aa r X i v : . [ m a t h . QA ] J a n A conjecture concerningthe q -Onsager algebra Paul Terwilliger
Abstract
The q -Onsager algebra O q is defined by two generators W , W and two relationscalled the q -Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBWbasis for O q with elements denoted { B nδ + α } ∞ n =0 , { B nδ + α } ∞ n =0 , { B nδ } ∞ n =1 . In their recent study of a current algebra A q , Baseilhac and Belliard conjecture thatthere exist elements { W − k } ∞ k =0 , { W k +1 } ∞ k =0 , { G k +1 } ∞ k =0 , { ˜ G k +1 } ∞ k =0 in O q that satisfy the defining relations for A q . In order to establish this conjecture, itis desirable to know how the elements on the second displayed line above are related tothe elements on the first displayed line above. In the present paper, we conjecture theprecise relationship and give some supporting evidence. This evidence consists of somecomputer checks on SageMath due to Travis Scrimshaw, and a proof of our conjecturefor a homomorphic image of O q called the universal Askey-Wilson algebra. Keywords . q -Onsager algebra; q -Dolan/Grady relations; PBW basis; tridiagonalpair. . Primary: 17B37. Secondary: 05E14,81R50. We will be discussing the q -Onsager algebra O q [3, 25]. This infinite-dimensional associativealgebra is defined by two generators W , W and two relations called the q -Dolan/Gradyrelations; see Definition 3.1 below. One can view O q as a q -analog of the universal envelopingalgebra of the Onsager Lie algebra O [14, 21, 22].The algebra O q originated in algebraic combinatorics [25]. There is a family of algebras calledtridiagonal algebras [25, Definition 3.9] that arise in the study of association schemes [24,Lemma 5.4] and tridiagonal pairs [15, Theorem 10.1], [25, Theorem 3.10]. The algebra O q isthe “most general” example of a tridiagonal algebra [17, Section 1.2]. A finite-dimensionalirreducible O q -module is essentially the same thing as a tridiagonal pair of q -Racah type [25,1heorem 3.10]. These tridiagonal pairs are classified up to isomorphism in [16, Theorem 3.3].To our knowledge the q -Dolan/Grady relations first appeared in [24, Lemma 5.4].The algebra O q has applications outside combinatorics. For instance, O q is used to studyboundary integrable systems [2–6, 8–10, 12]. The algebra O q can be realized as a left orright coideal subalgebra of the quantized enveloping algebra U q ( b sl ); see [4, 5, 18]. Thealgebra O q is the simplest example of a quantum symmetric pair coideal subalgebra of affinetype [18, Example 7.6]. A Drinfeld type presentation of O q is obtained in [19], and this is usedin [20] to realize O q as an ι Hall algebra of the projective line. There is an injective algebrahomomorphism from O q into the algebra (cid:3) q [27, Proposition 5.6], and a noninjective algebrahomomorphism from O q into the universal Askey-Wilson algebra ∆ q [26, Sections 9,10].In [11, Theorem 4.5], Baseilhac and Kolb obtain a Poincar´e-Birkhoff-Witt (or PBW) basisfor O q . They obtain this PBW basis by using a method of Damiani [13] along with twoautomorphisms of O q that are roughly analogous to the Lusztig automorphisms of U q ( b sl ).The PBW basis elements are denoted { B nδ + α } ∞ n =0 , { B nδ + α } ∞ n =0 , { B nδ } ∞ n =1 . (1)In [8] Baseilhac and Koizumi introduce a current algebra A q for O q , in order to solve bound-ary integrable systems with hidden symmetries. In [12, Definition 3.1] Baseilhac and Shigechigive a presentation of A q by generators and relations. The generators are denoted {W − k } ∞ k =0 , {W k +1 } ∞ k =0 , {G k +1 } ∞ k =0 , { ˜ G k +1 } ∞ k =0 and the relations are given in (20)–(30) below. One can view A q as a q -analog of theuniversal enveloping algebra of O [7, Definition 4.1, Theorem 2], so it is natural to ask how A q is related to O q . Baseilhac and Belliard investigate this issue in [6]; their results aresummarized as follows. In [6, line (3.7)] they show that W , W satisfy the q -Dolan/Gradyrelations. In [6, Section 3] they show that A q is generated by W , W together with thecentral elements { ∆ n } ∞ n =1 defined in [6, Lemma 2.1]. In [6, Section 3] they consider thequotient algebra of A q obtained by sending ∆ n to a scalar for all n ≥
1. The constructionyields an algebra homomorphism Ψ from O q onto this quotient. In [6, Conjecture 2] Baseilhacand Belliard conjecture that Ψ is an isomorphism. If the conjecture is true then there existsan algebra homomorphism A q → O q that sends W W and W W . In this case thereexist elements { W − k } ∞ k =0 , { W k +1 } ∞ k =0 , { G k +1 } ∞ k =0 , { ˜ G k +1 } ∞ k =0 (2)in O q that satisfy the relations (20)–(30). In order to make progress on the above conjecture,it is desirable to know how the elements (2) are related to the elements in (1). In thepresent paper, we conjecture the precise relationship and give some supporting evidence. Ourconjecture statement is Conjecture 6.1. Our supporting evidence consists of some computerchecks on SageMath (see [23]) due to Travis Scrimshaw, and a proof of the conjecture at thelevel of the algebra ∆ q mentioned above.The paper is organized as follows. Section 2 contains some preliminaries. In Section 3 werecall the algebra O q , and describe the PBW basis due to Baseilhac and Kolb. In Sections2, 5 we develop some results about generating functions that will be used in Conjecture 6.1.In Section 6 we state Conjecture 6.1 and explain its meaning. In Section 7 we present ourevidence supporting Conjecture 6.1. In Section 8 we give some comments. In AppendicesA, B we display in detail some equations from the main body of the paper. Throughout the paper, the following notational conventions are in effect. Recall the naturalnumbers N = { , , , . . . } and integers Z = { , ± , ± , . . . } . Let F denote a field. Everyvector space mentioned is over F . Every algebra mentioned is associative, over F , and has amultiplicative identity. Definition 2.1. (See [13, p. 299].) Let A denote an algebra. A Poincar´e-Birkhoff-Witt (or
PBW ) basis for A consists of a subset Ω ⊆ A and a linear order < on Ω such that thefollowing is a basis for the vector space A : a a · · · a n n ∈ N , a , a , . . . , a n ∈ Ω , a ≤ a ≤ · · · ≤ a n . We interpret the empty product as the multiplicative identity in A .We will be discussing generating functions. Let A denote an algebra and let t denote anindeterminate. For a sequence { a n } n ∈ N of elements in A , the corresponding generatingfunction is a ( t ) = X n ∈ N a n t n . The above sum is formal; issues of convergence are not considered. We call a ( t ) the generatingfunction over A with coefficients { a n } n ∈ N . For generating functions a ( t ) = P n ∈ N a n t n and b ( t ) = P n ∈ N b n t n over A , their product a ( t ) b ( t ) is the generating function P n ∈ N c n t n suchthat c n = P ni =0 a i b n − i for n ∈ N . The set of generating functions over A forms an algebra.Let a ( t ) = P n ∈ N a n t n denote a generating function over A . We say that a ( t ) is normalized whenever a = 1. If 0 = a ∈ F then define a ( t ) ∨ = a − a ( t ) , (3)and note that a ( t ) ∨ is normalized.Fix a nonzero q ∈ F that is not a root of unity. Recall the notation[ n ] q = q n − q − n q − q − n ∈ N . q -Onsager algebra O q In this section we recall the q -Onsager algebra O q . For elements X, Y in any algebra, definetheir commutator and q -commutator by[ X, Y ] = XY − Y X, [ X, Y ] q = qXY − q − Y X. X, [ X, [ X, Y ] q ] q − ] = X Y − [3] q X Y X + [3] q XY X − Y X . Definition 3.1. (See [3, Section 2], [25, Definition 3.9].) Define the algebra O q by generators W , W and relations [ W , [ W , [ W , W ] q ] q − ] = ( q − q − ) [ W , W ] , (4)[ W , [ W , [ W , W ] q ] q − ] = ( q − q − ) [ W , W ] . (5)We call O q the q -Onsager algebra . The relations (4), (5) are called the q -Dolan/Gradyrelations . Remark 3.2.
In [11] Baseilhac and Kolb define the q -Onsager algebra in a slightly moregeneral way that involves two scalar parameters c, q . Our O q is their q -Onsager algebra with c = q − ( q − q − ) .In [11], Baseilhac and Kolb obtain a PBW basis for O q that involves some elements { B nδ + α } ∞ n =0 , { B nδ + α } ∞ n =0 , { B nδ } ∞ n =1 . (6)These elements are recursively defined as follows. Writing B δ = q − W W − W W we have B α = W , B δ + α = W + q [ B δ , W ]( q − q − )( q − q − ) , (7) B nδ + α = B ( n − δ + α + q [ B δ , B ( n − δ + α ]( q − q − )( q − q − ) n ≥ B α = W , B δ + α = W − q [ B δ , W ]( q − q − )( q − q − ) , (9) B nδ + α = B ( n − δ + α − q [ B δ , B ( n − δ + α ]( q − q − )( q − q − ) n ≥ . (10)Moreover for n ≥ B nδ = q − B ( n − δ + α W − W B ( n − δ + α + ( q − − n − X ℓ =0 B ℓδ + α B ( n − ℓ − δ + α . (11)By [11, Proposition 5.12] the elements { B nδ } ∞ n =1 mutually commute. Lemma 3.3. (See [11, Theorem 4.5].)
Assume that q is transcendental over F . Then aPBW basis for O q is obtained by the elements (6) in any linear order. Definition 3.4.
We define a generating function in the indeterminate t : B ( t ) = X n ∈ N B nδ t n , B δ = q − − . (12)In Section 6 we will make a conjecture about B ( t ). In Sections 4, 5 we motivate the conjecturewith some comments about generating functions.4 Generating functions over a commutative algebra
Throughout this section the following notational conventions are in effect. We fix a commu-tative algebra A . Every generating function mentioned is over A .The following results are readily checked. Lemma 4.1.
A generating function a ( t ) = P n ∈ N a n t n is invertible if and only if a isinvertible in A . In this case ( a ( t )) − = P n ∈ N b n t n where b = a − and for n ≥ , b n = − a − n X k =1 a k b n − k . Lemma 4.2.
For generating functions a ( t ) = P n ∈ N a n t n and b ( t ) = P n ∈ N b n t n the followingare equivalent: (i) a ( t ) = b ( qt ) b ( q − t ) ; (ii) a n = P ni =0 b i b n − i q i − n for n ∈ N . Lemma 4.3.
For a normalized generating function a ( t ) = P n ∈ N a n t n , there exists a uniquenormalized generating function b ( t ) = P n ∈ N b n t n such that a ( t ) = b ( qt ) b ( q − t ) . Moreover for n ≥ , b n = a n − P n − i =1 b i b n − i q i − n q n + q − n . Definition 4.4.
Referring to Lemma 4.3, we call b ( t ) the q -square root of a ( t ). Lemma 4.5.
For generating functions a ( t ) = P n ∈ N a n t n and b ( t ) = P n ∈ N b n t n the followingare equivalent: (i) a ( t ) = b (cid:0) q + q − t + t − (cid:1) ; (ii) a = b and for n ≥ , a n = ⌊ ( n − / ⌋ X ℓ =0 ( − ℓ (cid:18) n − − ℓℓ (cid:19) [2] n − ℓq b n − ℓ . (13) Proof.
Note that for k ∈ N , (1 − t ) − k − = X ℓ ∈ N (cid:18) k + ℓℓ (cid:19) t ℓ . (14)5e have b (cid:18) q + q − t + t − (cid:19) = X n ∈ N (cid:18) q + q − t + t − (cid:19) n b n = b + X k ∈ N (cid:18) q + q − t + t − (cid:19) k +1 b k +1 . We have q + q − t + t − = [2] q t (1 + t ) − . By this and (14) we find that for k ∈ N , (cid:18) q + q − t + t − (cid:19) k +1 = [2] k +1 q t k +1 X ℓ ∈ N ( − ℓ (cid:18) k + ℓℓ (cid:19) t ℓ . By these comments b (cid:18) q + q − t + t − (cid:19) = b + X k,ℓ ∈ N ( − ℓ (cid:18) k + ℓℓ (cid:19) [2] k +1 q b k +1 t k +1+2 ℓ = b + ∞ X n =1 ⌊ ( n − / ⌋ X ℓ =0 ( − ℓ (cid:18) n − − ℓℓ (cid:19) [2] n − ℓq b n − ℓ t n . The result follows.
Lemma 4.6.
For a generating function a ( t ) = P n ∈ N a n t n , there exists a unique generatingfunction b ( t ) = P n ∈ N b n t n such that a ( t ) = b (cid:18) q + q − t + t − (cid:19) . (15) Moreover b = a and for n ≥ , b n = a n − P ⌊ ( n − / ⌋ ℓ =1 ( − ℓ (cid:0) n − − ℓℓ (cid:1) [2] n − ℓq b n − ℓ [2] nq . Proof.
This is a routine consequence of Lemma 4.5.
Definition 4.7.
Referring to Lemma 4.6, we call b ( t ) the q -symmetrization of a ( t ).We now combine the above constructions. Proposition 4.8.
Let a ( t ) = P n ∈ N a n t n denote a normalized generating function. Then fora generating function b ( t ) = P n ∈ N b n t n the following are equivalent: (i) b ( t ) is the q -symmetrization of the q -square root of the inverse of a ( t ) ; (ii) b ( t ) is normalized and a ( t ) b (cid:18) q + q − qt + q − t − (cid:19) b (cid:18) q + q − q − t + qt − (cid:19) = 1; (16)6iii) b ( t ) is normalized and a ( qt ) b (cid:18) q + q − q t + q − t − (cid:19) = a ( q − t ) b (cid:18) q + q − q − t + q t − (cid:19) ; (17)(iv) b = 1 and for n ≥ , n ] q a n + X j + k +2 ℓ +1= n,j,k,ℓ ≥ ( − ℓ (cid:18) k + ℓℓ (cid:19) [2 n − j ] q [2] k +1 q a j b k +1 . (18) Proof. (i) ⇒ (ii) Let a ( t ) denote the inverse of a ( t ), and let a ( t ) denote the q -square rootof a ( t ). By assumption b ( t ) is the q -symmetrization of a ( t ). The generating function a ( t )is normalized, so a ( t ) is normalized by Lemma 4.1. Now a ( t ) is normalized by Lemma 4.3and Definition 4.4. Now b ( t ) is normalized by Lemma 4.6 and Definition 4.7. By construction a ( t ) a ( t ) = 1 , a ( t ) = a ( qt ) a ( q − t ) , a ( t ) = b (cid:18) q + q − t + t − (cid:19) . Combining these equations we obtain (16).(ii) ⇒ (iii) In the equation (16), replace t by qt and also by q − t . Compare the two resultingequations to obtain (17).(iii) ⇒ (iv) Write each side of (17) as a power series in t , and compare coefficients.(iv) ⇒ (i) By assumption, the generating function b ( t ) is normalized and satisfies (18). Let b ′ ( t ) denote the the q -symmetrization of the q -square root of the inverse of a ( t ). Fromour earlier comments, the generating function b ′ ( t ) is normalized and satisfies (18). Theequations (18) admit a unique solution, so b ( t ) = b ′ ( t ). Definition 4.9.
Referring to Proposition 4.8, we call b ( t ) the q -expansion of a ( t ) wheneverthe equivalent conditions (i)–(iv) are satisfied. Lemma 4.10.
Let a ( t ) = P n ∈ N a n t n denote a normalized generating function. Let b ( t ) = P n ∈ N b n t n denote the q -expansion of a ( t ) . Then for n ≥ the following hold: (i) b n is a polynomial in a , a , . . . , a n that has coefficients in F and total degree n , wherewe view a k as having degree k for ≤ k ≤ n . In this polynomial the coefficient of a n is − [ n ] q [2 n ] − q [2] − nq . (ii) a n is a polynomial in b , b , . . . , b n that has coefficients in F and total degree n , wherewe view b k as having degree k for ≤ k ≤ n . In this polynomial the coefficient of b n is − [ n ] − q [2 n ] q [2] nq .Proof. (i) By (18) and induction on n .(ii) By (i) above and induction on n . 7 Generating functions over a noncommutative alge-bra
Throughout this section the following notational conventions are in effect. We fix an algebra B that is not necessarily commutative. Every generating function mentioned is over B . Definition 5.1.
A generating function a ( t ) = P n ∈ N a n t n is said to be commutative whenever { a n } n ∈ N mutually commute. Lemma 5.2.
For a commutative generating function a ( t ) = P n ∈ N a n t n there exists a com-mutative subalgebra A of B that contains a n for n ∈ N .Proof. Take A to be the subalgebra of B generated by { a n } n ∈ N .Referring to Lemma 5.2, we may view a ( t ) as a generating function over A . Definition 5.3.
Let a ( t ) = P n ∈ N a n t n denote a generating function that is commutative andnormalized. By the q -expansion of a ( t ) we mean the q -expansion of the generating function a ( t ) over A , where A is from Lemma 5.2. By (18) and Lemma 4.10, the q -expansion of a ( t )is independent of the choice of A . O q In the previous two sections we discussed generating functions. We now return our attentionto the q -Onsager algebra O q . Recall from Section 1 that in [6, Conjecture 2] Baseilhac andBelliard effectively conjecture that there exist elements { W − k } k ∈ N , { W k +1 } k ∈ N , { G k +1 } k ∈ N , { ˜ G k +1 } k ∈ N (19)in O q that satisfy the following relations. For k, ℓ ∈ N ,[ W , W k +1 ] = [ W − k , W ] = ( ˜ G k +1 − G k +1 ) / ( q + q − ) , (20)[ W , G k +1 ] q = [ ˜ G k +1 , W ] q = ρW − k − − ρW k +1 , (21)[ G k +1 , W ] q = [ W , ˜ G k +1 ] q = ρW k +2 − ρW − k , (22)[ W − k , W − ℓ ] = 0 , [ W k +1 , W ℓ +1 ] = 0 , (23)[ W − k , W ℓ +1 ] + [ W k +1 , W − ℓ ] = 0 , (24)[ W − k , G ℓ +1 ] + [ G k +1 , W − ℓ ] = 0 , (25)[ W − k , ˜ G ℓ +1 ] + [ ˜ G k +1 , W − ℓ ] = 0 , (26)[ W k +1 , G ℓ +1 ] + [ G k +1 , W ℓ +1 ] = 0 , (27)[ W k +1 , ˜ G ℓ +1 ] + [ ˜ G k +1 , W ℓ +1 ] = 0 , (28)[ G k +1 , G ℓ +1 ] = 0 , [ ˜ G k +1 , ˜ G ℓ +1 ] = 0 , (29)[ ˜ G k +1 , G ℓ +1 ] + [ G k +1 , ˜ G ℓ +1 ] = 0 . (30)8n the above equations ρ = − ( q − q − ) . For notational convenience define G = − ( q − q − )[2] q , ˜ G = − ( q − q − )[2] q . (31)It is desirable to know how the elements (19) are related to the elements (6). In this paperwe conjecture the precise relationship. We will state the conjecture shortly. Before statingthe conjecture, we discuss what is involved. Let us simplify things by writing the elements(19) in terms of W , W , { ˜ G k +1 } k ∈ N . To do this, we use (21), (22) to recursively obtain W − k , W k +1 for k ≥ W − = W − [ ˜ G , W ] q ( q − q − ) ,W = W − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) ,W − = W − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) ,W = W − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) ,W − = W − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) , · · · W = W − [ W , ˜ G ] q ( q − q − ) ,W − = W − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) ,W = W − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) ,W − = W − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) ,W = W − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) − [ ˜ G , W ] q ( q − q − ) − [ W , ˜ G ] q ( q − q − ) , · · · The recursion shows that for any integer k ≥
1, the generators W − k , W k +1 are given asfollows. For odd k = 2 r + 1, W − k = W − r X ℓ =0 [ ˜ G ℓ +1 , W ] q ( q − q − ) − r X ℓ =1 [ W , ˜ G ℓ ] q ( q − q − ) , (32) W k +1 = W − r X ℓ =0 [ W , ˜ G ℓ +1 ] q ( q − q − ) − r X ℓ =1 [ ˜ G ℓ , W ] q ( q − q − ) . (33)9or even k = 2 r , W − k = W − r − X ℓ =0 [ W , ˜ G ℓ +1 ] q ( q − q − ) − r X ℓ =1 [ ˜ G ℓ , W ] q ( q − q − ) , (34) W k +1 = W − r − X ℓ =0 [ ˜ G ℓ +1 , W ] q ( q − q − ) − r X ℓ =1 [ W , ˜ G ℓ ] q ( q − q − ) . (35)Next we use (20) to obtain the generators { G k +1 } k ∈ N : G k +1 = ˜ G k +1 + ( q + q − )[ W , W − k ] ( k ∈ N ) . (36)We have expressed the elements (19) in terms of W , W , { ˜ G k +1 } k ∈ N . Next, we would liketo know how the elements { ˜ G k +1 } k ∈ N are related to the elements (6). We will discuss thisrelationship using generating functions.Recall the generating function B ( t ) from Definition 3.4. The generating function B ( t ) iscommutative by Definition 5.1 and the comment above Lemma 3.3. By (12) the generatingfunction B ( t ) has constant term q − − − q − ( q − q − ), so by (3) we have B ( t ) ∨ = − q ( q − q − ) − B ( t ) . The generating function B ( t ) ∨ is commutative and normalized, so we may speak of its q -expansion as is Definition 5.3. Conjecture 6.1.
Define the elements { W − k } k ∈ N , { W k +1 } k ∈ N , { G k +1 } k ∈ N , { ˜ G k +1 } k ∈ N (37)in O q as follows:(i) the generating function ˜ G ( t ) ∨ is the q -expansion of B ( t ) ∨ , where ˜ G ( t ) = P n ∈ N ˜ G n t n and ˜ G is from (31);(ii) the elements { W − k } k ∈ N , { W k +1 } k ∈ N satisfy (32)–(35);(iii) the elements { G k +1 } k ∈ N satisfy (36).Then the elements (37) satisfy (20)–(30).We have some comments about the q -expansion of B ( t ) ∨ . We mentioned above that B ( t ) iscommutative, so by Lemma 5.2 there exists a commutative subalgebra A of O q that contains B nδ for n ∈ N . So B ( t ) is over A . The q -expansion of B ( t ) ∨ is over A , and described asfollows. For the moment let ˜ G ( t ) = P n ∈ N ˜ G n t n denote any generating function over A suchthat ˜ G satisfies (31). By Proposition 4.8 and Definitions 4.9, 5.3 we find that˜ G ( t ) ∨ is the q -expansion of B ( t ) ∨
10f and only if B ( t ) ˜ G (cid:18) q + q − qt + q − t − (cid:19) ˜ G (cid:18) q + q − q − t + qt − (cid:19) = − q − ( q − q − ) [2] q (38)if and only if B ( qt ) ˜ G (cid:18) q + q − q t + q − t − (cid:19) = B ( q − t ) ˜ G (cid:18) q + q − q − t + qt − (cid:19) (39)if and only if for n ≥ n ] q B nδ ˜ G + X j + k +2 ℓ +1= n,j,k,ℓ ≥ ( − ℓ (cid:18) k + ℓℓ (cid:19) [2 n − j ] q [2] k +1 q B jδ ˜ G k +1 . (40)In Appendix A we display (40) in detail for 1 ≤ n ≤ In this section we give some supporting evidence for Conjecture 6.1.Our first type of evidence is from checking via computer. The algebra O q has been im-plemented in the computer package SageMath (see [23]) by Travis Scimshaw. Using thispackage Scrimshaw defined the elements (37) for 0 ≤ k ≤ q [26, Defi-nition 1.2]. This algebra is defined by generators and relations. The generators are A, B, C .The relations assert that each of the following is central in ∆ q : A + qBC − q − CBq − q − , B + qCA − q − ACq − q − , C + qAB − q − BAq − q − . For the above three central elements, multiply each by q + q − to get α , β , γ . Thus A + qBC − q − CBq − q − = αq + q − , (41) B + qCA − q − ACq − q − = βq + q − , (42) C + qAB − q − BAq − q − = γq + q − . (43)Each of α , β , γ is central in ∆ q . By [26, Corollary 8.3] the center of ∆ q is generated by α, β, γ, Ω where Ω = qABC + q A + q − B + q C − qAα − q − Bβ − qCγ. (44)11he element Ω is called the Casimir element. By [26, Theorem 8.2] the elements α, β, γ, Ωare algebraically independent. We write F [ α, β, γ, Ω] for the center of ∆ q .Next we summarize from [26, Section 3] how the modular group PSL ( Z ) acts on ∆ q as agroup of automorphisms. By [1] the group PSL ( Z ) has a presentation by generators ̺ , σ and relations ̺ = 1, σ = 1. By [26, Theorems 3.1, 6.4] the group PSL ( Z ) acts on ∆ q as agroup of automorphisms in the following way: u A B C α β γ Ω ̺ ( u ) B C A β γ α Ω σ ( u ) B A C + [ A,B ] q − q − β α γ ΩFor notational convenience define C ′ = C + [ A, B ] q − q − . (45)Applying σ to (41)–(43) and using the above table, we obtain B + qAC ′ − q − C ′ Aq − q − = βq + q − , (46) A + qC ′ B − q − BC ′ q − q − = αq + q − , (47) C ′ + qBA − q − ABq − q − = γq + q − . (48)Next we explain how ∆ q is related to O q . By [26, Theorem 2.2] the algebra ∆ q has apresentation by generators A, B, γ and relations A B − [3] q A BA + [3] q ABA − BA = ( q − q − ) ( BA − AB ) , (49) B A − [3] q B AB + [3] q BAB − AB = ( q − q − ) ( AB − BA ) , (50) A B − B A + ( q + q − )( BABA − ABAB ) = ( q − q − ) ( BA − AB ) γ, (51) γA = Aγ, γB = Bγ. (52)The relations (49), (50) are the q -Dolan/Grady relations. Consequently there exists analgebra homomorphism ♮ : O q → ∆ q that sends W A and W B . This homomorphismis not injective by [26, Theorem 10.9].For the elements (6) and (37) we retain the same notation for their images under ♮ . We willshow that for ∆ q the elements (37) satisfy the relations (20)–(30).For the algebra ∆ q define Ψ( t ) = B ( t ) + 1 − q − , (53)where B ( t ) is from Definition 3.4. By (12) we have Ψ( t ) = P ∞ n =1 B nδ t n . By [28, Corollary 5.7]the elements { B nδ } ∞ n =1 are contained in the subalgebra of ∆ q generated by F [ α, β, γ, Ω] and12 . Consequently the elements { B nδ } ∞ n =1 commute with C , so Ψ( t ) commutes with C . Bythis and [28, Line (5.19)] we find thatΨ( t ) (cid:0) qt + q − t − + C (cid:1)(cid:0) q − t + qt − + C (cid:1) (54)is equal to 1 − q − timesΩ − ( t + t − ) αβ ( t − t − ) − α + β ( t − t − ) − ( t + t − ) γ + ( q + q − )( t + t − ) C + C . Upon eliminating Ψ( t ) from (54) using (53), we find that B ( t ) (cid:0) qt + q − t − + C (cid:1)(cid:0) q − t + qt − + C (cid:1) (55)is equal to 1 − q − timesΩ − ( t + t − ) αβ ( t − t − ) − α + β ( t − t − ) − ( t + t − ) γ − ( qt + q − t − )( q − t + qt − ) . Define N ( t ) = B ( t ) q − − qt + q − t − + Cqt + q − t − q − t + qt − + Cq − t + qt − . (56)By the above comments N ( t ) = 1 + N ( t )Ω + N ( t ) αβ + N ( t )( α + β ) + N ( t ) γ, (57)where N ( t ) = − qt + q − t − )( q − t + qt − ) , (58) N ( t ) = t + t − ( t − t − ) ( qt + q − t − )( q − t + qt − ) , (59) N ( t ) = 1( t − t − ) ( qt + q − t − )( q − t + qt − ) , (60) N ( t ) = t + t − ( qt + q − t − )( q − t + qt − ) . (61)Evaluating (58)–(61) using 1 qt + q − t − = X n ∈ N ( − n q n +1 t n +1 , q − t + qt − = X n ∈ N ( − n q − n − t n +1 , t − t − ) = X n ∈ N nt n we find that the functions N ( t ), N ( t ), N ( t ), N ( t ) are power series in t with zero con-stant term. By this and (57), we may view N ( t ) as a normalized generating function over F [ α, β, γ, Ω]. 13 efinition 7.1.
Define a generating function Z ( t ) = P n ∈ N Z n t n over F [ α, β, γ, Ω] such that Z = q − − q and Z ( t ) ∨ is the q -expansion of N ( t ).The notation Z ( t ) ∨ is explained in (3). The q -expansion concept is explained in Proposition4.8 and Definition 4.9. By these explanations and Definition 7.1, N ( t ) Z (cid:18) q + q − qt + q − t t − (cid:19) Z (cid:18) q + q − q − t + qt − (cid:19) = ( q − q − ) . (62) Proposition 7.2.
For the algebra ∆ q , ˜ G ( t ) = Z ( t )( q + q − + tC ) . (63) Proof.
Define the generating function ˜ G ( t ) = Z ( t )( q + q − + tC ). We show that ˜ G ( t ) =˜ G ( t ). Let A denote the subalgebra of ∆ q generated by F [ α, β, γ, Ω] and C . Note that A iscommutative. By construction ˜ G ( t ) is over A . By our comments below (53), the generatingfunction B ( t ) is over A . By the discussion around (38), it suffices to show that B ( t ) ˜ G (cid:18) q + q − qt + q − t − (cid:19) ˜ G (cid:18) q + q − q − t + qt − (cid:19) = − q − ( q − q − ) [2] q . (64)Using (56) and and (62), B ( t ) ˜ G (cid:18) q + q − qt + q − t − (cid:19) ˜ G (cid:18) q + q − q − t + qt − (cid:19) = [2] q B ( t ) Z (cid:18) q + q − qt + q − t − (cid:19) qt + q − t − + Cqt + q − t − Z (cid:18) q + q − q − t + qt − (cid:19) q − t + qt − + Cq − t + qt − = [2] q ( q − − N ( t ) Z (cid:18) q + q − qt + q − t − (cid:19) Z (cid:18) q + q − q − t + qt − (cid:19) = [2] q ( q − − q − q − ) = − q − ( q − q − ) [2] q . We have shown (64), and the result follows.Define the generating functions W − ( t ) = X n ∈ N W − n t n , W + ( t ) = X n ∈ N W n +1 t n . By (32)–(35) we obtain W + ( t ) = t [ ˜ G ( t ) , A ] q + [ B, ˜ G ( t )] q ( t − q − q − ) , (65) W − ( t ) = [ ˜ G ( t ) , A ] q + t [ B, ˜ G ( t )] q ( t − q − q − ) . (66)14 emma 7.3. For the algebra ∆ q , W + ( t ) = Z ( t ) ( q − q − )( α + βt ) − ( q − q − )( t − t − ) B ( q − q − ) ( t − t − ) , (67) W − ( t ) = Z ( t ) ( q − q − )( αt + β ) − ( q − q − )( t − t − ) A ( q − q − ) ( t − t − ) . (68) Proof.
To obtain (67), eliminate ˜ G ( t ) from (65) using (63), and evaluate the result using(41), (42). Equation (68) is similarly obtained.Define the generating function G ( t ) = X n ∈ N G n t n . Using (36) we obtain G ( t ) = ˜ G ( t ) + t ( q + q − )[ B, W − ( t )] . (69) Lemma 7.4.
For the algebra ∆ q we have G ( t ) = Z ( t )( q + q − + tC ′ ) , (70) where C ′ is from (45).Proof. Eliminate ˜ G ( t ) from (69) using (63). Eliminate W − ( t ) from (69) using (68), andevaluate the result using (45).Let s denote an indeterminate that commutes with t . Lemma 7.5.
For the algebra ∆ q we have [ A, W + ( t )] = [ W − ( t ) , B ] = t − ( ˜ G ( t ) − G ( t )) / ( q + q − ) , [ A, G ( t )] q = [ ˜ G ( t ) , A ] q = ρW − ( t ) − ρtW + ( t ) , [ G ( t ) , B ] q = [ B, ˜ G ( t )] q = ρW + ( t ) − ρtW − ( t ) , [ W − ( s ) , W − ( t )] = 0 , [ W + ( s ) , W + ( t )] = 0 , [ W − ( s ) , W + ( t )] + [ W + ( s ) , W − ( t )] = 0 ,s [ W − ( s ) , G ( t )] + t [ G ( s ) , W − ( t )] = 0 ,s [ W − ( s ) , ˜ G ( t )] + t [ ˜ G ( s ) , W − ( t )] = 0 ,s [ W + ( s ) , G ( t )] + t [ G ( s ) , W + ( t )] = 0 ,s [ W + ( s ) , ˜ G ( t )] + t [ ˜ G ( s ) , W + ( t )] = 0 , [ G ( s ) , G ( t )] = 0 , [ ˜ G ( s ) , ˜ G ( t )] = 0 , [ ˜ G ( s ) , G ( t )] + [ G ( s ) , ˜ G ( t )] = 0 , where ρ = − ( q − q − ) .Proof. These relations are routinely verified using Proposition 7.2 and Lemmas 7.3, 7.4 alongwith (41), (42), (46), (47).
Theorem 7.6.
In the algebra ∆ q the elements (37) satisfy the relations (20)–(30).Proof. This is a routine consequence of Lemma 7.5.15
Comments
In the previous section we gave some supporting evidence for Conjecture 6.1. In this sectionwe assume that Conjecture 6.1 is correct, and provide more information about how theelements (37) are related to the elements (6). We will give a variation on (32)–(35).Using Appendix A and B δ = q − W W − W W we obtain˜ G = − qB δ = [ W , W ] q . (71) Lemma 8.1.
For k ∈ N , (i) [ ˜ G k +1 , W ] q = ( q − q − ) W ˜ G k +1 − q [ B δ , W − k ] , (ii) [ W , ˜ G k +1 ] q = ( q − q − ) W ˜ G k +1 + [ B δ , W k +1 ] .Proof. (i) Observe that[ ˜ G k +1 , W ] q = ( q − q − ) W ˜ G k +1 + q [ ˜ G k +1 , W ] . By (26) and (71), [ ˜ G k +1 , W ] = [ ˜ G , W − k ] = − q [ B δ , W − k ] . The result follows.(ii) Observe that [ W , ˜ G k +1 ] q = ( q − q − ) W ˜ G k +1 − q − [ ˜ G k +1 , W ] . By (28) and (71), [ ˜ G k +1 , W ] = [ ˜ G , W k +1 ] = − q [ B δ , W k +1 ] . The result follows.
Lemma 8.2.
For n ≥ , W − n = W n − ( q − q − ) W ˜ G n ( q − q − ) + q [ B δ , W − n ]( q − q − ) , (72) W n +1 = W − n − ( q − q − ) W ˜ G n ( q − q − ) − [ B δ , W n ]( q − q − ) . (73) Proof.
Use the equations on the right in (21), (22) along with Lemma 8.1.We recall some notation from [11]. For a negative integer k define B kδ + α = B ( − k − δ + α , B kδ + α = B ( − k − δ + α . We have B rδ + α = B sδ + α ( r, s ∈ Z , r + s = − . (74)16 emma 8.3. For n ∈ Z , q [ B δ , B nδ + α ]( q − q − )( q − q − ) = B ( n +1) δ + α − B ( n − δ + α , (75) q [ B δ , B nδ + α ]( q − q − )( q − q − ) = B ( n − δ + α − B ( n +1) δ + α . (76) Proof.
Use (7)–(10) and (74).
Proposition 8.4.
For n ∈ N the following hold in O q : W − n = − ( q − q − ) − n X k =0 k X ℓ =0 (cid:18) kℓ (cid:19) q k − ℓ [2] − k − q B ( k − ℓ ) δ + α ˜ G n − k , (77) W n +1 = − ( q − q − ) − n X k =0 k X ℓ =0 (cid:18) kℓ (cid:19) q ℓ − k [2] − k − q B ( k − ℓ ) δ + α ˜ G n − k . (78) Proof.
We use induction on n . First assume that n = 0. Then (77), (78) hold. Next assumethat n ≥
1. To obtain (77), evaluate the right-hand side of (72) using induction along with(74), (75). To obtain (78), evaluate the right-hand side of (73) using induction along with(74), (76).In Appendix B we display (77), (78) in detail for 0 ≤ n ≤ G n − k as a polynomial in B δ , B δ , . . . , B ( n − k ) δ using (40), then we effectively write W − n and W n +1 in the PBW basis for O q given in Lemma3.3. Unfortunately the resulting formula are not pleasant. The author is deeply grateful to Travis Scrimshaw for performing the computer checks men-tioned at the beginning of Section 7. The author thanks Pascal Baseilhac and Nicolas Cramp´efor giving this paper a close reading and offering valuable suggestions.
10 Appendix A
For the q -Onsager algebra O q we use (40) to obtain ˜ G , ˜ G , . . . , ˜ G in terms of B δ , B δ , . . . , B δ .Recall that B δ = q − − , ˜ G = − ( q − q − )[2] q . ˜ G satisfies 0 = [2] q B δ [1] q B δ ˜ G q ˜ G G satisfies 0 = [4] q B δ [3] q B δ [2] q B δ ˜ G q ˜ G q ˜ G G satisfies 0 = [6] q B δ [5] q B δ [4] q B δ [3] q B δ ˜ G q ˜ G − q ˜ G q ˜ G G satisfies 0 = [8] q B δ [7] q B δ [6] q B δ [5] q B δ [4] q B δ ˜ G q ˜ G − q ˜ G − q ˜ G q ˜ G G satisfies 0 = [10] q B δ [9] q B δ [8] q B δ [7] q B δ [6] q B δ [5] q B δ ˜ G q ˜ G − q ˜ G − q ˜ G − q ˜ G q ˜ G G satisfies0 = [12] q B δ [11] q B δ [10] q B δ [9] q B δ [8] q B δ [7] q B δ [6] q B δ ˜ G q ˜ G − q ˜ G − q ˜ G − q ˜ G − q ˜ G q ˜ G G satisfies0 = [14] q B δ [13] q B δ [12] q B δ [11] q B δ [10] q B δ [9] q B δ [8] q B δ [7] q B δ ˜ G q ˜ G − − q ˜ G − q ˜ G − q ˜ G − q ˜ G − q ˜ G q ˜ G G satisfies 0 =[16] q B δ [15] q B δ [14] q B δ [13] q B δ [12] q B δ [11] q B δ [10] q B δ [9] q B δ [8] q B δ ˜ G q ˜ G − − q ˜ G − − q ˜ G − q ˜ G
10 0 − q ˜ G − q ˜ G − q ˜ G q ˜ G
11 Appendix B
For the q -Onsager algebra O q we use (77), (78) to obtain { W − n } n =0 and { W n +1 } n =0 in termsof { B nδ + α } n =0 , { B nδ + α } n =0 , { ˜ G n } n =0 . Recall that ˜ G = − ( q − q − )[2] q .We have W = B α = − ( q − q − ) − [2] − q B α ˜ G . W − is equal to − ( q − q − ) − [2] − q times ˜ G [2] q ˜ G q − B α B α qB δ + α − is equal to − ( q − q − ) − [2] − q times ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B α B α qB δ + α q B δ + α W − is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B α B α qB δ + α q B δ + α q B δ + α W − is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B α B α qB δ + α q B δ + α q B δ + α q B δ + α W − is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B δ + α q − B α
10 0 3 0 1 0 B α qB δ + α
10 0 3 0 1 0 q B δ + α q B δ + α q B δ + α q B δ + α − is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B δ + α q − B δ + α
15 0 4 0 1 0 0 q − B α B α
20 0 6 0 2 0 1 qB δ + α q B δ + α
15 0 4 0 1 0 0 q B δ + α q B δ + α q B δ + α q B δ + α W − is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B δ + α q − B δ + α
21 0 5 0 1 0 0 0 q − B δ + α q − B α
35 0 10 0 3 0 1 0 B α qB δ + α
35 0 10 0 3 0 1 0 q B δ + α q B δ + α
21 0 5 0 1 0 0 0 q B δ + α q B δ + α q B δ + α q B δ + α W = B α = − ( q − q − ) − [2] − q B α ˜ G . W is equal to − ( q − q − ) − [2] − q times ˜ G [2] q ˜ G q − B δ + α B α qB α is equal to − ( q − q − ) − [2] − q times ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α B α qB α q B δ + α W is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α B α qB α q B δ + α q B δ + α W is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B δ + α B α qB α q B δ + α q B δ + α q B δ + α W is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B δ + α q − B δ + α
10 0 3 0 1 0 B α qB α
10 0 3 0 1 0 q B δ + α q B δ + α q B δ + α q B δ + α is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B δ + α q − B δ + α
15 0 4 0 1 0 0 q − B δ + α B α
20 0 6 0 2 0 1 qB α q B δ + α
15 0 4 0 1 0 0 q B δ + α q B δ + α q B δ + α q B δ + α W is equal to − ( q − q − ) − [2] − q times˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G [2] q ˜ G q − B δ + α q − B δ + α q − B δ + α q − B δ + α q − B δ + α
21 0 5 0 1 0 0 0 q − B δ + α q − B δ + α
35 0 10 0 3 0 1 0 B α qB α
35 0 10 0 3 0 1 0 q B δ + α q B δ + α
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936 (2018) 306–319; arXiv:1808.09901 .Paul TerwilligerDepartment of MathematicsUniversity of Wisconsin480 Lincoln DriveMadison, WI 53706-1388 USAemail: [email protected]@math.wisc.edu