A compact presentation for the alternating central extension of the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

This paper concerns the positive part $U^+_q$ of the quantum group $U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U^+_q$ has a presentation involving two generators that satisfy the cubic $q$-Serre relations. We recently introduced an algebra $\mathcal U^+_q$ called the alternating central extension of $U^+_q$. We presented $\mathcal U^+_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal U^+_q$ that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal U^+_q$.


Introduction
The algebra U q ( sl 2 ) is well known in representation theory [15] and statistical mechanics [21].This algebra has a subalgebra U + q called the positive part.The algebra U + q has a presentation involving 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.In the applications, either q is not a root of unity, or q is a root of unity with exponent large enough to not interfere with the rest of the application.To keep things simple, throughout the paper we will assume that q is not a root of unity.
Our first application has to do with tridiagonal pairs [20].A tridiagonal pair is roughly described as an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on the eigenspaces of the other one in a block-tridiagonal fashion [20, Definition 1.1].There is a type of tridiagonal pair said to be q-geometric [18,Definition 2.6]; for this type of tridiagonal pair the eigenvalues of each map form a q 2 -geometric progression.A finite-dimensional irreducible U + q -module on which the standard generators are not nilpotent, is essentially the same thing as a tridiagonal pair of q-geometric type [18,Theorem 2.7]; these U + q -modules are described in [18, Section 1].See [13], [25] for more background on tridiagonal pairs.Our next application has to do with distance-regular graphs [1], [14], [16].Consider a distance-regular graph Γ that has diameter d ≥ 3 and classical parameters (d, b, α, β) [14, p. 193] with b = q 2 and α = q 2 − 1.The condition on α implies that Γ is formally self-dual in the sense of [14, p. 49].Let A denote the adjacency matrix of Γ, and let A * denote the dual adjacency matrix with respect to any vertex of Γ [19, Section 7].Then by [19,Lemma 9.4], there exist complex numbers r, s, r * , s * with r, r * nonzero such that rA+sI, r * A * +s * I satisfy the q-Serre relations.As mentioned in [19,Example 8.4], the above parameter restriction is satisfied by the bilinear forms graph [14, p. 280], the alternating forms graph [14, p. 282], the Hermitean forms graph [14, p. 285], the quadratic forms graph [14, p. 290], the affine E 6 graph [14, p. 340], and the extended ternary Golay code graph [14, p. 359].
Our next application has to do with uniform posets [24], [28].Let GF(b) denote a finite field with b elements, and let N, M denote positive integers.Let H denote a vector space over GF(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) [22], [28,Example 3.1].By [28,Theorem 3.2] the poset A b (N, M) is uniform in the sense of [28,Definition 2.2].It is shown in [22,Lemma 3.3] that for A b (N, M) the raising matrix R and the lowering matrix L satisfy the q-Serre relations, provided that b = q 2 .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 with generators 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 1 v 2 • • • v n .The words in V form a basis for the vector space V .In [26,27] M. Rosso introduced an algebra structure on V , called the q-shuffle algebra.For letters u, v their q-shuffle product is [26,Theorem 13], in the q-shuffle algebra V the elements x, y satisfy the q-Serre relations.Consequently there exists an algebra homomorphism ♮ from U + q into the q-shuffle algebra V , that sends the standard generators of U + q to x, y.By [27,Theorem 15] the map ♮ is injective.Next we recall the alternating elements in U + q [30]. is called alternating whenever it is the ♮-preimage of an alternating word.For example, the standard generators of U + q are alternating because they are the ♮-preimages of the alternating words x, y.It is shown in [30,Lemma 5.12] that for each row in the above display, the corresponding alternating elements mutually commute.A naming scheme for alternating elements is introduced in [30, Definition 5.2].
Next we recall the alternating central extension of U + q [31].In [30] we displayed two types of relations among the alternating elements of U + q ; the first type is [30, Propositions 5.7, 5.10, 5.11] and the second type is [30,Propositions 6.3,8.1].The relations in [30,Proposition 5.11] are redundant; they follow from the relations in [30,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 [30,Proposition 6.3] are also redundant; they follow from the relations in [30,Propositions 5.7,5.10] as shown in the proof of [30,Proposition 6.3].By [30,Lemma 8.4] and the previous comments, the algebra U + q is presented by its alternating elements and the relations in [30,Propositions 5.7,5.10,8.1].For this presentation it is natural to ask what happens if the relations in [30,Proposition 8.1] are removed.To answer this question, in [31,Definition 3.1] we defined an algebra U + q by generators and relations in the following way.The generators, said to be alternating, are in bijection with the alternating elements of U + q .The relations are the ones in [30,Propositions 5.7,5.10].By construction there exists a surjective algebra homomorphism U + q → U + q that sends each alternating generator of U + q to the corresponding alternating element of U + q .In [31, Lemma 3.6, Theorem 5.17] we adjusted this homomorphism to get an algebra isomorphism are mutually commuting indeterminates.By [31,Theorem 10.2] the alternating generators form 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 the relations in [30,Propositions 5.7,5.10].This presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy.In this paper we obtain a presentation of U + q that involves a small subset of the original set of generators and a very manageable set of relations.This presentation is given in Definition 3.1 below; we call it the compact presentation of U + q .At first glance, it is not clear that the algebra presented in Definition 3.1 is equal to U + q .So we denote by U the algebra presented in Definition 3.1, and eventually 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 inspired by some recent developments in statistical mechanics, concerning the q-Onsager algebra O q .In [8] Baseilhac and Koizumi introduce a current algebra 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 and relations.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 relationship between A q and O q was conjectured in [7, Conjectures 1, 2] and [29,Conjectures 4.5,4.6,4.8],before being settled in [32,Theorems 9.14,10.2,10.3,10.4].The articles [2,3,[6][7][8][9][10][11][12] contain background information on O q and A q .
Earlier in this section, we indicated how U + q has applications to tridiagonal pairs, distanceregular graphs, and uniform posets.Possibly U + q appears in these applications, and this possibility should be investigated in the future.This paper is organized as follows.In Section 2 we review some facts about U + q .In Section 3, 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 these generators.In Sections 5, 6 we use these formulas and generating functions to show that the alternating generators for U satisfy the relations in [30,Propositions 5.7,5.10].Using this result, we prove that U = U + q .Theorem 6.2 and Corollary 6.5 are the main results of the paper.In Section 7 we describe some features of U + q that are illuminated by the presentation in Definition 3.1.Appendix A contains a list of relations involving the generating functions from Section 5.

The algebra U + q
We now begin our formal argument.For the rest of the paper, the following notational conventions are in effect.Recall the natural numbers N = {0, 1, 2, . ..}.Let F denote a field.Every vector space and tensor product mentioned is over F. Every algebra mentioned is associative, over F, and has a multiplicative identity.Fix a nonzero q ∈ F that is not a root of unity.Recall the notation For elements X, Y in any algebra, define their commutator and q-commutator by ) Define the algebra U + q by generators W 0 , W 1 and relations We call U + q the positive part of U q ( sl 2 ).The generators W 0 , W 1 are called standard.The relations (1) are called the q-Serre relations.
We will use the following concept.Definition 2.2.(See [17, p. 299].)Let A denote an algebra.A Poincaré-Birkhoff-Witt (or PBW) basis for A consists of a subset Ω ⊆ A and a linear order < on Ω such that the following is a basis for the vector space A: We interpret the empty product as the multiplicative identity in A.
In [17, p. 299] Damiani obtains a PBW basis for U + q that involves some elements These elements are defined recursively as follows: and for n ≥ 1, Proposition 2.3.(See [17, p. 308].)A PBW basis for U + q is obtained by the elements (2) in the linear order The elements (2) satisfy many relations [17].We mention a few for later use.
Lemma 2.4.(See [17, p. 300].)For i, j ∈ N with i > j the following hold in U + q .

An extension of U + q
In this section we introduce the algebra U.In Section 6 we will show that U coincides with the alternating central extension U + q of U + q .
In Corollary 6.7 we will show that ♭ is injective.Let W 0 , W 1 denote the subalgebra of U generated by For the elements (2) of U + q , the same notation will be used for their ♭-images in W 0 , W 1 .
4 Augmenting the generating set for U Some of the relations in Definition 3.1 are nonlinear.Our next goal is to linearize the relations by adding more generators.Definition 4.1.We define some elements in U as follows.For k ∈ N, For notational convenience define G 0 = 1.
Lemma 4.2.For k ∈ N the following hold in U: Proof.These are reformulations of ( 10) and (11).
The following is a generating set for U: The elements of this set will be called alternating.We seek a presentation of U, that has the above generating set and all relations linear.We will obtain this presentation in Theorem 6.2.
Next we obtain some formulas that will help us prove Theorem 6.2.We will show that for n ∈ N, We will prove ( 14), ( 15) by induction on n.Note that ( 14), ( 15) hold for n = 0, since W 1 = E α 1 and W 0 = E α 0 .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 4.8.The following relations hold in U.For n ∈ N, Proof.By Lemmas 4.3-4.6 and Proposition 4.7.
Lemma 4.9.The following relations hold in U.For k ∈ N,

Generating functions
The alternating generators of U are displayed in (13).In the previous section we described how these generators are related to W 0 and W 1 .Our next goal is to describe how the alternating 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), Lemma 5.2.For the algebra U, and 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.For the algebra U, and also Proof.We refer to the generating functions A(s, t), B(s, t), . . ., S(s, t) from Appendix A. The present lemma asserts that for the algebra U these generating functions are all zero.To verify this assertion, we refer to the canonical relations in Appendix A. 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).
It is apparent from the proof of Theorem 6.2 that the relations in Lemma 6.1 are redundant in the following sense.
Proof.By Lemma 6.1 the relations ( 20)-( 36 and the relations ( 20)- (30).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 in Definition 3.1.By the expanded presentation of U + q we mean the presentation given in Theorem 6. 7 The subalgebra of U + q generated by { Gk+1 } k∈N Let G denote the subalgebra of U + q generated by { Gk+1 } k∈N .In this section we describe G and its relationship to W 0 , W 1 .
Proof.By Theorem 6.2 and the nature of the relations in Lemma 6.1.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 W 0 , W 1 .
Shortly we will describe how G is related to W 0 , W 1 .This description involves the center Z of U + q .To prepare for this description, we have some comments about Z.In [31, Sections 5, 6] we introduced some algebraically independent elements Z 1 , Z 2 , . . .that generate the algebra Z.For notational convenience define Z 0 = 1.Using {Z n } n∈N we obtain a basis for Z that is described as follows.For n ∈ N, a partition of n is a sequence λ The elements {Z λ } λ∈Λ form a basis for the vector space Z. Next we describe a grading for Z.For n ∈ N let Z n denote the subspace of Z with basis {Z λ } λ∈Λn .For example Z 0 = F1.The sum Z = 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 W 0 , W 1 .Lemma 7.4.(See [31,Proposition 6.5].)The multiplication map For n ∈ N let U n denote the image of W 0 , W 1 ⊗ Z n under the multiplication map.By construction the sum U + q = n∈N U n is direct.In the next two lemmas we describe how G is related to Z.
For λ ∈ Λ define Gλ = ∞ i=1 Gλ i .By Corollary 7.2 the elements { Gλ } λ∈Λ form a basis for the vector space G. Lemma 7.6.For n ∈ N and λ ∈ Λ n , 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 W 0 , W 1 .
Proposition 7.7.The multiplication map W 0 , W 1 ⊗ 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 Lemma 7.3 and since U + q is generated by W 0 , W 1 , G. We now show that the multiplicaton map is injective.Consider a vector v ∈ W 0 , W 1 ⊗ G that is sent to zero by the multiplication map.We show that v = 0. Write v = λ∈Λ a λ ⊗ Gλ , where a λ ∈ W 0 , W 1 for λ ∈ Λ and a λ = 0 for 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 λ∈Λn a λ ⊗ Z λ is nonzero.By Lemma 7.4, λ∈Λn a λ Z λ = 0.

Paul Terwilliger Department of Mathematics
University of Wisconsin 480 Lincoln Drive Madison, WI 53706-1388 USA email: terwilli@math.wisc.edu