + ε k ) , i < k , (i) ( α, ν, β, γ ) = ( ε i + ε j , ε i − ε j , ε i + ε k , ε i − ε k ) , i < j < k . In this case, g ∈ { so n , so n +1 } . (j) ( α, ν, β, γ ) = ( ε i + ε j , ε i − ε j , ε i , ε i ) . In this case, g = so n +1 .Proof. We write α + ν = P nl =1 b l ε l since we are assuming α, ν ∈ R + . Note that α + ν 6∈ R + . By (2.1) α + ν ∈ I ∪ I , where I = { ε i ± ε k , ε i − ε k − ε j , ε i + ε k − ε j , ε i ± ε l } and I = { ε k + ε i − ε l − ε j , ε k + ε i + ε l − ε j , ε k + ε i + ε l + ε j , ε i + ε k + ε j , ε i + ε k − ε j , ε k + 2 ε i , ε k + ε i + ε j , ε k + ε i − ε l , ε i } . Since any element in I can be uniquely decomposed into the summation of two positive roots, α + ν / ∈ I . When α + ν ∈ I , one can check (a)-(j) via (2.17) directly. For example, if α + ν = ε k + ε i − ε l − ε j , we can assume α = ε k − ε l and ν = ε i − ε j without loss of any gen-erality. In this case, if α + ν = β + γ satisfying α > β ≥ γ > ν , then β = ε k − ε j , γ = ε i − ε j . and i < k < j < l . This verifies (a). Since (b)-(j) can be checked in a similar way, we omit details. (cid:3) Lemma A.3. Suppose α, ν ∈ R + and α > ν . If α + ν = P ji =1 γ i , for some j ≥ and γ i ∈ R + , suchthat α > γ i > ν , then { α, ν } is one of pairs in Lemma A.2(f )-(h). In this case, g = sp n . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Proof. At moment, let R + g be the set of positive roots with respect to g . Thanks to (2.1) and (2.17), R + so n ⊂ R + sp n and α > ν in R + sp n if α > ν in R + so n . So, it is enough to consider g ∈ { so n +1 , sp n } .Suppose α + ν = j X i =1 γ i = n X l =1 b l ε l (A.1)where γ i ’s are required positive roots. By (2.1), P nl =1 | b l | ≤ 4. Let A = { γ , γ , . . . , γ j } .Suppose g = sp n . Since j ≥ P ji =1 γ i = P nl =1 b l ε l , there is a t such that { ε k − ε t , x + ε t } ⊂ A for some k < t and some x ∈ {± ε s | ≤ s ≤ n, x + ε t ∈ R + sp n } . Otherwise, P nl =1 | b l | ≥ 6, acontradiction.Without loss of any generality, we assume { γ , γ } = { ε k − ε t , x + ε t } ⊂ R + sp n as sets and γ < γ .So, x = − ε k and hence γ ′ = γ + γ = ε k + x ∈ R sp n . However, x + ε k is a summation of two positiveroots, it can not be a negative root. So, γ ′ ∈ R + sp n and γ < γ ′ < γ . So, α + ν is a summation of j − α and ν . Using the above arguments repeatedly, wehave λ + λ + λ = α + ν (A.2)where α > λ i > ν and λ + λ = β ∈ R + sp n . Without loss of any generality, we can assume λ < β < λ and ( λ , λ ) is a minimal pair of β . Since β + λ = α + ν and α > λ > ν and α > β > ν , { α, ν } hasto be one of pairs in Lemmas A.1-A.2. Since(2 ε k ) + ( ε i − ε j ) = ( ε k − ε n ) + ( ε k − ε j ) + ( ε i + ε n ) , ( ε k + ε j ) + 2 ε i = ( ε i − ε j ) + ( ε k + ε j ) + ( ε i + ε j ) , (2 ε k ) + (2 ε i ) = ( ε k − ε j ) + ( ε i + ε j ) + ( ε i + ε k ) , { α, ν } can be one of pairs in Lemmas A.2(f)-(h). Otherwise, we know the exact information on β . Weuse Corollary 2.5 to conclude that there is no required minimal pair ( λ , λ ) of β such that ν < λ i < α .This contradicts to (A.2). We give an example as follows. One can check other cases in a similar way.If { α, ν } is the pair in Lemma A.2(a), then { β, λ } = { ε k − ε j , ε i − ε l } , i < k < j < l . In this case, ε i − ε j < ε k − ε m < ε k − ε l < ε m − ε j for each minimal pair ( ε m − ε j , ε k − ε m ) of ε k − ε j in Corollary 2.5and either ε i − ε m ≤ ε i − ε j or ε m − ε l > ε k − ε l for each minimal pair ( ε m − ε l , ε i − ε m ) of ε i − ε l inCorollary 2.5. We can not find the required minimal pair ( λ , λ ) of β . This is a contradiction.Suppose g = so n +1 . Thanks to (2.1) and (A.1), b l ≤ ≤ l ≤ n . We will get a contradictionin any case. First, we assume b s = 2 for some 1 ≤ s ≤ n . By (2.1), α, ν ∈ { ε s , ε s ± ε j , ε l + ε s | l < s < j } .Suppose ν ∈ { ε s , ε s ± ε j | s < j } . Since α > ν , we have α = ε l + ε s if l < s . Thanks to (2.17),the coefficient of ε s in the expression of γ i is 1 for α > γ i > ν . Since j ≥ P ji =1 γ i = P nl =1 b l ε l , b s ≥ 3, a contradiction.Suppose ν = ε l + ε s for some l < s . Then α ∈ { ε k + ε s , ε s , ε s − ε j } and either k > s or k < s .Firstly, if α = ε s , then α + ν = P ji =1 γ i = ε l + 2 ε s and either ε s + ε t ∈ A or ε s − ε f ∈ A for some t and f . Otherwise, b s = 2.(a) Suppose ε s + ε t ∈ A . Since ν < ε s + ε t < α , by (2.17), l < t < s . Further, ε k − ε t ∈ A for some k < t . Otherwise, b t = 0. If k ≤ l , by (2.17) ε k − ε t < ε l + ε s , a contradiction. Suppose t > k > l . Since b k = 0, there is a k such that ε k − ε k ∈ A . If k < l , we get a contradictionvia (2.17). Otherwise, we repeat above arguments so that we can assume k < l . This leads toa contradiction, too.(b) Suppose ε s − ε f ∈ A for some f > s . Since b f = 0 and α > γ i > ν , ε j + ε f ∈ A for some l < j < s . Further, b j = 0 and ε j − ε j ∈ A for some j < j . If j ≤ l , by (2.17), ε j − ε j < ν ,a contradiction. If j > l , this also leads to a contradiction by using arguments in (a). Secondly, if α = ε s − ε j , then α + ν = ε l + 2 ε s − ε j , l < s < j . So, b l = 1. Note that γ i > ν = ε l + ε s .We have ε l + ε t ∈ A for some l < t < s . Consequently, b t = 0 and ε f − ε t ∈ A for some f < t . Now,comparing f and l and using arguments in (a) repeatedly leads to a contradiction.Finally, if α = ε k + ε s , then α + ν = ε k + 2 ε s + ε l with l < s . Suppose k < s . Since α > ν , we have l < k < s . Note that b s = 2 and ν < γ i < α . By (2.17), ε t + ε s ∈ A for some l < t < k . So, b t = 0and ε j − ε t ∈ A for some j < t . Now, comparing f and l and using arguments in (a) repeatedly leadsto a contradiction. When k > s , b k = 1. One can get a contradiction by arguments on the case k < s .We have proved that b s = 2 for any admissible s . It remains to deal with the case b s < ≤ s ≤ n . We claim that one of the following cases has to happen:(1) ε f , ε g ∈ A for some f = g ,(2) ε k − ε t , x + ε t ∈ A for some x ∈ {± ε s , | ≤ s ≤ n, x + ε t ∈ R + so n +1 } and 1 ≤ k < t ≤ n .It is enough to prove that (1) happens if we assume (2) does not happen. In fact, since j ≥ 3, thereis a pair { γ i , γ t } such that γ i = ε f and γ t = ε g , for some f and g . Otherwise, P ni =1 | b i | ≥ 5, acontradiction. Suppose f = g . Since we are assuming b f < ε k − ε f ∈ A and { ε k − ε f , ε f } ⊂ A .This shows that (2) happens, a contradiction. So, f = g and (1) happens.In the first case, we have ε f + ε g ∈ R + so n +1 since f = g . In the second case, if x = ε k , wehave ε k − ε t + x + ε t = x + ε k ∈ R + so n +1 . If x = ε k . Since b k < ε l − ε k ∈ A for some l and ε l − ε k + ε k − ε t = ε l − ε t ∈ R + so n +1 . In any of these cases, we can assume that γ ′ = γ + γ ∈ R + so n +1 , and γ ′ + j X i =3 γ i = α + ν , ν < γ < γ ′ < γ < α . By induction on j , we need to deal with the case λ + λ + λ = α + ν and λ + λ ∈ R + so n +1 . Via Corollary 2.5, one can check that this will never happen by arguments similarto those for the case g = sp n . (cid:3) Corollary A.4. Suppose α, ν ∈ R + such that α > ν and α + ν / ∈ R + . If there are no β, γ ∈ R + suchthat α > β ≥ γ > ν and α + ν = β + γ , then x − ν x − α = v ( α | ν ) x − α x − ν .Proof. By assumption and Lemma A.3, α + ν = P ji =1 γ i , where α > γ i > ν , γ i ∈ R + and j ≥ α + ν / ∈ R + , by Lemma 2.2, we have x − ν x − α = v ( α | ν ) x − α x − ν . (cid:3) Thanks to (2.5), q = v if g = so n +1 and q = v if g ∈ { so n , sp n } . Recall that [ k ] = v k − v − k v − v − and z q = q − q − . We are going to deal with pairs { α, ν } in (2)-(3). First, we list some equalities whichfollow from Lemma 2.4. Recall that Υ β is the set of minimal pairs of β ∈ R + . Corollary A.5. Suppose g ∈ { so n +1 , so n , sp n } . (1) [ x − ε i − ε k , x − ε k − ε j ] v = − q − x − ε i − ε j , if ≤ i < k < j ≤ n , (2) [ x − ε i + ε j +1 , x − ε j − ε j +1 ] v = − q − x − ε i + ε j , if ≤ i < j < n , (3) [ x − ε i , x − ε n ] v = − [2] x − ε i + ε n , if ≤ i < n and g = so n +1 , (4) [ x − ε i − ε k , x − ε k ] v = − q − x − ε i , if ≤ i < k ≤ n and g = so n +1 , (5) [ x − ε i − ε k , x − ε k + ε j ] v = − q − x − ε i + ε j , if ≤ i < k < j ≤ n , (6) [ x − ε i − ε n , x − ε n ] v = − q − x − ε i + ε n , if ≤ i < n and g = sp n , (7) [ x − ε i − ε n , x − ε i + ε n ] v = − [2] x − ε i , if ≤ i < n and g = sp n .Proof. In (1)-(7), we compute [ x − ν , x − α ] ν for some admissible α, ν ∈ R + . Thanks to Corollary 2.5, such( α, ν )’s are minimal pairs. So, (1)-(7) follow from Lemma 2.4, immediately. (cid:3) HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Lemma A.6. For all admissible i < j < k , let a k = [ x − ε i + ε k , x − ε j − ε k ] v . Then a k = ( − q ) j − k x − ε i + ε j − z q k − X t = j +1 ( − q ) t − k x − ε j − ε t x − ε i + ε t . Proof. The required formula for a j +1 follows from Corollary A.5(2). Suppose k > j + 1. Then a j = − qx − ε i + ε k [ x − ε j − ε k − , x − ε k − − ε k ] v − q − x − ε j − ε k x − ε i + ε k , by Corollary A.5(1)= 1 q ( z q x − ε j − ε k − x − ε i + ε k − − [ x − ε i + ε k − , x − ε j − ε k − ] v − x − ε j − ε k x − ε i + ε k ) − [ x − ε j − ε k − , x − ε k − − ε k ] v x − ε i + ε k = − q − a k − + q − z q x − ε j − ε k − x − ε i + ε k − , by Corollary A.5(1)= ( − q ) j +1 − k a j +1 − k − X t = j +1 ( − q ) t − k z q x − ε j − ε t x − ε i + ε t , by induction assumption on k, = ( − q ) j − k x − ε i + ε j − z q k − X t = j +1 ( − q ) t − k x − ε j − ε t x − ε i + ε t , by Corollary A.5(2),where the second and third equalities follow from [ x − ε i + ε k , x − ε j − ε k − ] v = 0 (see Corollary A.4) andCorollary A.5(2). So, we have the required formula on a k in general. (cid:3) Lemma A.7. For any j , i < j ≤ n , let a j = [ x − ε i − ε j , x − ε i + ε j ] v . Then a j − n X t = j +1 ( − q ) j − t z q x − ε i + ε t x − ε i − ε t = , if g = so n , − ( − q ) j − n [2] x − ε i , if g = sp n , ( − q ) j − n [2] − (1 − q − )( x − ε i ) , if g = so n +1 .Proof. Suppose j < n . a j = − qx − ε i − ε j [ x − ε i + ε j +1 , x − ε j − ε j +1 ] v − x − ε i + ε j x − ε i − ε j , by Corollary A.5(2)= 1 q ( x − ε j − ε j +1 x − ε i − ε j − x − ε i − ε j +1 ) x − ε i + ε j +1 − qx − ε i + ε j +1 ( x − ε j − ε j +1 x − ε i − ε j − x − ε i − ε j +1 ) − x − ε i + ε j x − ε i − ε j = − q [ x − ε i + ε j +1 , x − ε j − ε j +1 ] v x − ε i − ε j − q − [ x − ε i − ε j +1 , x − ε i + ε j +1 ] v + z q x − ε i + ε j +1 x − ε i − ε j +1 − x − ε i + ε j x − ε i − ε j = − q − a j +1 + z q x − ε i + ε j +1 x − ε i − ε j +1 , by Corollary A.5(2)= ( − q ) j − n a n + n X t = j +1 ( − q ) j − t z q x − ε i + ε t x − ε i − ε t , by induction on n − j, where the second and third equalities follow from [ x − ε i − ε j , x − ε i + ε j +1 ] v = 0 (see Corollary A.4) andCorollary A.5(1). Therefore, the result follows from the corresponding result on a n .When g ∈ { so n , sp n } , the required formulae on a n follow from Corollary A.4 and Corollary A.5(7), respectively. If g = so n +1 , then a n = − [2] − x − ε i − ε n [ x − ε i , x − ε n ] v − x − ε i + ε n x − ε i − ε n , by Corollary A.5(3)= [2] − ( q − ( x − ε n x − ε i − ε n − x − ε i ) x − ε i − x − ε i ( x − ε n x − ε i − ε n − x − ε i )) − x − ε i + ε n x − ε i − ε n = [2] − ( − [ x − ε i , x − ε n ] v x − ε i − ε n − q − ( x − ε i ) + ( x − ε i ) ) − x − ε i + ε n x − ε i − ε n = [2] − (1 − q − )( x − ε i ) , by Corollary A.5(3) , where the second and third equalities follow from [ x − ε i − ε n , x − ε i ] v = 0 (see Corollary A.4) and Corol-lary A.5(4). (cid:3) Lemma A.8. Suppose g = so n +1 . For all admissible i < j < n , we have [ x − ε i , x − ε j ] v = − [2]( − q ) j − n x − ε i + ε j + [2] n X t = j +1 ( − q ) t − n z q x − ε j − ε t x − ε i + ε t . Proof. We have[ x − ε i , x − ε j ] v = − qx − ε i [ x − ε j − ε n , x − ε n ] v − x − ε j x − ε i , by Corollary A.5(4)= ( x − ε n x − ε i − [2] x − ε i + ε n ) x − ε j − ε n − qx − ε j − ε n ( x − ε n x − ε i − [2] x − ε i + ε n ) − x − ε j x − ε i = − q [ x − ε j − ε n , x − ε n ] v x − ε i − [2][ x − ε i + ε n , x − ε j − ε n ] v + [2] z q x − ε j − ε n x − ε i + ε n − x − ε j x − ε i = − [2][ x − ε i + ε n , x − ε j − ε n ] v + [2] z q x − ε j − ε n x − ε i + ε n , by Corollary A.5(4) , = − [2]( − q ) j − n x − ε i + ε j + [2] n X t = j +1 ( − q ) t − n z q x − ε j − ε t x − ε i + ε t , by Lemma A.6,where the second and third equalities follow from [ x − ε i , x − ε j − ε n ] v = 0 (see Corollary A.4) and Corol-lary A.5(3). (cid:3) Lemma A.9. If ( α, ν, β, γ ) is one of pairs in Lemma A.2(a)-(e). Then [ x − ν , x − α ] v = z q x − β x − γ .Proof. Suppose ( α, ν, β, γ ) is given in Lemma A.2(a). Then[ x − ε i − ε j , x − ε k − ε l ] v = − qx − ε i − ε j [ x − ε k − ε j , x − ε j − ε l ] v − x − ε k − ε l x − ε i − ε j , by Corollary A.5(1)= q − ( x − ε j − ε l x − ε i − ε j − x − ε i − ε l ) x − ε k − ε j − qx − ε k − ε j ( x − ε j − ε l x − ε i − ε j − x − ε i − ε l ) − x − ε k − ε l x − ε i − ε j = − q [ x − ε k − ε j , x − ε j − ε l ] v x − ε i − ε j + qx − ε k − ε j x − ε i − ε l − q − x − ε i − ε l x − ε k − ε j − x − ε k − ε l x − ε i − ε j = z q x − ε k − ε j x − ε i − ε l , by Corollary A.5(1) , where the second and third equalities follow from [ x − ε i − ε j , x − ε k − ε j ] v = 0, [ x − ε i − ε l , x − ε k − ε j ] v = 0 (seeCorollary A.4) and Corollary A.5(1).Suppose ( α, ν, β, γ ) is given in Lemma A.2(b). Then[ x − ε i + ε l , x − ε k + ε j ] v = − q [ x − ε i − ε k , x − ε k + ε l ] v x − ε k + ε j − x − ε k + ε j x − ε i + ε l , by Corollary A.5(5)= − q ( x − ε k + ε j x − ε i − ε k − x − ε i + ε j ) x − ε k + ε l + q − x − ε k + ε l ( x − ε k + ε j x − ε i − ε k − x − ε i + ε j ) − x − ε k + ε j x − ε i + ε l = − qx − ε k + ε j [ x − ε i − ε k , x − ε k + ε l ] v + qx − ε i + ε j x − ε k + ε l − q − x − ε k + ε l x − ε i + ε j − x − ε k + ε j x − ε i + ε l = z q x − ε k + ε l x − ε i + ε j , by Corollary A.5(5) , (A.3)where the second and third equalities follow from [ x − ε k + ε l , x − ε h + ε j ] v = 0, if h ∈ { i, k } (see Corollary A.4)and Corollary A.5(5).Suppose ( α, ν, β, γ ) is given in Lemma A.2(c). So ( α, ν, β, γ ) = ( ε k + ε j , ε i − ε l , ε k − ε l , ε i + ε j ), i < k < j , i < k < l and j = l . If j > l , then[ x − ε i − ε l , x − ε k + ε j ] v = − qx − ε i − ε l [ x − ε k − ε l , x − ε l + ε j ] v − x − ε k + ε j x − ε i − ε l , by Corollary A.5(5)= q − ( x − ε l + ε j x − ε i − ε l − x − ε i + ε j ) x − ε k − ε l − qx − ε k − ε l ( x − ε l + ε j x − ε i − ε l − x − ε i + ε j ) − x − ε k + ε j x − ε i − ε l = − q [ x − ε k − ε l , x − ε l + ε j ] v x − ε i − ε l + qx − ε k − ε l x − ε i + ε j − q − x − ε i + ε j x − ε k − ε l − x − ε k + ε j x − ε i − ε l = z q x − ε k − ε l x − ε i + ε j , by Corollary A.5(5) , (A.4)where the second and third equalities follow from [ x − ε i − ε l , x − ε k − ε l ] v = 0, [ x − ε i + ε j , x − ε k − ε l ] v = 0 (seeCorollary A.4) and Corollary A.5(5). If j < l , then[ x − ε i − ε l , x − ε k + ε j ] v = − q [ x − ε i − ε k , x − ε k − ε l ] v x − ε k + ε j − x − ε k + ε j x − ε i − ε l , by Corollary A.5(1)= q − x − ε k − ε l ( x − ε k + ε j x − ε i − ε k − x − ε i + ε j ) − q ( x − ε k + ε j x − ε i − ε k − x − ε i + ε j ) x − ε k − ε l − x − ε k + ε j x − ε i − ε l = − qx − ε k + ε j [ x − ε i − ε k , x − ε k − ε l ] v − q − x − ε k − ε l x − ε i + ε j + qx − ε i + ε j x − ε k − ε l − x − ε k + ε j x − ε i − ε l = z q x − ε k − ε l x − ε i + ε j , by Corollary A.5(1) , (A.5)where the second and third equalities follow from [ x − ε k − ε l , x − ε k + ε j ] v = 0, [ x − ε i + ε j , x − ε k − ε l ] v = 0 (seeCorollary A.4) and Corollary A.5(5). HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Suppose ( α, ν, β, γ ) is given in Lemma A.2(d). Then[ x − ε i − ε j , x − ε k ] v = − q [ x − ε i − ε k , x − ε k − ε j ] v x − ε k − x − ε k x − ε i − ε j , by Corollary A.5(1)= − q ( x − ε k x − ε i − ε k − x − ε i ) x − ε k − ε j + q − x − ε k − ε j ( x − ε k x − ε i − ε k − x − ε i ) − x − ε k x − ε i − ε j = − qx − ε k [ x − ε i − ε k , x − ε k − ε j ] v + qx − ε i x − ε k − ε j − q − x − ε k − ε j x − ε i − x − ε k x − ε i − ε j = z q x − ε k − ε j x − ε i , by Corollary A.5(1) , (A.6)where the second and third equalities follow from [ x − ε k − ε j , x − ε k ] v = 0, [ x − ε i , x − ε k − ε j ] v = 0 (see Corol-lary A.4) and Corollary A.5(4).Finally, suppose ( α, ν, β, γ ) is given in Lemma A.2(e). We have[ x − ε i , x − ε k + ε j ] v = − q [ x − ε i − ε k , x − ε k ] v x − ε k + ε j − x − ε k + ε j x − ε i , by Corollary A.5(4)= − q ( x − ε k + ε j x − ε i − ε k − x − ε i + ε j ) x − ε k + q − x − ε k ( x − ε k + ε j x − ε i − ε k − x − ε i + ε j ) − x − ε k + ε j x − ε i = − qx − ε k + ε j [ x − ε i − ε k , x − ε k ] v + qx − ε i + ε j x − ε k − q − x − ε k x − ε i + ε j − x − ε k + ε j x − ε i = z q x − ε k x − ε i + ε j , by Corollary A.5(4) , where the second and third equalities follow from [ x − ε k , x − ε k + ε j ] v = 0, [ x − ε i + ε j , x − ε k ] v = 0 (see Corol-lary A.4) and Corollary A.5(5). (cid:3) Lemma A.10. Suppose ≤ i < j < n . For any k, j < k ≤ n , let a k = [ x − ε i − ε k , x − ε j + ε k ] v . Then a k + n X t = k +1 ( − q ) k − t z q x − ε j + ε t x − ε i − ε t − z q x − ε j − ε k x − ε i + ε k + z q ( − q ) k − n x − ε j − ε n x − ε i + ε n = ( − q ) k − n { z q x − ε j − ε n x − ε i + ε n + [2] − q − ( z q x − ε j x − ε i − [ x − ε i , x − ε j ] v ) } , if g = so n +1 , ( − q ) k − n [ x − ε i + ε n , x − ε j − ε n ] v , if g = so n , ( − q ) k − n { (1 + q − ) z q x − ε j − ε n x − ε i + ε n − q − [ x − ε i + ε n , x − ε j − ε n ] v } , if g = sp n .Moreover, [ x − ε i + ε n , x − ε j − ε n ] v and [ x − ε i , x − ε j ] v have been computed in Lemma A.6 and Lemma A.8, re-spectively.Proof. Suppose k < n . Then a k = − qx − ε i − ε k [ x − ε j + ε k +1 , x − ε k − ε k +1 ] v , by Corollary A.5(2)= q − ( x − ε k − ε k +1 x − ε i − ε k − x − ε i − ε k +1 ) x − ε j + ε k +1 − q − x − ε j + ε k x − ε i − ε k − q ( x − ε j + ε k +1 x − ε i − ε k + z q x − ε j − ε k x − ε i + ε k +1 ) x − ε k − ε k +1 = q − x − ε k − ε k +1 ( x − ε j + ε k +1 x − ε i − ε k + z q x − ε j − ε k x − ε i + ε k +1 ) − q − x − ε i − ε k +1 x − ε j + ε k +1 − x − ε j + ε k +1 ( x − ε k − ε k +1 x − ε i − ε k − x − ε i − ε k +1 ) − qz q x − ε j − ε k x − ε i + ε k +1 x − ε k − ε k +1 − q − x − ε j + ε k x − ε i − ε k = − [ x − ε j + ε k +1 , x − ε k − ε k +1 ] v x − ε i − ε k + q − z q x − ε k − ε k +1 x − ε j − ε k x − ε i + ε k +1 − q − [ x − ε i − ε k +1 , x − ε j + ε k +1 ] v + (1 − q − ) x − ε j + ε k +1 x − ε i − ε k +1 − qz q x − ε j − ε k x − ε i + ε k +1 x − ε k − ε k +1 − q − x − ε j + ε k x − ε i − ε k = z q q ( x − ε j + ε k +1 x − ε i − ε k +1 + x − ε j − ε k +1 x − ε i + ε k +1 + qx − ε j − ε k x − ε i + ε k ) − q a k +1 , by Corollary A.5(1)(2)= ( − q ) k − n a n − z q n X t = k +1 ( − q ) k − t x − ε j + ε t x − ε i − ε t + z q x − ε j − ε k x − ε i + ε k − z q ( − q ) k − n x − ε j − ε n x − ε i + ε n , where the second and third equalities follow from (A.4) and Corollary A.5(1) and, the last equalityfollows from induction assumption on n − k . In order to complete the proof, it remain to show that a n has the required formula. If g = so n , then a n = − q [ x − ε i − ε j , x − ε j − ε n ] v x − ε j + ε n − q − x − ε j + ε n x − ε i − ε n , by Corollary A.5(1)= q − x − ε j − ε n ( x − ε j + ε n x − ε i − ε j − x − ε i + ε n ) − ( x − ε j + ε n x − ε i − ε j − x − ε i + ε n ) x − ε j − ε n − q − x − ε j + ε n x − ε i − ε n = − x − ε j + ε n [ x − ε i − ε j , x − ε j − ε n ] v + x − ε i + ε n x − ε j − ε n − q − x − ε j − ε n x − ε i + ε n − q − x − ε j + ε n x − ε i − ε n = [ x − ε i + ε n , x − ε j − ε n ] v , by Corollary A.5(1) , where the second and third equalities follow from [ x − ε j − ε n , x − ε j + ε n ] v = 0 (see Corollary A.4) andCorollary A.5(5). If g = so n +1 , then a n = 1[2] ( 1 q ( x − ε n x − ε i − ε n − x − ε i ) x − ε j − ( x − ε j x − ε i − ε n + z q x − ε j − ε n x − ε i ) x − ε n ) − q x − ε j + ε n x − ε i − ε n , by (A.6)= [2] − ( q − x − ε n ( x − ε j x − ε i − ε n + z q x − ε j − ε n x − ε i ) − q − x − ε i x − ε j − q − x − ε j ( x − ε n x − ε i − ε n − x − ε i )) − z q [2] − x − ε j − ε n x − ε i x − ε n − q − x − ε j + ε n x − ε i − ε n , by Corollary A.5(4),= 1 q [2] ( z q x − ε n x − ε j − ε n x − ε i − [ x − ε j , x − ε n ] v x − ε i − ε n − [ x − ε i , x − ε j ] v − z q qx − ε j − ε n x − ε i x − ε n ) − q x − ε j + ε n x − ε i − ε n = − q [2] ([ x − ε i , x − ε j ] v − z q x − ε n x − ε j − ε n x − ε i ) − z q q [2] ( x − ε n x − ε j − ε n − x − ε j ) x − ε i + z q x − ε j − ε n x − ε i + ε n = − q − [2] − [ x − ε i , x − ε j ] v + z q x − ε j − ε n x − ε i + ε n + q − [2] − z q x − ε j x − ε i , where the fourth equality follows from Corollary A.5(3)-(4). If g = sp n , then a n = − q x − ε i − ε n [ x − ε j − ε n , x − ε n ] v − q − x − ε j + ε n x − ε i − ε n , by Corollary A.5(6)= q − ( x − ε n x − ε i − ε n − x − ε i + ε n ) x − ε j − ε n − qx − ε j − ε n ( x − ε n x − ε i − ε n − x − ε i + ε n ) − q − x − ε j + ε n x − ε i − ε n = − q [ x − ε j − ε n , x − ε n ] v x − ε i − ε n − q − [ x − ε i + ε n , x − ε j − ε n ] v + q − ([2] z q x − ε j − ε n x − ε i + ε n − x − ε j + ε n x − ε i − ε n )= − q − [ x − ε i + ε n , x − ε j − ε n ] v + q − [2] z q x − ε j − ε n x − ε i + ε n , by Corollary A.5(6) , where the second and third equalities follow from [ x − ε i − ε n , x − ε j − ε n ] v = 0 (see Corollary A.4) andCorollary A.5(6). (cid:3) Lemma A.11. Suppose g = sp n and i < j < n . We have [ x − ε i − ε j , x − ε j ] v = − ( − q ) j − n − x − ε i + ε j + z q n X t = j +1 ( − q ) t − n − x − ε j − ε t x − ε i + ε t . Proof. We have[ x − ε i − ε j , x − ε j ] v = [2] − ( − x − ε i − ε j [ x − ε j − ε n , x − ε j + ε n ] v ) − q − x − ε j x − ε i − ε j , by Corollary A.5(7)=[2] − q − (( x − ε j + ε n x − ε i − ε j − x − ε i + ε n ) x − ε j − ε n − ( x − ε j − ε n x − ε i − ε j − x − ε i − ε n ) x − ε j + ε n ) − q − x − ε j x − ε i − ε j =[2] − ( q − x − ε j + ε n ( x − ε j − ε n x − ε i − ε j − x − ε i − ε n ) − q − x − ε i + ε n x − ε j − ε n ) − q − [2] − x − ε j − ε n × ( x − ε j + ε n x − ε i − ε j − x − ε i + ε n ) + [2] − q − x − ε i − ε n x − ε j + ε n − q − x − ε j x − ε i − ε j , by Corollary A.5(1),(5)=[2] − q − ([ x − ε i − ε n , x − ε j + ε n ] v − q − [ x − ε j − ε n , x − ε j + ε n ] v x − ε i − ε j − [ x − ε i + ε n , x − ε j − ε n ] v ) − q − x − ε j x − ε i − ε j =[2] − ( q − [ x − ε i − ε n , x − ε j + ε n ] v − q − [ x − ε i + ε n , x − ε j − ε n ] v ) , by Corollary A.5(7)=[2] − ( q − ( − q − [ x − ε i + ε n , x − ε j − ε n ] v + (1 + q − ) z q x − ε j − ε n x − ε i + ε n )) − [2] − q − [ x − ε i + ε n , x − ε j − ε n ] v , = − q − [ x − ε i + ε n , x − ε j − ε n ] v + q − z q x − ε j − ε n x − ε i + ε n , where the second equality follows from Corollary A.5(1),(5) and the sixth equality follows from theformula on [ x − ε i − ε n , x − ε j + ε n ] v in Lemma A.10. (cid:3) HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Lemma A.12. For all admissible i < l < j , we have [ x − ε i − ε j , x − ε l ] v = − ( − q ) l − n − z q x − ε l − ε j x − ε i + ε l − z q n X t = j +1 ( − q ) t − n x − ε l − ε t x − ε l − ε j x − ε i + ε t + z q ( − q ) j − n − ( x − ε l − ε j ) x − ε i + ε j + z q j − X t = l +1 ( − q ) t − n − x − ε l − ε j x − ε l − ε t x − ε i + ε t . Proof. We have[ x − ε i − ε j , x − ε l ] v = − q [ x − ε i − ε l , x − ε l − ε j ] v x − ε l − x − ε l x − ε i − ε j , by Corollary A.5(1)= x − ε l − ε j ( q − x − ε l x − ε i − ε l + [ x − ε i − ε l , x − ε l ] v ) − ( qx − ε l x − ε i − ε l + q [ x − ε i − ε l , x − ε l ] v ) x − ε l − ε j − x − ε l x − ε i − ε j = − qx − ε l [ x − ε i − ε l , x − ε l − ε j ] v + x − ε l − ε j [ x − ε i − ε l , x − ε l ] v − q [ x − ε i − ε l , x − ε l ] v x − ε l − ε j − x − ε l x − ε i − ε j = x − ε l − ε j [ x − ε i − ε l , x − ε l ] v − q [ x − ε i − ε l , x − ε l ] v x − ε l − ε j , by Corollary A.5(1)=( − q ) l − n − ( x − ε l − ε j x − ε i + ε l − q x − ε i + ε l x − ε l − ε j ) + z q X l +1 ≤ t ≤ n,t = j ( − q ) t − n +1 x − ε l − ε t x − ε i + ε t x − ε l − ε j + z q n X t = l +1 ( − q ) t − n − x − ε l − ε j x − ε l − ε t x − ε i + ε t + z q ( − q ) j − n +1 x − ε l − ε j ([ x − ε i + ε j , x − ε l − ε j ] v + q − x − ε l − ε j x − ε i + ε j ) , by Lemma A.11=( − q ) l − n − ( q − x − ε l − ε j x − ε i + ε l − z q n X t = j +1 ( − q ) t − n x − ε l − ε t x − ε l − ε j x − ε i + ε t + z q j − X t = l +1 ( − q ) t − n − (1 − q ) x − ε l − ε j x − ε l − ε t x − ε i + ε t + z q ( − q ) j − n − ( x − ε l − ε j ) x − ε i + ε j − z q ( − q ) j − n x − ε l − ε j ( x − ε l − ε j x − ε i + ε j − ( − q ) l +1 − j x − ε i + ε l − qz q j − X t = l +1 ( − q ) t − j x − ε l − ε t x − ε i + ε t )= − ( − q ) l − n − z q x − ε l − ε j x − ε i + ε l − z q n X t = j +1 ( − q ) t − n x − ε l − ε t x − ε l − ε j x − ε i + ε t + z q ( − q ) j − n − ( x − ε l − ε j ) x − ε i + ε j + z q j − X t = l +1 ( − q ) t − n − x − ε l − ε j x − ε l − ε t x − ε i + ε t . The second and the third equalities follow from [ x − ε l − ε j , x − ε l ] v = 0 (see Corollary A.4). The sixthequality follows from Lemma A.6 and [ x − ε i + ε t , x − ε l − ε j ] v = 0, [ x − ε l − ε t , x − ε l − ε t ] v = 0 where l ≤ t ≤ n , t = j and l < t < t ≤ n . Such formulae follows from Corollary A.4. (cid:3) Lemma A.13. Suppose g = sp n . For all admissible i < k < j , let a j = [ x − ε i , x − ε k + ε j ] v . Then a j = − ( − q ) k − n − z q x − ε i + ε k x − ε i + ε j + z q j X t = k +1 ( − q ) t − n − x − ε k − ε t x − ε i + ε t x − ε i + ε j − z q n X t = j +1 ( − q ) t − n x − ε k − ε t x − ε i + ε j x − ε i + ε t . Proof. We have a n = − [2] − [ x − ε i − ε n , x − ε i + ε n ] v x − ε k + ε n − x − ε k + ε n x − ε i , by Corollary A.5(7)=[2] − x − ε i + ε n ([ x − ε i − ε n , x − ε k + ε n ] v + q − x − ε k + ε n x − ε i − ε n ) − x − ε k + ε n x − ε i − [2] − q ([ x − ε i − ε n , x − ε k + ε n ] v + q − x − ε k + ε n x − ε i − ε n ) x − ε i + ε n =[2] − ( x − ε i + ε n [ x − ε i − ε n , x − ε k + ε n ] v − q [ x − ε i − ε n , x − ε k + ε n ] v x − ε i + ε n )= q − z q x − ε i + ε n x − ε k − ε n x − ε i + ε n − z q x − ε k − ε n ( x − ε i + ε n ) = q − z q [ x − ε i + ε n , x − ε k − ε n ] v x − ε i + ε n − q − z q x − ε k − ε n ( x − ε i + ε n ) . The third equality follows from [ x − ε i + ε n , x − ε k + ε n ] v = 0 (see Corollary A.4) and Corollary A.5(7). Thefourth equality follow from Lemma A.10 and [ x − ε i + ε n , x − ε i + ε h ] v = 0, [ x − ε i + ε n , x − ε k − ε t ] v = 0, k < t < n , k ≤ h < n . Such formulae follow from Corollary A.4. Thanks to the formula of [ x − ε i + ε n , x − ε k − ε n ] v inLemma A.6, a n has the required formula. Suppose j < n . Then a j = − [2] − [ x − ε i − ε n , x − ε i + ε n ] v x − ε k + ε j − x − ε k + ε j x − ε i , by Corollary A.5(7)=[2] − x − ε i + ε n ( x − ε k + ε j x − ε i − ε n + z q x − ε k − ε n x − ε i + ε j ) − x − ε k + ε j x − ε i − [2] − x − ε i − ε n ( x − ε k + ε j x − ε i + ε n + z q x − ε k + ε n x − ε i + ε j ) , by (A.3), (A.5)=[2] − (( x − ε k + ε j x − ε i + ε n + z q x − ε k + ε n x − ε i + ε j ) x − ε i − ε n + z q x − ε i + ε n x − ε k − ε n x − ε i + ε j ) − x − ε k + ε j x − ε i − [2] − (( x − ε k + ε j x − ε i − ε n + z q x − ε k − ε n x − ε i + ε j ) x − ε i + ε n + z q x − ε i − ε n x − ε k + ε n x − ε i + ε j ) , by (A.3), (A.5)= 1[2] ( z q ([ x − ε i + ε n , x − ε k − ε n ] v − [ x − ε i − ε n , x − ε k + ε n ] v ) x − ε i + ε j − x − ε k + ε j [ x − ε i − ε n , x − ε i + ε n ] v ) − x − ε k + ε j x − ε i =[2] − z q ([ x − ε i + ε n , x − ε k − ε n ] v − [ x − ε i − ε n , x − ε k + ε n ] v ) x − ε i + ε j , by Corollary A.5(7)= n − X t = k +1 ( − q ) t − n − z q x − ε k − ε t x − ε i + ε t x − ε i + ε j − z q x − ε k − ε n x − ε i + ε j x − ε i + ε n − z q ( − q ) k − n − x − ε i + ε k x − ε i + ε j = z q j X t = k +1 ( − q ) t − n − x − ε k − ε t x − ε i + ε t x − ε i + ε j − ( − q ) k − n − z q x − ε i + ε k x − ε i + ε j − z q n X t = j +1 ( − q ) t − n x − ε k − ε t x − ε i + ε j x − ε i + ε t . The fourth equality follows from [ x − ε i ± ε n , x − ε i + ε j ] v = 0 (see Corollary A.4). The sixth equality followsfrom Lemmas A.6, A.10 and the last equality follows from [ x − ε i ± ε t , x − ε i + ε j ] v = 0 if j < t < n (seeCorollary A.4). (cid:3) Lemma A.14. Suppose g = sp n . For all admissible i < j , let a j = [ x − ε i , x − ε j ] v . Then a j = − [2] − ( − q ) j − n − z q ( x − ε i + ε j ) − z q n − X t = j +1 ( − q ) t − n − x − ε j − ε n x − ε j − ε t x − ε i + ε t x − ε i + ε n + z q n X t = j +1 ( − q ) j + t − n − x − ε j − ε t x − ε i + ε j x − ε i + ε t + [2] − z q n X t = j +1 ( − q ) t − n − ( x − ε j − ε t ) ( x − ε i + ε t ) − z q n − X t = j +1 n − X s = t +1 ( − q ) s + t − n − x − ε j − ε s x − ε j − ε t x − ε i + ε t x − ε i + ε s . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Proof. Suppose j = n . Then a n = − [2] − [ x − ε i − ε n , x − ε i + ε n ] v x − ε n − x − ε n x − ε i , by Corollary A.5(7)= [2] − ( q − x − ε i + ε n ( x − ε n x − ε i − ε n − x − ε i + ε n ) − ( x − ε n x − ε i − ε n − x − ε i + ε n ) x − ε i + ε n ) − x − ε n x − ε i = [2] − ( − x − ε n [ x − ε i − ε n , x − ε i + ε n ] v + (1 − q − )( x − ε i + ε n ) ) − x − ε n x − ε i , by Corollary A.5(6)= q − [2] − z q ( x − ε i + ε n ) , by Corollary A.5(7) , where the second equality follows from [ x − ε i + ε n , x − ε n ] v = 0 (see Corollary A.4). So, we have therequired formula for j = n follows.If j < n , then a j = − [2] − x − ε i [ x − ε j − ε n , x − ε j + ε n ] v − x − ε j x − ε i , by Corollary A.5(7)= [2] − ([ x − ε i , x − ε j + ε n ] v x − ε j − ε n − x − ε j − ε n [ x − ε i , x − ε j + ε n ] v ) , (A.7)where the last equation follows from [ x − ε i , x − ε j − ε n ] v = 0 (see Corollary A.4) and Corollary A.5(7). Weuse Lemma A.13 to rewrite [ x − ε i , x − ε j + ε n ] v x − ε j − ε n as follows:[ x − ε i , x − ε j + ε n ] v x − ε j − ε n = − ( − q ) j − n − z q x − ε i + ε j ( q − x − ε j − ε n x − ε i + ε n + [ x − ε i + ε n , x − ε j − ε n ] v )+ z q n X t = j +1 ( − q ) t − n − x − ε j − ε t x − ε i + ε t ( q − x − ε j − ε n x − ε i + ε n + [ x − ε i + ε n , x − ε j − ε n ] v )= − ( − q ) j − n − z q ( x − ε j − ε n x − ε i + ε j x − ε i + ε n + x − ε i + ε j [ x − ε i + ε n , x − ε j − ε n ] v )+ z q n − X t = j +1 ( − q ) t − n − ( x − ε j − ε n x − ε j − ε t x − ε i + ε t x − ε i + ε n + x − ε j − ε t x − ε i + ε t [ x − ε i + ε n , x − ε j − ε n ] v ) − q − z q x − ε j − ε n (cid:16) x − ε i + ε n [ x − ε i + ε n , x − ε j − ε n ] v + ( q − x − ε j − ε n x − ε i + ε n + q − [ x − ε i + ε n .x − ε j − ε n ] v ) x − ε i + ε n (cid:17) . The last equality follows from [ x − ε i + ε j , x − ε j − ε n ] v = 0, [ x − ε j − ε k , x − ε j − ε n ] v = 0, and [ x − ε i + ε k , x − ε j − ε n ] = 0where j < k < n . Such equalities follows from Corollary A.4. We also use Lemma A.13 to rewrite x − ε j − ε n [ x − ε i , x − ε j + ε n ] v in (A.7). So,[ x − ε i , x − ε j ] v = [2] − ( − ( − q ) j − n − z q x − ε i + ε j + z q n X t = j +1 ( − q ) t − n − x − ε j − ε t x − ε i + ε t )[ x − ε i + ε n , x − ε j − ε n ] v + [2] − q − z q ( x − ε j − ε n ) ( x − ε i + ε n ) − [2] − q − z q x − ε j − ε n [ x − ε i + ε n , x − ε j − ε n ] v x − ε i + ε n . In the following, we compute x − ε i + ε j [ x − ε i + ε n , x − ε j − ε n ] v , and x − ε j − ε t x − ε i + ε t [ x − ε i + ε n , x − ε j − ε n ] v , j < t ≤ n ,and x − ε j − ε n [ x − ε i + ε n , x − ε j − ε n ] v x − ε i + ε n . In any case, we rewrite [ x − ε i + ε n , x − ε j − ε n ] via Lemma A.6. So, wehave the formula on x − ε j − ε n [ x − ε i + ε n , x − ε j − ε n ] v x − ε i + ε n directly. In other cases, we need extra commutativerelations as follows:(a) [ x − ε i + ε j , x − ε j − ε h ] v = 0, if j < h ≤ n ,(b) [ x − ε i + ε n , x − ε i + ε s ] v = 0, [ x − ε i + ε n , x − ε j − ε h ] v = 0, if i < s < n , j < h < n ,(c) [ x − ε i + ε h , x − ε j − ε h ] v = 0, [ x − ε i + ε h , x − ε i + ε h ] v = 0, [ x − ε j − ε h , x − ε j − ε h ] v = 0, if h < h and h i > j for all 1 ≤ i ≤ x − ε i + ε j [ x − ε i + ε n , x − ε j − ε n ] v ( a ) = ( − q ) j − n ( x − ε i + ε j ) + z q n − X t = j +1 ( − q ) t − n +1 x − ε j − ε t x − ε i + ε j x − ε i + ε t ,x − ε j − ε n x − ε i + ε n [ x − ε i + ε n , x − ε j − ε n ] v ( b ) = ( − q ) j − n x − ε j − ε n ( x − ε i + ε j + z q n − X t = j +1 ( − q ) t − j +1 x − ε j − ε t x − ε i + ε t ) x − ε i + ε n . Finally, we simplify x − ε j − ε t x − ε i + ε t [ x − ε i + ε n , x − ε j − ε n ] v for any j < t < n . x − ε j − ε t x − ε i + ε t [ x − ε i + ε n , x − ε j − ε n ] v = − ( − q ) j − n +1 x − ε j − ε t x − ε i + ε j x − ε i + ε t + z q t − X s = j +1 ( − q ) s − n +1 x − ε j − ε t x − ε j − ε s x − ε i + ε s x − ε i + ε t + z q n − X s = t +1 ( − q ) s − n +1 x − ε j − ε s x − ε j − ε t x − ε i + ε t x − ε i + ε s − ( − q ) t − n z q x − ε j − ε t x − ε i + ε t x − ε j − ε t x − ε i + ε t , by (c)= − ( − q ) j − n +1 x − ε j − ε t x − ε i + ε j x − ε i + ε t + z q t − X s = j +1 ( − q ) s − n +1 x − ε j − ε t x − ε j − ε s x − ε i + ε s x − ε i + ε t + z q n − X s = t +1 ( − q ) s − n +1 x − ε j − ε s x − ε j − ε t x − ε i + ε t x − ε i + ε s + ( − q ) t − n − z q ( x − ε j − ε t ) ( x − ε i + ε t ) − z q ( − q ) j − n x − ε j − ε t x − ε i + ε j x − ε i + ε t + z q t − X s = j +1 ( − q ) s − n x − ε j − ε t x − ε j − ε s x − ε i + ε s x − ε i + ε t . We remark that the last equality can be checked directly by rewriting x − ε i + ε t x − ε j − ε t via Lemma A.6and (c). Now, we rewrite [ x − ε i , x − ε j ] v via (A.7). We have a j = − ( − q ) j − n − [2] − z q ( x − ε i + ε j ) − z q n − X t = j +1 ( − q ) t − n − x − ε j − ε n x − ε j − ε t x − ε i + ε t x − ε i + ε n + z q n X t = j +1 ( − q ) j + t − n − x − ε j − ε t x − ε i + ε j x − ε i + ε t + [2] − z q n X t = j +1 ( − q ) t − n − ( x − ε j − ε t ) ( x − ε i + ε t ) + [2] − z q n − X t = j +1 n − X s = t +1 ( − q ) s + t − n x − ε j − ε s x − ε j − ε t x − ε i + ε t x − ε i + ε s + [2] − z q n − X t = j +1 t − X s = j +1 ( − q ) s + t − n − x − ε j − ε t x − ε j − ε s x − ε i + ε s x − ε i + ε t = − [2] − ( − q ) j − n − z q ( x − ε i + ε j ) − z q n − X t = j +1 ( − q ) t − n − x − ε j − ε n x − ε j − ε t x − ε i + ε t x − ε i + ε n + z q n X t = j +1 ( − q ) j + t − n − x − ε j − ε t x − ε i + ε j x − ε i + ε t + [2] − z q n X t = j +1 ( − q ) t − n − ( x − ε j − ε t ) ( x − ε i + ε t ) − z q n − X t = j +1 n − X s = t +1 ( − q ) s + t − n − x − ε j − ε s x − ε j − ε t x − ε i + ε t x − ε i + ε s , proving the required formula about a j . (cid:3) “ Proof of Proposition 2.6 ” We have described explicitly possible pairs { α, ν } in (1)-(4). Thanksto Lemma 2.4, c − r ∈ A | r |− if it appears in Lemma 2.2(1) for the expression of x − ν x − α − v ( ν | α ) x − α x − ν suchthat { α, ν } appears in (1). If { α, ν } appears in (4), the corresponding result follows from Corollary A.4.If { α, ν } appears in (2), the corresponding result follows from Lemmas A.6, A.8, A.10–A.11 and thosefor g = sp n in Lemma A.7. Finally, if { α, ν } appears in (3), the corresponding result follows fromLemmas A.9, A.12–A.14 and those for g ∈ { so n , so n +1 } in Lemma A.7. This completes the proof ofProposition 2.6 for c − r . Applying τ yields the results on c + r . Appendix B. Proof of Proposition 2.7 Later on, we will freely use k λ x − ν = v − ( λ | ν ) x − ν k λ , ∀ ( ν, λ ) ∈ R + × P . HE AFFINE KAUFFMANN CATEGORY AND THE CYCLOTOMIC KAUFFMANN CATEGORY Lemma B.1. Suppose that g ∈ { so n , sp n , so n +1 } and ≤ i < j ≤ n . Then (1) ∆( x − ε i − ε j ) = x − ε i − ε j ⊗ k − ( ε i − ε j ) + 1 ⊗ x − ε i − ε j − qz q P i Suppose that g = so n +1 and ≤ j ≤ n . Then (1) ∆( x − ε j ) = x − ε j ⊗ k − ε j + 1 ⊗ x − ε j − qz q P j 0. ByCorollary A.5(4), Ψ( x − ε j ) = Ψ( x − ε n )Ψ( x − ε j − ε n ) − q Ψ( x − ε j − ε n )Ψ( x − ε n ) , Ψ ∈ { ∆ , ∆ } . Rewriting Ψ( x − ε j − ε n ) via Lemma B.1 and using x − ε j − ε t x − ε n = x − ε n x − ε j − ε t if j < t < n (seeCorollary A.4) yield the following two equations:∆( x − ε j ) = x − ε j ⊗ k − ε j + 1 ⊗ x − ε j − qz q x − ε n ⊗ k − ε n x − ε j − ε n + q z q X j Suppose that g ∈ { so n , sp n , so n +1 } and ≤ h < l ≤ n . Then (1) ∆( x − ε h + ε l ) = x − ε h + ε l ⊗ k − ( ε h + ε l ) + 1 ⊗ x − ε h + ε l + P I,γ h I,γ x − I ⊗ k − wt ( I ) x − γ , (2) ∆( x − ε h + ε l ) = x − ε h + ε l ⊗ k ε h + ε l + 1 ⊗ x − ε h + ε l + P I,γ g I,γ x − γ ⊗ k γ x − I ,where I ranges over non-empty sequence of positive roots and γ ∈ D ε h + ε l such that wt ( I ) + γ = ε h + ε l and, both h I,γ and g I,γ are in A ℓ ( I ) .Proof. Let l = n . First, we assume g = sp n . By Corollary A.5(6),Ψ( x − ε h + ε n ) = Ψ( x − ε n )Ψ( x − ε h − ε n ) − q Ψ( x − ε h − ε n )Ψ( x − ε n ) , Ψ ∈ { ∆ , ∆ } . By Corollary A.4, we have x − ε h − ε t x − ε n = x − ε n x − ε h − ε t , if h < t < n . (B.1) Rewriting Ψ( x − ε h − ε n ) and Ψ( x − ε n ) via Lemma B.1, (2.6)-(2.8) and using (B.1) yield the following twoequations: ∆( x − ε h + ε n ) = x − ε h + ε n ⊗ k − ( ε h + ε n ) + 1 ⊗ x − ε h + ε n + (1 − q ) x − ε n ⊗ k − ε n x − ε h − ε n + q z q X h 1, then x − ε h + ε l = x − n . In this case, (1) and (2) followfrom (2.6)–(2.8). In general, by Corollary A.5(5),Ψ( x − ε h + ε n ) = Ψ( x − ε n − + ε n )Ψ( x − ε h − ε n − ) − q Ψ( x − ε h − ε n − )Ψ( x − ε n − + ε n ) , Ψ ∈ { ∆ , ∆ } .By Corollary A.4), we have x − ε h − ε t x − ε n − + ε n = x − ε n − + ε n x − ε h − ε t , if h < t < n − 1. (B.4)Rewriting Ψ( x − ε h − ε n ) and Ψ( x − ε n − + ε n ) via Lemma B.1 and (2.6)-(2.8) and using (B.4), we have thefollowing two equations:∆( x − ε h + ε n ) = x − ε h + ε n ⊗ k − ε h − ε n + 1 ⊗ x − ε h + ε n − qz q x − ε n − + ε n ⊗ k − ε n − − ε n x − ε h − ε n − + q z q X h 0. By Corollary A.5(2), for Ψ ∈ { ∆ , ∆ } ,Ψ( x − ε h + ε l ) = Ψ( x − ε l − ε l +1 )Ψ( x − ε h + ε l +1 ) − q Ψ( x − ε h + ε l +1 )Ψ( x − ε l − ε l +1 ) . Note that Ψ( x − ε l − ε l +1 ) can be computed by via (2.6)-(2.8), and Ψ( x − ε h + ε l +1 ) can be computed byinduction assumption on n − ( l + 1). We have∆( x − ε h + ε l +1 ) = x − ε h + ε l +1 ⊗ k − ( ε h + ε l +1 ) + 1 ⊗ x − ε h + ε l +1 + X I ,γ h I ,γ x − I ⊗ k − wt ( I ) x − γ , ∆( x − ε h + ε l +1 ) = x − ε h + ε l +1 ⊗ k ε h + ε l +1 + 1 ⊗ x − ε h + ε l +1 + X I ,γ g I ,γ x − γ ⊗ k γ x − I , where I ranges over non-empty sequence of positive roots and γ ∈ D ε h + ε l +1 such that wt ( I ) + γ = ε h + ε l +1 and h I ,γ , g I ,γ ∈ A ℓ ( I ) . Suppose l + 1 < t ≤ n , h < j < l or l + 1 < j ≤ n .By Corollary A.4, we have x − ε h + ε t x − ε l − ε l +1 = x − ε l − ε l +1 x − ε h + ε t , x − ε h − ε j x − ε l − ε l +1 = x − ε l − ε l +1 x − ε h − ε j ,x − ε h x − ε l − ε l +1 = x − ε l − ε l +1 x − ε h , x − ε h − ε l +1 x − ε l − ε l +1 = qx − ε l − ε l +1 x − ε h − ε l +1 . (B.6)Using (B.6) to simplify two equations above, we have∆( x − ε h + ε l ) − x − ε h + ε l ⊗ k − ( ε h + ε l ) − ⊗ x − ε h + ε l + qz q x − ε l − ε l +1 ⊗ k − ( ε l − ε l +1 ) x − ε h + ε l +1 + q X I h I ,ε h − ε l x − I ⊗ k − wt ( I ) [ x − ε h − ε l , x − ε l − ε l +1 ] v = X I ,γ h I ,γ ( x − ε l − ε l +1 x − I − q − ( γ ,ε l − ε l +1 ) x − I x − ε l − ε l +1 ) ⊗ k − wt ( I ) − ( ε l − ε l +1 ) x − γ (B.7)and ∆( x − ε h + ε l ) − x − ε h + ε l ⊗ k ε h + ε l − ⊗ x − ε h + ε l + z q x − ε h + ε l +1 ⊗ k ε h + ε l +1 x − ε l − ε l +1 + q X I g I ,ε h − ε l [ x − ε h − ε l , x − ε l − ε l +1 ] v ⊗ k ε h − ε l +1 x − I = q ( γ ,ε l − ε l +1 ) X I ,γ g I ,γ x − γ ⊗ k γ ( x − ε l − ε l +1 x − I − q − ( γ ,ε l − ε l +1 ) x − I x − ε l − ε l +1 ) . (B.8)Thanks to Corollary A.5(1), we replace [ x − ε h − ε l , x − ε l − ε l +1 ] v by − q − x − ε h − ε l +1 in those two equations. So,we only need to deal with the (RHS) of (B.7)–(B.8). Since h I ,γ , g I ,γ ∈ A ℓ ( I ) , by Proposition 2.6(1),for each pair { I , γ } there exists c ∈ Z such that: D ( x − ε l − ε l +1 x − I − q − ( γ ,ε l − ε l +1 ) x − I x − ε l − ε l +1 ) = D (1 − q c ) x − ε l − ε l +1 x − I + X J c J x − J , D ∈ { h I ,γ , g I ,γ } , where J is non-empty sequence of positive roots such that wt ( J ) = wt ( I ) and c J ∈ A ℓ ( J ) . We rewritethe (RHS) of (B.7)–(B.8) via the above equation. Now, (1)–(2) follow, since (1 − q c ) D ∈ A ℓ ( I )+1 , nomater whether c = 0 or not and, x − I = x − ε l − ε l +1 x − I where I = ( I , ε l − ε l +1 ), ℓ ( I ) = ℓ ( I ) + 1. (cid:3) Lemma B.4. Suppose that g = sp n and ≤ h ≤ n . Then (1) ∆( x − ε h ) = x − ε h ⊗ k − ε h + 1 ⊗ x − ε h + P I,J h I,J x − I ⊗ k − wt ( I ) x − J , (2) ∆( x − ε h ) = x − ε h ⊗ k ε h + 1 ⊗ x − ε h + P I,J g I,J x − I ⊗ k wt ( I ) x − J ,where I , J are non-empty sequences of positive roots such that wt ( I ) + wt ( J ) = 2 ε h and, both h I,J and g I,J are in A max { ℓ ( I ) ,ℓ ( J ) } .Proof. If h = n , then x − ε h = x − n . In this case, (1)-(2) follow from (2.6)–(2.8). Suppose h < n . ByCorollary A.5(7), Ψ( x − ε h ) = [2] − Ψ( x − ε h + ε n )Ψ( x − ε h − ε n ) − [2] − Ψ( x − ε h − ε n )Ψ( x − ε h + ε n ) , Ψ ∈ { ∆ , ∆ } . By Corollary A.4, x − ε h − ε t x − ε h ± ε n = qx − ε h ± ε n x − ε h − ε t , if h < t < n . (B.9)Rewriting Ψ( x − ε h − ε n ) and Ψ( x − ε h + ε n ) via Lemma B.1 and (B.2)-(B.3) and using (B.9), we have∆( x − ε h ) = x − ε h ⊗ k − ε h + 1 ⊗ x − ε h + q z q x − ε n ⊗ k − ε n ( x − ε h − ε n ) − z q x − ε h + ε n ⊗ k − ( ε h + ε n ) x − ε h − ε n + q z q X h S. 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School of Mathematical Science, Tongji University, Shanghai, 200092, China E-mail address ::