A compact presentation for the alternating central extension of the positive part of U q ( sl ˆ 2 )
aa r X i v : . [ m a t h . QA ] N ov A compact presentation for thealternating central extension of thepositive part of U q ( b sl ) Paul Terwilliger
Abstract
This paper concerns the positive part U + q of the quantum group U q ( b sl ). Thealgebra U + q has a presentation involving two generators that satisfy the cubic q -Serrerelations. We recently introduced an algebra U + q called the alternating central extensionof U + q . We presented U + q by generators and relations. The presentation is attractive,but the multitude of generators and relations makes the presentation unwieldy. In thispaper we obtain a presentation of U + q that involves a small subset of the original setof generators and a very manageable set of relations. We call this presentation thecompact presentation of U + q . Keywords . q -Onsager algebra; q -Serre relations; q -shuffle algebra; tridiagonal pair. . Primary: 17B37. Secondary: 05E14,81R50. The algebra U q ( b sl ) is well known in representation theory [14] and statistical mechanics [19].This algebra has a subalgebra U + q called the positive part. The algebra U + q has a presentationinvolving two generators (said to be standard) and two relations, called the q -Serre relations.The presentation is given in Definition 2.1 below.Our interest in U + q is motivated by some applications to linear algebra and combinatorics;these will be described shortly. Before going into detail, we have a comment about q . Inthe applications, either q is not a root of unity, or q is a root of unity with exponent largeenough to not interfere with the rest of the application. To keep things simple, throughoutthe paper we will assume that q is not a root of unity.Our first application has to do with tridiagonal pairs [18]. A tridiagonal pair is roughly de-scribed as an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vectorspace, that each act on the eigenspaces of the other one in a block-tridiagonal fashion [18, Def-inition 1.1]. There is a type of tridiagonal pair said to be q -geometric [16, Definition 2.6];for this type of tridiagonal pair the eigenvalues of each map form a q -geometric progression.1 finite-dimensional irreducible U + q -module on which the standard generators are not nilpo-tent, is essentially the same thing as a tridiagonal pair of q -geometric type [16, Theorem 2.7];these U + q -modules are described in [16, Section 1].Our next application has to do with distance-regular graphs [1], [13]. Consider a distance-regular graph Γ that has diameter d ≥ d, b, α, β ) [13, p. 193] with b = q and α = q −
1. The condition on α implies that Γ is formally self-dual in the senseof [13, p. 49]. Let A denote the adjacency matrix of Γ, and let A ∗ denote the dual adjacencymatrix with respect to any vertex of Γ [17, Section 7]. Then by [17, Lemma 9.4], thereexist complex numbers r, s, r ∗ , s ∗ with r, r ∗ nonzero such that rA + sI , r ∗ A ∗ + s ∗ I satisfythe q -Serre relations. As mentioned in [17, Example 8.4], the above parameter restriction issatisfied by the bilinear forms graph [13, p. 280], the alternating forms graph [13, p. 282],the Hermitean forms graph [13, p. 285], the quadratic forms graph [13, p. 290], the affine E graph [13, p. 340], and the extended ternary Golay code graph [13, p. 359].Our next application has to do with uniform posets [24]. Let GF( b ) denote a finite fieldwith b elements, and let N, M denote positive integers. Let H denote a vector space overGF( b ) that has dimension N + M . Let h denote a subspace of H with dimension M .Let P denote the set of subspaces of H that have zero intersection with h . For x, y ∈ P define x ≤ y whenever x ⊆ y . The relation ≤ is a partial order on P , and the poset P is ranked with rank N . The poset P is called an attenuated space poset, and denoted by A b ( N, M ) [20], [24, Example 3.1]. By [24, Theorem 3.2] the poset A b ( N, M ) is uniform inthe sense of [24, Definition 2.2]. It is shown in [20, Lemma 3.3] that for A b ( N, M ) the raisingmatrix R and the lowering matrix L satisfy the q -Serre relations, provided that b = q .Our last application has to do with q -shuffle algebras. Let F denote a field, and let x, y denote noncommuting indeterminates. Let V denote the free associative F -algebra withgenerators x, y . By a letter in V we mean x or y . For an integer n ≥
0, by a word of length n in V we mean a product of letters v v · · · v n . The words in V form a basis for the vectorspace V . In [22, 23] M. Rosso introduced an algebra structure on V , called the q -shufflealgebra. For letters u, v their q -shuffle product is u ⋆ v = uv + q h u,v i vu , where h u, v i = 2(resp. h u, v i = −
2) if u = v (resp. u = v ). By [22, Theorem 13], in the q -shuffle algebra V the elements x , y satisfy the q -Serre relations. Consequently there exists an algebrahomomorphism ♮ from U + q into the q -shuffle algebra V , that sends the standard generatorsof U + q to x , y . By [23, Theorem 15] the map ♮ is injective.Next we recall the alternating elements in U + q [26]. Let v v · · · v n denote a word in V . Thisword is called alternating whenever n ≥ v i − = v i for 2 ≤ i ≤ n . Thus the alternatingwords have the form · · · xyxy · · · . The alternating words are displayed below: x, xyx, xyxyx, xyxyxyx, . . .y, yxy, yxyxy, yxyxyxy, . . .yx, yxyx, yxyxyx, yxyxyxyx, . . .xy, xyxy, xyxyxy, xyxyxyxy, . . . By [26, Theorem 8.3] each alternating word is contained in the image of ♮ . An element of U + q is called alternating whenever it is the ♮ -preimage of an alternating word. For example, the2tandard generators of U + q are alternating because they are the ♮ -preimages of the alternatingwords x, y . It is shown in [26, Lemma 5.12] that for each row in the above display, thecorresponding alternating elements mutually commute. A naming scheme for alternatingelements is introduced in [26, Definition 5.2].Next we recall the alternating central extension of U + q [27]. In [26] we displayed twotypes of relations among the alternating elements of U + q ; the first type is [26, Proposi-tions 5.7, 5.10, 5.11] and the second type is [26, Propositions 6.3, 8.1]. The relations in [26,Proposition 5.11] are redundant; they follow from the relations in [26, Propositions 5.7, 5.10]as pointed out in [4, Propositions 3.1, 3.2] and [5, Remark 2.5]; see also Corollary 6.3 below.The relations in [26, Proposition 6.3] are also redundant; they follow from the relations in [26,Propositions 5.7, 5.10] as shown in the proof of [26, Proposition 6.3]. By [26, Lemma 8.4]and the previous comments, the algebra U + q is presented by its alternating elements andthe relations in [26, Propositions 5.7, 5.10, 8.1]. For this presentation it is natural to askwhat happens if the relations in [26, Proposition 8.1] are removed. To answer this question,in [27, Definition 3.1] we defined an algebra U + q by generators and relations in the followingway. The generators, said to be alternating, are in bijection with the alternating elements of U + q . The relations are the ones in [26, Propositions 5.7, 5.10]. By construction there exists asurjective algebra homomorphism U + q → U + q that sends each alternating generator of U + q tothe corresponding alternating element of U + q . In [27, Lemma 3.6, Theorem 5.17] we adjustedthis homomorphism to get an algebra isomorphism U + q → U + q ⊗ F [ z , z , . . . ], where { z n } ∞ n =1 are mutually commuting indeterminates. By [27, Theorem 10.2] the alternating generatorsform a PBW basis for U + q . The algebra U + q is called the alternating central extension of U + q .We mentioned above that the algebra U + q is presented by its alternating generators and therelations in [26, Propositions 5.7, 5.10]. This presentation is attractive, but the multitudeof generators and relations makes the presentation unwieldy. In this paper we obtain apresentation of U + q that involves a small subset of the original set of generators and a verymanageable set of relations. This presentation is given in Definition 3.1 below; we call itthe compact presentation of U + q . At first glance, it is not clear that the algebra presented inDefinition 3.1 is equal to U + q . So we denote by U the algebra presented in Definition 3.1, andeventually prove that U = U + q . After this result is established, we describe some features of U + q that are illuminated by the presentation in Definition 3.1.Our investigation of U + q is motivated by some recent developments in statistical mechanics,concerning the q -Onsager algebra O q . In [8] Baseilhac and Koizumi introduce a currentalgebra A q for O q , in order to solve boundary integrable systems with hidden symmetries.In [12, Definition 3.1] Baseilhac and Shigechi give a presentation of A q by generators andrelations. This presentation and the discussion in [12, Section 4] suggest that A q is related to O q in roughly the same way that U + q is related to U + q . The precise relationship between A q and O q is presently unknown, but see [7, Conjectures 1, 2] and [25, Conjectures 4.5, 4.6, 4.8].The articles [2, 3, 6–12] contain background information on O q and A q .This paper is organized as follows. In Section 2 we review some facts about U + q . In Section3, we introduce the algebra U and give an algebra homomorphism U + q → U . In Section 4,we introduce the alternating generators for U and establish some formulas involving these3enerators. In Sections 5, 6 we use these formulas and generating functions to show that thealternating generators for U satisfy the relations in [26, Propositions 5.7, 5.10]. Using thisresult, we prove that U = U + q . Theorem 6.2 and Corollary 6.5 are the main results of thepaper. In Section 7 we describe some features of U + q that are illuminated by the presentationin Definition 3.1. U + q We now begin our formal argument. For the rest of the paper, the following notationalconventions are in effect. Recall the natural numbers N = { , , , . . . } . Let F denote a field.Every vector space and tensor product mentioned is over F . Every algebra mentioned isassociative, over F , and has a multiplicative identity. Fix a nonzero q ∈ F that is not a rootof unity. Recall the notation[ n ] q = q n − q − n q − q − n ∈ N . For elements
X, Y in any algebra, define their commutator and q -commutator by[ X, Y ] = XY − Y X, [ X, Y ] q = qXY − q − Y X.
Note that [ X, [ X, [ X, Y ] q ] q − ] = X Y − [3] q X Y X + [3] q XY X − Y X . Definition 2.1. (See [21, Corollary 3.2.6].) Define the algebra U + q by generators W , W and relations[ W , [ W , [ W , W ] q ] q − ] = 0 , [ W , [ W , [ W , W ] q ] q − ] = 0 . (1)We call U + q the positive part of U q ( b sl ). The generators W , W are called standard . Therelations (1) are called the q -Serre relations .We will use the following concept. Definition 2.2. (See [15, 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 .In [15, p. 299] Damiani obtains a PBW basis for U + q that involves some elements { E nδ + α } ∞ n =0 , { E nδ + α } ∞ n =0 , { E nδ } ∞ n =1 . (2)4hese elements are defined recursively as follows: E α = W , E α = W , E δ = q − W W − W W (3)and for n ≥ E nδ + α = [ E δ , E ( n − δ + α ] q + q − , E nδ + α = [ E ( n − δ + α , E δ ] q + q − , (4) E nδ = q − E ( n − δ + α W − W E ( n − δ + α . (5) Proposition 2.3. (See [15, p. 308].)
A PBW basis for U + q is obtained by the elements (2) in the linear order E α < E δ + α < E δ + α < · · · < E δ < E δ < E δ < · · · < E δ + α < E δ + α < E α . The elements (2) satisfy many relations [15]. We mention a few for later use.
Lemma 2.4.
The following hold in U + q : (i) (See [15, p. 307].) For positive i, j ∈ N , E iδ E jδ = E jδ E iδ . (6)(ii) (See [15, p. 307].) For i, j ∈ N , [ E iδ + α , E jδ + α ] q = − qE ( i + j +1) δ . (7)(iii) (See [15, p. 300].) For i ∈ N , [ W , E iδ + α ] q q − q − = i X ℓ =0 E ℓδ + α E ( i − ℓ ) δ + α , (8)[ E iδ + α , W ] q q − q − = i X ℓ =0 E ℓδ + α E ( i − ℓ ) δ + α . (9)5 An extension of U + q In this section we introduce the algebra U . In Section 6 we will show that U coincides withthe alternating central extension U + q of U + q . Definition 3.1.
Define the algebra U by generators W , W , { ˜ G k +1 } k ∈ N and relations(i) [ W , [ W , [ W , W ] q ] q − ] = 0,(ii) [ W , [ W , [ W , W ] q ] q − ] = 0,(iii) [ ˜ G , W ] = q [[ W , W ] q , W ] q − q − , (iv) [ W , ˜ G ] = q [ W , [ W , W ] q ] q − q − , (v) for k ≥
1, [ ˜ G k +1 , W ] = [[[ ˜ G k , W ] q , W ] q , W ](1 − q − )( q − q − ) , (vi) for k ≥
1, [ W , ˜ G k +1 ] = [ W , [ W , [ W , ˜ G k ] q ] q ](1 − q − )( q − q − ) , (vii) for k, ℓ ∈ N , [ ˜ G k +1 , ˜ G ℓ +1 ] = 0 . For notational convenience define ˜ G = 1. Note 3.2.
Referring to Definition 3.1, the relation (iii) (resp. (iv)) is obtained from (v)(resp. (vi)) by setting k = 0. Lemma 3.3.
There exists a unique algebra homomorphism ♭ : U + q → U that sends W W and W W .Proof. Compare Definitions 2.1, 3.1.In Corollary 6.7 we will show that ♭ is injective. Let h W , W i denote the subalgebra of U generated by W , W . Of course h W , W i is the ♭ -image of U + q . For the elements (2) of U + q ,the same notation will be used for their ♭ -images in h W , W i .6 Augmenting the generating set for U Some of the relations in Definition 3.1 are nonlinear. Our next goal is to linearize the relationsby adding more generators.
Definition 4.1.
We define some elements in U as follows. For k ∈ N , W − k = [ ˜ G k , W ] q q − q − , (10) W k +1 = [ W , ˜ G k ] q q − q − , (11) G k +1 = ˜ G k +1 + [ W , W − k ]1 − q − . (12)For notational convenience define G = 1. Lemma 4.2.
For k ∈ N the following hold in U : ˜ G k W = q − W ˜ G k + (1 − q − ) W − k , ˜ G k W = q W ˜ G k + (1 − q ) W k +1 . Proof.
These are reformulations of (10) and (11).The following is a generating set for U : { W − k } k ∈ N , { W k +1 } k ∈ N , { G k +1 } k ∈ N , { ˜ G k +1 } k ∈ N . (13)The elements of this set will be called alternating . We seek a presentation of U , that has theabove generating set and all relations linear. We will obtain this presentation in Theorem6.2.Next we obtain some formulas that will help us prove Theorem 6.2. We will show that for n ∈ N , W n +1 = n X k =0 E kδ + α ˜ G n − k ( − k q k ( q − q − ) k , (14) W − n = n X k =0 E kδ + α ˜ G n − k ( − k q k ( q − q − ) k . (15)We will prove (14), (15) by induction on n . Note that (14), (15) hold for n = 0, since W = E α and W = E α . We will give the main induction argument after a few lemmas.For the rest of this section k and ℓ are understood to be in N . Lemma 4.3.
Pick n ∈ N , and assume that (14) , (15) hold for n, n − , . . . , , . Then [ W , W n +1 ] = [ W − n , W ] . (16)7 roof. The commutator [ W , W n +1 ] is equal to W W n +1 − W n +1 W = n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k − n X k =0 E kδ + α ˜ G n − k W ( − k q k ( q − q − ) k = n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k − n X k =0 E kδ + α (cid:0) q − W ˜ G n − k + (1 − q − ) W k − n (cid:1) ( − k q k ( q − q − ) k = n X k =0 (cid:0) W E kδ + α − q − E kδ + α W (cid:1) ˜ G n − k ( − k q k ( q − q − ) k − n X k =0 E kδ + α W k − n ( − k q k − ( q − q − ) k − = − n X k =0 E ( k +1) δ ˜ G n − k ( − k q k ( q − q − ) k − n X k =0 E kδ + α W k − n ( − k q k − ( q − q − ) k − = − n X k =0 E ( k +1) δ ˜ G n − k ( − k q k ( q − q − ) k − n X k =0 E kδ + α ( − k q k − ( q − q − ) k − n − k X ℓ =0 E ℓδ + α ˜ G n − k − ℓ ( − ℓ q ℓ ( q − q − ) ℓ = − n X p =0 E ( p +1) δ ˜ G n − p ( − p q p ( q − q − ) p − n X p =0 X k + ℓ = p q ℓ E kδ + α E ℓδ + α ! ˜ G n − p ( − p q p − ( q − q − ) p − . The commutator [ W − n , W ] is equal to W − n W − W W − n = n X k =0 E kδ + α ˜ G n − k W ( − k q k ( q − q − ) k − n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k = n X k =0 E kδ + α (cid:0) q W ˜ G n − k + (1 − q ) W n − k +1 (cid:1) ( − k q k ( q − q − ) k − n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k = n X k =0 (cid:0) q E kδ + α W − W E kδ + α (cid:1) ˜ G n − k ( − k q k ( q − q − ) k − n X k =0 E kδ + α W n − k +1 ( − k q k +1 ( q − q − ) k − = − n X k =0 E ( k +1) δ ˜ G n − k ( − k q k +2 ( q − q − ) k − n X k =0 E kδ + α W n − k +1 ( − k q k +1 ( q − q − ) k − = − n X k =0 E ( k +1) δ ˜ G n − k ( − k q k +2 ( q − q − ) k − n X k =0 E kδ + α ( − k q k +1 ( q − q − ) k − n − k X ℓ =0 E ℓδ + α ˜ G n − k − ℓ ( − ℓ q ℓ ( q − q − ) ℓ = − n X p =0 E ( p +1) δ ˜ G n − p ( − p q p +2 ( q − q − ) p − n X p =0 X k + ℓ = p q k E kδ + α E ℓδ + α ! ˜ G n − p ( − p q p +1 ( q − q − ) p − . By these comments [ W − n , W ] − [ W , W n +1 ] = n X p =0 C p ˜ G n − p ( − p q p ( q − q − ) p , ≤ p ≤ n , C p = E ( p +1) δ + q − ( q − q − ) X k + ℓ = p q ℓ E kδ + α E ℓδ + α − q p +2 E ( p +1) δ − q ( q − q − ) X k + ℓ = p q k E kδ + α E ℓδ + α = (1 − q p +2 ) E ( p +1) δ − (1 − q ) X k + ℓ = p q ℓ (cid:0) q − E kδ + α E ℓδ + α − E ℓδ + α E kδ + α (cid:1) = (1 − q p +2 ) E ( p +1) δ − (1 − q ) X k + ℓ = p q ℓ E ( p +1) δ = (cid:18) − q p +2 − (1 − q ) p X ℓ =0 q ℓ (cid:19) E ( p +1) δ = 0 . The result follows.
Lemma 4.4.
Pick n ∈ N , and assume that (14) , (15) hold for n, n − , . . . , , . Then [ ˜ G n , E δ ] = 0 . (17) Proof.
Using Lemma 4.3, 0 = ( q − q − ) (cid:0) [ W − n , W ] − [ W , W n +1 ] (cid:1) = [[ ˜ G n , W ] q , W ] − [ W , [ W , ˜ G n ] q ]= [ ˜ G n , [ W , W ] q ]= − q [ ˜ G n , E δ ] . Lemma 4.5.
Pick n ∈ N , and assume that (14) , (15) hold for n, n − , . . . , , . Then [ W − n , W ] = 0 . (18)9 roof. The commutator [ W − n , W ] is equal to W − n W − W W − n = n X k =0 E kδ + α ˜ G n − k W ( − k q k ( q − q − ) k − n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k = n X k =0 E kδ + α (cid:0) q − W ˜ G n − k + (1 − q − ) W k − n (cid:1) ( − k q k ( q − q − ) k − n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k = n X k =0 [ W , E kδ + α ] q ˜ G n − k ( − k − q k − ( q − q − ) k + n X k =0 E kδ + α W k − n ( − k q k − ( q − q − ) k − = n X k =0 [ W , E kδ + α ] q ˜ G n − k ( − k − q k − ( q − q − ) k + n X k =0 E kδ + α ( − k q k − ( q − q − ) k − n − k X ℓ =0 E ℓδ + α ˜ G n − k − ℓ ( − ℓ q ℓ ( q − q − ) ℓ = n X p =0 [ W , E pδ + α ] q ˜ G n − p ( − p − q p − ( q − q − ) p + n X p =0 X k + ℓ = p E kδ + α E ℓδ + α ! ˜ G n − p ( − p q p − ( q − q − ) p − . By these comments [ W − n , W ] = n X p =0 S p ˜ G n − p ( − p − q p − ( q − q − ) p − where S p = [ W , E pδ + α ] q q − q − − X k + ℓ = p E kδ + α E ℓδ + α (0 ≤ p ≤ n ) . By (8) we have S p = 0 for 0 ≤ p ≤ n . The result follows. Lemma 4.6.
Pick n ∈ N , and assume that (14) , (15) hold for n, n − , . . . , , . Then [ W n +1 , W ] = 0 . (19) Proof.
The commutator [ W n +1 , W ] is equal to W n +1 W − W W n +1 = n X k =0 E kδ + α ˜ G n − k W ( − k q k ( q − q − ) k − n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k = n X k =0 E kδ + α (cid:0) q W ˜ G n − k + (1 − q ) W n − k +1 (cid:1) ( − k q k ( q − q − ) k − n X k =0 W E kδ + α ˜ G n − k ( − k q k ( q − q − ) k = n X k =0 [ E kδ + α , W ] q ˜ G n − k ( − k q k +1 ( q − q − ) k − n X k =0 E kδ + α W n − k +1 ( − k q k +1 ( q − q − ) k − = n X k =0 [ E kδ + α , W ] q ˜ G n − k ( − k q k +1 ( q − q − ) k − n X k =0 E kδ + α ( − k q k +1 ( q − q − ) k − n − k X ℓ =0 E ℓδ + α ˜ G n − k − ℓ ( − ℓ q ℓ ( q − q − ) ℓ = n X p =0 [ E pδ + α , W ] q ˜ G n − p ( − p q p +1 ( q − q − ) p − n X p =0 X k + ℓ = p E kδ + α E ℓδ + α ! ˜ G n − p ( − p q p +1 ( q − q − ) p − .
10y these comments [ W n +1 , W ] = n X p =0 T p ˜ G n − p ( − p q p +1 ( q − q − ) p − where T p = [ E pδ + α , W ] q q − q − − X k + ℓ = p E kδ + α E ℓδ + α (0 ≤ p ≤ n ) . By (9) we have T p = 0 for 0 ≤ p ≤ n . The result follows. Proposition 4.7.
The equations (14) , (15) hold for n ∈ N .Proof. The proof is by induction on n . We assume that (14), (15) hold for n, n − , . . . , , , and show that (14), (15) hold for n + 1. Concerning (14), W n +2 = qW ˜ G n +1 − q − ˜ G n +1 W q − q − by (11)= W ˜ G n +1 − q − [ ˜ G n +1 , W ] q − q − = W ˜ G n +1 − [[[ ˜ G n , W ] q , W ] q , W ]( q − q − ) ( q − q − ) by Definition 3.1(v)= W ˜ G n +1 − [[ W − n , W ] q , W ]( q − q − )( q − q − ) by (10)= W ˜ G n +1 − [[ W − n , W ] , W ] q ( q − q − )( q − q − )= W ˜ G n +1 − [[ W , W n +1 ] , W ] q ( q − q − )( q − q − ) by Lemma 4.3= W ˜ G n +1 − [[ W , W ] q , W n +1 ]( q − q − )( q − q − ) by Lemma 4.6= W ˜ G n +1 + q [ E δ , W n +1 ]( q − q − )( q − q − ) by (3)= W ˜ G n +1 + q n X k =0 [ E δ , E kδ + α ˜ G n − k ]( − k q k ( q − q − ) k +1 ( q − q − ) by (14) and induction= W ˜ G n +1 + q n X k =0 [ E δ , E kδ + α ] ˜ G n − k ( − k q k ( q − q − ) k +1 ( q − q − ) by Lemma 4.4= W ˜ G n +1 + n X k =0 E ( k +1) δ + α ˜ G n − k ( − k +1 q k +1 ( q − q − ) k +2 by (4)= E α ˜ G n +1 + n +1 X k =1 E kδ + α ˜ G n +1 − k ( − k q k ( q − q − ) k = n +1 X k =0 E kδ + α ˜ G n +1 − k ( − k q k ( q − q − ) k .
11e have shown that (14) holds for n + 1. Concerning (15), W − n − = q ˜ G n +1 W − q − W ˜ G n +1 q − q − by (10)= W ˜ G n +1 − q [ W , ˜ G n +1 ] q − q − = W ˜ G n +1 − q [ W , [ W , [ W , ˜ G n ] q ] q ]( q − q − ) ( q − q − ) by Definition 3.1(vi)= W ˜ G n +1 − q [ W , [ W , W n +1 ] q ]( q − q − )( q − q − ) by (11)= W ˜ G n +1 − q [ W , [ W , W n +1 ]] q ( q − q − )( q − q − )= W ˜ G n +1 − q [ W , [ W − n , W ]] q ( q − q − )( q − q − ) by Lemma 4.3= W ˜ G n +1 − q [ W − n , [ W , W ] q ]( q − q − )( q − q − ) by Lemma 4.5= W ˜ G n +1 + q [ W − n , E δ ]( q − q − )( q − q − ) by (3)= W ˜ G n +1 + q n X k =0 [ E kδ + α ˜ G n − k , E δ ]( − k q k ( q − q − ) k +1 ( q − q − ) by (15) and induction= W ˜ G n +1 + q n X k =0 [ E kδ + α , E δ ] ˜ G n − k ( − k q k ( q − q − ) k +1 ( q − q − ) by Lemma 4.4= W ˜ G n +1 + n X k =0 E ( k +1) δ + α ˜ G n − k ( − k +1 q k +3 ( q − q − ) k +2 by (4)= E α ˜ G n +1 + n +1 X k =1 E kδ + α ˜ G n +1 − k ( − k q k ( q − q − ) k = n +1 X k =0 E kδ + α ˜ G n +1 − k ( − k q k ( q − q − ) k . We have shown that (15) holds for n + 1. Lemma 4.8.
For n ∈ N , [ W , W n +1 ] = [ W − n , W ] , [ ˜ G n , E δ ] = 0 , [ W − n , W ] = 0 , [ W n +1 , W ] = 0 . Proof.
By Lemmas 4.3–4.6 and Proposition 4.7.
Lemma 4.9.
For k ∈ N , (i) [ G k +1 , W ] q = [ W , ˜ G k +1 ] q ; W , G k +1 ] q = [ ˜ G k +1 , W ] q .Proof. (i) We have[ G k +1 , W ] q − [ W , ˜ G k +1 ] q = (cid:20) ˜ G k +1 + [ W , W − k ]1 − q − , W (cid:21) q − [ W , ˜ G k +1 ] q = ( q + q − )[ ˜ G k +1 , W ] − [[ W − k , W ] , W ] q − q − = ( q + q − )[ ˜ G k +1 , W ] − [[ W − k , W ] q , W ]1 − q − = ( q + q − )[ ˜ G k +1 , W ] − [[[ ˜ G k , W ] q , W ] q , W ](1 − q − )( q − q − )= 0 . (ii) We have[ W , G k +1 ] q − [ ˜ G k +1 , W ] q = (cid:20) W , ˜ G k +1 + [ W k +1 , W ]1 − q − (cid:21) q − [ ˜ G k +1 , W ] q = ( q + q − )[ W , ˜ G k +1 ] − [ W , [ W , W k +1 ]] q − q − = ( q + q − )[ W , ˜ G k +1 ] − [ W , [ W , W k +1 ] q ]1 − q − = ( q + q − )[ W , ˜ G k +1 ] − [ W , [ W , [ W , ˜ G k ] q ] q ](1 − q − )( q − q − )= 0 . The alternating generators of U are displayed in (13). In the previous section we describedhow these generators are related to W and W . Our next goal is to describe how thealternating generators are related to each other. It is convenient to use generating functions. Definition 5.1.
We define some generating functions in an indeterminate t . Referring to(13), G ( t ) = X n ∈ N G n t n , ˜ G ( t ) = X n ∈ N ˜ G n t n ,W − ( t ) = X n ∈ N W − n t n , W + ( t ) = X n ∈ N W n +1 t n . emma 5.2. We have [ W , G ( t )] q q − q − = W − ( t ) , [ ˜ G ( t ) , W ] q q − q − = W − ( t ) , [ W , W − ( t )] = 0 , [ W , W + ( t )]1 − q − = t − ( ˜ G ( t ) − G ( t )) and [ G ( t ) , W ] q q − q − = W + ( t ) , [ W , ˜ G ( t )] q q − q − = W + ( t ) , [ W , W + ( t )] = 0 , [ W , W − ( t )]1 − q − = t − ( G ( t ) − ˜ G ( t )) . Proof.
Use Definition 4.1 and Lemmas 4.8, 4.9.For the rest of this section, let s denote an indeterminate that commutes with t . Lemma 5.3.
We have [ 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 and also [ W − ( s ) , G ( t )] q = [ W − ( t ) , G ( s )] q , [ G ( s ) , W + ( t )] q = [ G ( t ) , W + ( s )] q , [ ˜ G ( s ) , W − ( t )] q = [ ˜ G ( t ) , W − ( s )] q , [ W + ( s ) , ˜ G ( t )] q = [ W + ( t ) , ˜ G ( s )] q ,t − [ G ( s ) , ˜ G ( t )] − s − [ G ( t ) , ˜ G ( s )] = q [ W − ( t ) , W + ( s )] q − q [ W − ( s ) , W + ( t )] q ,t − [ ˜ G ( s ) , G ( t )] − s − [ ˜ G ( t ) , G ( s )] = q [ W + ( t ) , W − ( s )] q − q [ W + ( s ) , W − ( t )] q , [ G ( s ) , ˜ G ( t )] q − [ G ( t ) , ˜ G ( s )] q = qt [ W − ( t ) , W + ( s )] − qs [ W − ( s ) , W + ( t )] , [ ˜ G ( s ) , G ( t )] q − [ ˜ G ( t ) , G ( s )] q = qt [ W + ( t ) , W − ( s )] − qs [ W + ( s ) , W − ( t )] . roof. Define A ( s, t ) = [ W − ( s ) , W − ( t )] ,B ( s, t ) = [ W + ( s ) , W + ( t )] ,C ( s, t ) = [ W − ( s ) , W + ( t )] + [ W + ( s ) , W − ( t )] ,D ( s, t ) = s [ W − ( s ) , G ( t )] + t [ G ( s ) , W − ( t )] ,E ( s, t ) = s [ W − ( s ) , ˜ G ( t )] + t [ ˜ G ( s ) , W − ( t )] ,F ( s, t ) = s [ W + ( s ) , G ( t )] + t [ G ( s ) , W + ( t )] ,G ( s, t ) = s [ W + ( s ) , ˜ G ( t )] + t [ ˜ G ( s ) , W + ( t )] ,H ( s, t ) = [ G ( s ) , G ( t )] ,I ( s, t ) = [ ˜ G ( s ) , ˜ G ( t )] ,J ( s, t ) = [ ˜ G ( s ) , G ( t )] + [ G ( s ) , ˜ G ( t )]and also K ( s, t ) = [ W − ( s ) , G ( t )] q − [ W − ( t ) , G ( s )] q ,L ( s, t ) = [ G ( s ) , W + ( t )] q − [ G ( t ) , W + ( s )] q ,M ( s, t ) = [ ˜ G ( s ) , W − ( t )] q − [ ˜ G ( t ) , W − ( s )] q ,N ( s, t ) = [ W + ( s ) , ˜ G ( t )] q − [ W + ( t ) , ˜ G ( s )] q ,P ( s, t ) = t − [ G ( s ) , ˜ G ( t )] − s − [ G ( t ) , ˜ G ( s )] − q [ W − ( t ) , W + ( s )] q + q [ W − ( s ) , W + ( t )] q ,Q ( s, t ) = t − [ ˜ G ( s ) , G ( t )] − s − [ ˜ G ( t ) , G ( s )] − q [ W + ( t ) , W − ( s )] q + q [ W + ( s ) , W − ( t )] q ,R ( s, t ) = [ G ( s ) , ˜ G ( t )] q − [ G ( t ) , ˜ G ( s )] q − qt [ W − ( t ) , W + ( s )] + qs [ W − ( s ) , W + ( t )] ,S ( s, t ) = [ ˜ G ( s ) , G ( t )] q − [ ˜ G ( t ) , G ( s )] q − qt [ W + ( t ) , W − ( s )] + qs [ W + ( s ) , W − ( t )] . We will show that each of A ( s, t ) , B ( s, t ) , . . . , S ( s, t ) is zero. By linear algebra, C ( s, t ) = ( q + q − )( P ( s, t ) + Q ( s, t )) − ( s − + t − )( R ( s, t ) + S ( s, t ))( q − s − t )( q − st − ) q − , (20) J ( s, t ) = ( q + q − )( R ( s, t ) + S ( s, t )) − ( s + t )( P ( s, t ) + Q ( s, t ))( q − s − t )( q − st − ) q − . (21)Using Lemma 5.2 we routinely obtain[ W , A ( s, t )] = 0 , [ W , B ( s, t )]1 − q − = G ( s, t ) − F ( s, t ) st , [ W , C ( s, t )]1 − q − = E ( s, t ) − D ( s, t ) st , [ W , D ( s, t )] q q − q − = ( s + t ) A ( s, t ) , [ E ( s, t ) , W ] q q − q − = ( s + t ) A ( s, t ) , [ W , F ( s, t )] q − q − = S ( s, t ) − ( q + q − ) H ( s, t ) , [ G ( s, t ) , W ] q − q − = S ( s, t ) − ( q + q − ) I ( s, t ) , [ W , H ( s, t )] q q − q − = K ( s, t ) , [ I ( s, t ) , W ] q q − q − = M ( s, t ) , [ W , J ( s, t )] q − q − = M ( s, t ) − K ( s, t )15nd [ W , K ( s, t )] q q − q − = A ( s, t ) , [ W , L ( s, t )] q q − q − = P ( s, t ) − ( s − + t − ) H ( s, t ) , [ M ( s, t ) , W ] q q − q − = A ( s, t ) , [ N ( s, t ) , W ] q q − q − = Q ( s, t ) − ( s − + t − ) I ( s, t ) , [ P ( s, t ) , W ] q − q − = ( s − + t − ) K ( s, t ) − ( q + q − ) s − t − E ( s, t ) , [ W , Q ( s, t )] q − q − = ( s − + t − ) M ( s, t ) − ( q + q − ) s − t − D ( s, t ) , [ W , R ( s, t )] q − q − = ( s − + t − )( E ( s, t ) − D ( s, t )) , [ W , S ( s, t )] q − q − = M ( s, t ) − K ( s, t )and [ W , A ( s, t )]1 − q − = D ( s, t ) − E ( s, t ) st , [ W , B ( s, t )] = 0 , [ W , C ( s, t )]1 − q − = F ( s, t ) − G ( s, t ) st , [ D ( s, t ) , W ] q − q − = R ( s, t ) − ( q + q − ) H ( s, t ) , [ W , E ( s, t )] q − q − = R ( s, t ) − ( q + q − ) I ( s, t ) , [ F ( s, t ) , W ] q q − q − = ( s + t ) B ( s, t ) , [ W , G ( s, t )] q q − q − = ( s + t ) B ( s, t ) , [ H ( s, t ) , W ] q q − q − = L ( s, t ) , [ W , I ( s, t )] q q − q − = N ( s, t ) , [ W , J ( s, t )] q − q − = L ( s, t ) − N ( s, t )and [ K ( s, t ) , W ] q q − q − = P ( s, t ) − ( s − + t − ) H ( s, t ) , [ L ( s, t ) , W ] q q − q − = B ( s, t ) , [ W , M ( s, t )] q q − q − = Q ( s, t ) − ( s − + t − ) I ( s, t ) , [ W , N ( s, t )] q q − q − = B ( s, t ) , [ W , P ( s, t )] q − q − = ( s − + t − ) L ( s, t ) − ( q + q − ) s − t − G ( s, t ) , [ Q ( s, t ) , W ] q − q − = ( s − + t − ) N ( s, t ) − ( q + q − ) s − t − F ( s, t ) , [ W , R ( s, t )] q − q − = L ( s, t ) − N ( s, t ) , [ W , S ( s, t )] q − q − = ( s − + t − )( F ( s, t ) − G ( s, t )) . We just listed 38 relations, including (20), (21). These 38 relations are called canonical .Using the canonical relations, we can easily show that each of A ( s, t ) , B ( s, t ) , . . . , S ( s, t ) iszero. We will use induction with respect to the linear order I ( s, t ) , M ( s, t ) , N ( s, t ) , A ( s, t ) , B ( s, t ) , Q ( s, t ) , D ( s, t ) , E ( s, t ) , F ( s, t ) ,G ( s, t ) , R ( s, t ) , S ( s, t ) , H ( s, t ) , K ( s, t ) , L ( s, t ) , P ( s, t ) , C ( s, t ) , J ( s, t ) . I ( s, t ), there exists a canonical relation thatexpresses the given element in terms of the previous elements in the linear order. So thegiven element is zero, provided that every previous element is zero. Note that I ( s, t ) = 0 byDefinition 3.1(vii). By these comments and induction, every element in the linear order iszero. We have shown that each of A ( s, t ), B ( s, t ) , . . . , S ( s, t ) is zero. In this section we present our main results, which are Theorem 6.2 and Corollary 6.5. Recallthe alternating generators (13) for U . Lemma 6.1.
The following relations hold in U . For k, ℓ ∈ N we have [ W , W k +1 ] = [ W − k , W ] = (1 − q − )( ˜ G k +1 − G k +1 ) , (22)[ W , G k +1 ] q = [ ˜ G k +1 , W ] q = ( q − q − ) W − k − , (23)[ G k +1 , W ] q = [ W , ˜ G k +1 ] q = ( q − q − ) W k +2 , (24)[ W − k , W − ℓ ] = 0 , [ W k +1 , W ℓ +1 ] = 0 , (25)[ W − k , W ℓ +1 ] + [ W k +1 , W − ℓ ] = 0 , (26)[ W − k , G ℓ +1 ] + [ G k +1 , W − ℓ ] = 0 , (27)[ W − k , ˜ G ℓ +1 ] + [ ˜ G k +1 , W − ℓ ] = 0 , (28)[ W k +1 , G ℓ +1 ] + [ G k +1 , W ℓ +1 ] = 0 , (29)[ W k +1 , ˜ G ℓ +1 ] + [ ˜ G k +1 , W ℓ +1 ] = 0 , (30)[ G k +1 , G ℓ +1 ] = 0 , [ ˜ G k +1 , ˜ G ℓ +1 ] = 0 , (31)[ ˜ G k +1 , G ℓ +1 ] + [ G k +1 , ˜ G ℓ +1 ] = 0 (32) and also [ W − k , G ℓ ] q = [ W − ℓ , G k ] q , [ G k , W ℓ +1 ] q = [ G ℓ , W k +1 ] q , (33)[ ˜ G k , W − ℓ ] q = [ ˜ G ℓ , W − k ] q , [ W ℓ +1 , ˜ G k ] q = [ W k +1 , ˜ G ℓ ] q , (34)[ G k , ˜ G ℓ +1 ] − [ G ℓ , ˜ G k +1 ] = q [ W − ℓ , W k +1 ] q − q [ W − k , W ℓ +1 ] q , (35)[ ˜ G k , G ℓ +1 ] − [ ˜ G ℓ , G k +1 ] = q [ W ℓ +1 , W − k ] q − q [ W k +1 , W − ℓ ] q , (36)[ G k +1 , ˜ G ℓ +1 ] q − [ G ℓ +1 , ˜ G k +1 ] q = q [ W − ℓ , W k +2 ] − q [ W − k , W ℓ +2 ] , (37)[ ˜ G k +1 , G ℓ +1 ] q − [ ˜ G ℓ +1 , G k +1 ] q = q [ W ℓ +1 , W − k − ] − q [ W k +1 , W − ℓ − ] . (38) Proof.
The relations (22)–(24) are from Definition 4.1 and Lemmas 4.8, 4.9. The relations(25)–(38) follow from Definition 5.1 and Lemma 5.3.
Theorem 6.2.
The algebra U has a presentation by generators { W − k } k ∈ N , { W k +1 } k ∈ N , { G k +1 } k ∈ N , { ˜ G k +1 } k ∈ N and the relations in Lemma 6.1. roof. It suffices to show that the relations in Definition 3.1 are implied by the relations inLemma 6.1. The relation (iii) in Definition 3.1 is obtained from the equation on the left in(24) at k = 0, by eliminating G using [ W , W ] = (1 − q − )( ˜ G − G ). The relation (iv)in Definition 3.1 is obtained from the equation on the left in (23) at k = 0, by eliminating G using [ W , W ] = (1 − q − )( ˜ G − G ). For k ≥ G k +1 using [ W − k , W ] =(1 − q − )( ˜ G k +1 − G k +1 ) and evaluating the result using [ ˜ G k , W ] q = ( q − q − ) W − k . For k ≥ G k +1 using [ W , W k +1 ] = (1 − q − )( ˜ G k +1 − G k +1 ) and evaluating the result using[ W , ˜ G k ] q = ( q − q − ) W k +1 . The relation (vii) in Definition 3.1 is from (31). The relation(i) in Definition 3.1 is obtained from [ W , W − ] = 0, by eliminating W − using [ ˜ G , W ] q =( q − q − ) W − and evaluating the result using Definition 3.1(iv). The relation (ii) in Definition3.1 is obtained from [ W , W ] = 0, by eliminating W using [ W , ˜ G ] q = ( q − q − ) W andevaluating the result using Definition 3.1(iii).It is apparent from the proof of Theorem 6.2 that the relations in Lemma 6.1 are redundantin the following sense. Corollary 6.3.
The relations in Lemma 6.1 are implied by the relations listed in (i)–(iii)below:(i) (22)–(24);(ii) (25) with k = 0 and ℓ = 1;(iii) the relations on the right in (31). Proof.
By Lemma 6.1 the relations (22)–(38) are implied by the relations in Definitions 3.1,4.1. The relations listed in (i)–(iii) are used in the proof of Theorem 6.2 to obtain therelations in Definition 3.1. The relations listed in (i) imply the relations in Definition 4.1.The result follows.The relations in Lemma 6.1 first appeared in [26, Propositions 5.7, 5.10, 5.11]. It wasobserved in [4, Propositions 3.1, 3.2] and [5, Remark 2.5] that the relations (22)–(32) implythe relations (33)–(38). This observation motivated the following definition.
Definition 6.4. (See [27, Definition 3.1].) Define the algebra U + q by generators { W − k } k ∈ N , { W k +1 } k ∈ N , { G k +1 } k ∈ N , { ˜ G k +1 } k ∈ N and the relations (22)–(32). The algebra U + q is called the alternating central extension of U + q . Corollary 6.5.
We have U = U + q .Proof. By Theorem 6.2, Corollary 6.3, and Definition 6.4.
Definition 6.6.
By the compact presentation of U + q we mean the presentation given inDefinition 3.1. By the expanded presentation of U + q we mean the presentation given inTheorem 6.2. Corollary 6.7.
The map ♭ from Lemma 3.3 is injective.Proof. By Corollary 6.5 and [27, Proposition 6.4].18
The subalgebra of U + q generated by { ˜ G k +1 } k ∈ N Let ˜ G denote the subalgebra of U + q generated by { ˜ G k +1 } k ∈ N . In this section we describe ˜ G and its relationship to h W , W i .The following notation will be useful. Let z , z , . . . denote mutually commuting indetermi-nates. Let F [ z , z , . . . ] denote the algebra consisting of the polynomials in z , z , . . . thathave all coefficients in F . For notational convenience define z = 1. Lemma 7.1. (See [27, Lemma 3.5].)
There exists an algebra homomorphism U + q → F [ z , z , . . . ] that sends W − n , W n +1 , G n z n , ˜ G n z n for n ∈ N .Proof. By Theorem 6.2 and the nature of the relations in Lemma 6.1.
Corollary 7.2. (See [27, Theorem 10.2].)
The generators { ˜ G k +1 } k ∈ N of ˜ G are algebraicallyindependent.Proof. By Lemma 7.1 and since { z k +1 } k ∈ N are algebraically independent.The following result will help us describe how ˜ G is related to h W , W i . Lemma 7.3.
For n ∈ N , ˜ G n W = W ˜ G n + n X k =1 E kδ + α ˜ G n − k ( − k +1 q k +1 ( q − q − ) k − , (39)˜ G n W = W ˜ G n + n X k =1 E kδ + α ˜ G n − k ( − k q k − ( q − q − ) k − . (40) Proof.
To obtain (39), eliminate W n +1 from (14) using (11), and solve the resulting equationfor ˜ G n W . To obtain (40), eliminate W − n from (15) using (10), and solve the resultingequation for ˜ G n W .Shortly we will describe how ˜ G is related to h W , W i . This description involves the center Z of U + q . To prepare for this description, we have some comments about Z . In [27, Sections 5, 6]we introduced some algebraically independent elements Z , Z , . . . that generate the algebra Z . For notational convenience define Z = 1. Using { Z n } n ∈ N we obtain a basis for Z thatis described as follows. For n ∈ N , a partition of n is a sequence λ = { λ i } ∞ i =1 of naturalnumbers such that λ i ≥ λ i +1 for i ≥ n = P ∞ i =1 λ i . The set Λ n consists of the partitionsof n . Define Λ = ∪ n ∈ N Λ n . For λ ∈ Λ define Z λ = Q ∞ i =1 Z λ i . The elements { Z λ } λ ∈ Λ form abasis for the vector space Z . Next we describe a grading for Z . For n ∈ N let Z n denotethe subspace of Z with basis { Z λ } λ ∈ Λ n . For example Z = F
1. The sum Z = P n ∈ N Z n is direct. Moreover Z r Z s ⊆ Z r + s for r, s ∈ N . By these comments the subspaces {Z n } n ∈ N form a grading of Z . Note that Z n ∈ Z n for n ∈ N . Next we describe how Z is related to h W , W i . 19 emma 7.4. (See [27, Proposition 6.5].) The multiplication map h W , W i ⊗ Z → U + q w ⊗ z wz is an algebra isomorphism. For n ∈ N let U n denote the image of h W , W i ⊗ Z n under the multiplication map. Byconstruction the sum U + q = P n ∈ N U n is direct.In the next two lemmas we describe how ˜ G is related to Z . Lemma 7.5. (See [27, Lemmas 3.6, 5.9].)
For n ∈ N , ˜ G n ∈ n X k =0 h W , W i Z k , ˜ G n − Z n ∈ n − X k =0 h W , W i Z k . For λ ∈ Λ define ˜ G λ = Q ∞ i =1 ˜ G λ i . By Corollary 7.2 the elements { ˜ G λ } λ ∈ Λ form a basis forthe vector space ˜ G . Lemma 7.6.
For n ∈ N and λ ∈ Λ n , ˜ G λ ∈ n X k =0 U k , ˜ G λ − Z λ ∈ n − X k =0 U k . Proof.
By Lemma 7.5 and our comments above Lemma 7.4 about the grading of Z .Next we describe how ˜ G is related to h W , W i . Proposition 7.7.
The multiplication map h W , W i ⊗ ˜ G → U + q w ⊗ g wg is an isomorphism of vector spaces.Proof. The multiplication map is F -linear. The multiplication map is surjective by Lemma7.3 and since U + q is generated by W , W , ˜ G . We now show that the multiplicaton map isinjective. Consider a vector v ∈ h W , W i ⊗ ˜ G that is sent to zero by the multiplication map.We show that v = 0. Write v = P λ ∈ Λ a λ ⊗ ˜ G λ , where a λ ∈ h W , W i for λ ∈ Λ and a λ = 0for all but finitely many λ ∈ Λ. To show that v = 0, we must show that a λ = 0 for all λ ∈ Λ.Suppose that there exists λ ∈ Λ such that a λ = 0. Let C denote the set of natural numbers m such that Λ m contains a partition λ with a λ = 0. The set C is nonempty and finite. Let n denote the maximal element of C . By construction P λ ∈ Λ n a λ ⊗ Z λ is nonzero. By Lemma7.4, X λ ∈ Λ n a λ Z λ = 0 . (41)20y construction 0 = X λ ∈ Λ a λ ˜ G λ = n X k =0 X λ ∈ Λ k a λ ˜ G λ = X λ ∈ Λ n a λ ˜ G λ + n − X k =0 X λ ∈ Λ k a λ ˜ G λ . (42)Using (42), X λ ∈ Λ n a λ Z λ = X λ ∈ Λ n a λ ( Z λ − ˜ G λ ) − n − X k =0 X λ ∈ Λ k a λ ˜ G λ . (43)The left-hand side of (43) is contained in U n . By Lemma 7.6 the right-hand side of (43) iscontained in P n − k =0 U k . The subspaces U n and P n − k =0 U k have zero intersection because thesum P nk =0 U k is direct. This contradicts (41), so a λ = 0 for λ ∈ Λ. Consequently v = 0, asdesired. We have shown that the multiplication map is injective. By the above commentsthe multiplication map is an isomorphism of vector spaces. The author thanks Pascal Baseilhac for many conversations about U + q and its central exten-sion U + q . The author thanks Kazumasa Nomura for giving this paper a close reading andoffering many valuable comments. References [1] E. Bannai and T. Ito.
Algebraic Combinatorics, I. Association schemes.
Ben-jamin/Cummings, Menlo Park, CA, 1984.[2] P. Baseilhac. An integrable structure related with tridiagonal algebras.
Nuclear Phys.B
705 (2005) 605–619; arXiv:math-ph/0408025 .[3] P. Baseilhac. Deformed Dolan-Grady relations in quantum integrable models.
NuclearPhys. B
709 (2005) 491–521; arXiv:hep-th/0404149 .[4] P. Baseilhac. The positive part of U q ( b sl ) and tensor product representations; preprint .[5] P. Baseilhac. The alternating presentation of U q ( c gl ) from Freidel-Maillet algebras; arXiv:2011.01572 .[6] P. Baseilhac and S. Belliard. The half-infinite XXZ chain in Onsager’s approach. NuclearPhys. B
873 (2013) 550–584; arXiv:1211.6304 .[7] P. Baseilhac and S. Belliard. An attractive basis for the q -Onsager algebra; arXiv:1704.02950 .[8] P. Baseilhac and K. Koizumi. A new (in)finite dimensional algebra for quantum inte-grable models. Nuclear Phys. B
720 (2005) 325–347; arXiv:math-ph/0503036 .219] P. Baseilhac and K. Koizumi. A deformed analogue of Onsager’s symmetry in the
XXZ open spin chain.
J. Stat. Mech. Theory Exp. arXiv:hep-th/0507053 .[10] P. Baseilhac and K. Koizumi. Exact spectrum of the
XXZ open spin chain from the q -Onsager algebra representation theory. J. Stat. Mech. Theory Exp. arXiv:hep-th/0703106 .[11] P. Baseilhac and S. Kolb. Braid group action and root vectors for the q -Onsager algebra. Transform. Groups
25 (2020) 363–389; arXiv:1706.08747 .[12] P. Baseilhac and K. Shigechi. A new current algebra and the reflection equation.
Lett.Math. Phys.
92 (2010) 47–65; arXiv:0906.1482v2 .[13] A. E. Brouwer, A. Cohen, A. Neumaier.
Distance-Regular Graphs.
Springer-Verlag,Berlin, 1989.[14] V. Chari and A. Pressley. Quantum affine algebras.
Commun. Math. Phys.
142 (1991)261–283.[15] I. Damiani. A basis of type Poincare-Birkoff-Witt for the quantum algebra of b sl . J.Algebra
161 (1993) 291–310.[16] T. Ito and P. Terwilliger. Two non-nilpotent linear transformations that satisfy thecubic q -Serre relations. J. Algebra Appl. arXiv:math/0508398 .[17] T. Ito and P. Terwilliger. Distance-regular graphs and the q -tetrahedron algebra. Eu-ropean J. Combin.
30 (2009) 682–697; arXiv:math.CO/0608694 .[18] T. Ito, K. Tanabe, and P. Terwilliger. Some algebra related to P - and Q -polynomialassociation schemes, in: Codes and Association Schemes (Piscataway NJ, 1999) , Amer.Math. Soc., Providence RI, 2001, pp. 167–192; arXiv:math.CO/0406556 .[19] M. Jimbo and T. Miwa. Algebraic analysis of solvable lattice models. CBMS Re-gional Conference Series in Mathematics, 85. Published for the Conference Board ofthe Mathematical Sciences, Washington, DC; by the American Mathematical Society,Providence, RI, 1995.[20] W. Liu. The attenuated space poset A q ( M, N ). Linear Algebra Appl.
506 (2016) 244–273; arXiv:1605.00625 .[21] G. Lusztig.
Introduction to quantum groups . Progress in Mathematics, 110. Birkhauser,Boston, 1993.[22] M. Rosso. Groupes quantiques et alg`ebres de battage quantiques.
C. R. Acad. Sci.Paris
320 (1995) 145–148.[23] M. Rosso. Quantum groups and quantum shuffles.
Invent. Math
133 (1998) 399–416.2224] P. Terwilliger. The incidence algebra of a uniform poset, in
Coding Theory and DesignTheory, Part I , pp. 193–212. IMA Vol. Math. Appl. 20, Springer, New York, 1990.[25] P. Terwilliger. An action of the free product Z ⋆ Z ⋆ Z on the q -Onsager algebra andits current algebra. Nuclear Phys. B
936 (2018) 306–319; arXiv:1808.09901 .[26] P. Terwilliger. The alternating PBW basis for the positive part of U q ( b sl ). J. Math.Phys.
60 (2019) 071704; arXiv:1902.00721 .[27] P. Terwilliger. The alternating central extension for the positive part of U q ( b sl ). NuclearPhys. B
947 (2019) 114729; arXiv:1907.09872 .Paul TerwilligerDepartment of MathematicsUniversity of Wisconsin480 Lincoln DriveMadison, WI 53706-1388 USAemail: [email protected]@math.wisc.edu