An infinite-dimensional q-module obtained from the q-shuffle algebra for affine sl 2

Let F denote a field, and pick a nonzero q ∈ F that is not a root of unity. Let Z4 = Z/4Z denote the cyclic group of order 4. Define a unital associative F-algebra q by generators {xi}i∈Z4 and relations qxixi+1 − q xi+1xi q − q−1 = 1, xixi+2 − [3]qx 2 ixi+2xi + [3]qxixi+2x 2 i − xi+2x 3 i = 0, where [3]q = (q 3 − q−3)/(q − q−1). Let V denote a q-module. A vector ξ ∈ V is called NIL whenever x1ξ = 0 and x3ξ = 0 and ξ 6= 0. The q-module V is called NIL whenever V is generated by a NIL vector. We show that up to isomorphism there exists a unique NIL q-module, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description an important role is played by the q-shuffle algebra for affine sl2.


Introduction
The algebra q was introduced in [29]. It is associative, infinite-dimensional, and noncommutative. It is defined by generators and relations. There are four generators, and it is natural to identify these with the edges of an oriented four-cycle. The relations are roughly described as follows. Each pair of adjacent edges satisfy a q-Weyl relation. Each pair of opposite edges satisfy the q-Serre relations associated with affine sl 2 ; these have degree 3 in one variable and degree 1 in the other variable. The cyclic group Z 4 = Z/4Z acts on the oriented four-cycle as a group of rotational symmetries, and this induces a Z 4 -action on q as a group of automorphisms.
In the theory of quantum groups, there is an algebra U + q called the positive part of U q sl 2 . The algebra U + q is defined by two generators, subject to the above q-Serre relations [20, Corollary 3.2.6]. The algebras q and U + q are related as follows. In the algebra q , each pair of opposite edges generate a subalgebra that is isomorphic to U + q [29,Proposition 5.5]. This gives two subalgebras of q that are isomorphic to U + q ; consider their tensor product. There is a map from this tensor product to q , given by multiplication in q . The map is an isomorphism of vector spaces [29,Proposition 5.5]. Thus the vector space q is isomorphic to U + q ⊗ U + q . Next we discuss how q is related to the q-Onsager algebra O q . The algebra O q originated in algebraic combinatorics [24,Lemma 5.4], [26] and statistical mechanics [1,Section 1], [2,Section 2]. Research on O q is presently active in both areas; see [10,15,16,17,25,27,28,29,30,31,32] and [1,2,3,4,5,6,7,8,9,11,12]. The algebra O q is defined by two generators, subject to the q-Dolan/Grady relations [29,Definition 4.1]. The q-Dolan/Grady relations resemble the above q-Serre relations, but are slightly more complicated. The algebras q and O q are related as follows. By [29,Proposition 5.6] there exists an injective algebra homomorphism O q → q that sends one O q -generator to a linear combination of two adjacent q -generators, and the other O q -generator to a linear combination of the remaining two q -generators. The only constraint on the four coefficients is that the first two are reciprocals and the last two are reciprocals. We just explained how q and O q are related. This relationship is our primary motivation for investigating q .
The finite-dimensional q -modules are investigated in [33,34]. By [34,Proposition 5.2], each q -generator is invertible on every nonzero finite-dimensional q -module. This result gets used in [34,Sections 8 and 9] to obtain some remarkable q-exponential formulas. In [33], the finite-dimensional irreducible q -modules are classified up to isomorphism. This classification is summarized as follows. There is a family of finite-dimensional irreducible q -modules, said to have type 1 [33,Definition 6.8]. Any finite-dimensional irreducible q -module can be normalized to have type 1, by twisting it via an appropriate automorphism of q [33, Note 6.9]. Let V denote a finite-dimensional irreducible q -module of type 1. In [33,Definition 8.6] V gets attached to a polynomial P V in one variable z that has constant coefficient 1; this P V is called the Drinfel'd polynomial of V . By [33,Proposition 1.4], the map V → P V induces a bijection between the following two sets: (i) the isomorphism classes of finite-dimensional irreducible q -modules of type 1; (ii) the polynomials in the variable z that have constant coefficient 1 and do not vanish at z = 1.
In the present paper, our topic is a set of infinite-dimensional q -modules, said to be NIL. A NIL q -module is generated by a vector that is sent to zero by a pair of opposite q -generators. We show that up to isomorphism there exists a unique NIL q -module, and it is irreducible and infinite-dimensional. We then describe this module from 16 points of view. In this description an important role is played by the q-shuffle algebra for affine sl 2 . This algebra was introduced by Rosso [21] and described further by Green [13].
We now summarize our results in more detail. Let F denote a field. All vector spaces discussed in this paper are over F. All algebras discussed in this paper are associative, over F, and have a multiplicative identity. Fix a nonzero q ∈ F that is not a root of unity. Define the algebra q by generators {x i } i∈Z 4 and relations where [3] The relations (1.1) and (1.2) are called the q-Weyl and q-Serre relations, respectively. Let V denote a q -module. A vector ξ ∈ V is called NIL whenever x 1 ξ = 0 and x 3 ξ = 0 and ξ = 0. The q -module V is called NIL whenever V is generated by a NIL vector.
In this paper we obtain the following results. Up to isomorphism, there exists a unique NIL q -module, which we denote by U. The q -module U is irreducible, and isomorphic to U + q as a vector space. Recall the natural numbers N = {0, 1, 2, . . .}. The q -module U has a unique sequence of subspaces {U n } n∈N such that (i) U 0 = 0; (ii) the sum U = n∈N U n is direct; (iii) for n ∈ N, where U −1 = 0. The sequence {U n } n∈N is described as follows. The subspace U 0 has dimension 1. The nonzero vectors in U 0 are precisely the NIL vectors in U, and each of these vectors generates U. Let ξ denote a NIL vector in U. Then for n ∈ N, the subspace U n is spanned by the vectors u 1 u 2 · · · u n ξ such that u i ∈ {x 0 , x 2 } for 1 ≤ i ≤ n.
We will state some more results after a few comments. Let V denote the free algebra on two generators A, B. For n ∈ N, a word of length n in V is a product v 1 v 2 · · · v n such that v i ∈ {A, B} for 1 ≤ i ≤ n. We interpret the word of length zero to be the multiplicative identity of V; this word is called trivial and denoted by 1. The standard basis for V consists of the words. There exists a symmetric bilinear form ( , ) : V × V → F with respect to which the standard basis is orthonormal. The algebra End(V) consists of the F-linear maps V → V with the following property: the matrix that represents the map with respect to the standard basis for V has finitely many nonzero entries in each row. We define an invertible K ∈ End(V) as follows. The map K is the automorphism of the free algebra V that sends A → q 2 A and We define four maps in End(V), denoted We have been discussing the free algebra V. There is another algebra structure on V, called the q-shuffle algebra [13,21,22]. We will follow the approach of [13], which is well suited to our purpose. The q-shuffle product will be denoted by . In the main body of the paper we will describe this product in detail, and for now just make a few points. We have It turns out that K is an automorphism of the q-shuffle algebra V. We define four maps in End(V), denoted We recall the concept of an adjoint. For X ∈ End(V) there exists a unique X * ∈ End(V) such that (Xu, v) = (u, X * v) for all u, v ∈ V. The element X * is called the adjoint of X with respect to ( , ). For example K * = K. We will consider give the Kashiwara operators for the negative part of U q sl 2 . In [18,Lemma 3.4.2] the above maps e 0 , e 1 and f 0 = A L , f 1 = B L are used to obtain a module for the reduced q-analog B q sl 2 . The maps A * r , B * r are discussed in [13,Definition 4.2], where they are called ∆ 0 , ∆ 1 . The map A * R (resp. B * R ) is called ∂ 0 (resp. ∂ 1 ) in [13, Section 3.1] and e 0 (resp. e 1 ) in [19, p. 696]. In the present paper, we will put the well known maps (1.3), (1.4) and (1.5), (1.6) to a new use.
Let J denote the 2-sided ideal of the free algebra V generated by As we will see, the ideal J is invariant under K ±1 and (1.3), (1.6). The quotient algebra V/J is often denoted by U + q and called the positive part of U q sl 2 ; see for example [14, p. 40] or [20,Corollary 3.2.6]. Let U denote the subalgebra of the q-shuffle algebra V generated by A, B. As we will see, the algebra U is invariant under K ±1 and (1.4), (1.5). It is well known that the algebra U is isomorphic to U + q ; see [22,Theorem 15] or [19, p. 696]. We are now ready to state some more results. For notational convenience define Q = 1 − q 2 . Theorem 1.1. For each row in the tables below, the vector space V/J becomes a q -module on which the generators {x i } i∈Z 4 act as indicated.
module label x 0 Each q -module in the tables is isomorphic to U. Theorem 1.2. For each row in the tables below, the vector space U becomes a q -module on which the generators {x i } i∈Z 4 act as indicated.
We will state some additional results shortly. We recall the concept of a derivation. Let A denote an algebra, and let ϕ, φ denote automorphisms of A. By a (ϕ, φ)-derivation of A we mean an F-linear map δ : The derivation concept is well known in the theory of quantum groups and q-shuffle algebras. For instance, by [13,Theorem 3.2] the maps A * R and B * R act on the q-shuffle algebra V as a (I, K)-derivation and I, K −1 -derivation, respectively. By [13,Section 4.2] the maps A * r and B * r act on the free algebra V as a (I, K)-derivation and I, K −1 -derivation, respectively. Concerning q we have the following results. Theorem 1.3. For each q -module in Theorem 1.1, the elements x 1 and x 3 act on the algebra V/J as a derivation of the following sort: module label For each q -module in Theorem 1.2, the elements x 1 and x 3 act on the algebra U as a derivation of the following sort: module label This paper is organized as follows. Section 2 contains some preliminaries. In Section 3 we recall the free algebra V on two generators. In Section 4, we endow V with a bilinear form and discuss the corresponding adjoint map. In Section 5, we describe two automorphisms and one antiautomorphism of V that will play a role in our main results. In Section 6, we recall the q-shuffle product on V, and embed U + q into the q-shuffle algebra V. In Sections 7-9, we give a detailed description of how the free algebra V is related to the q-shuffle algebra V. In Section 10 we introduce an algebra ∨ q that is a homomorphic preimage of q . In Section 11 we give sixteen ∨ q -module structures on V. In Section 12 we describe many homomorphisms between our sixteen ∨ q -modules. In Section 13 we use our sixteen ∨ q -modules to obtain sixteen q -modules. In Section 14 we show that these sixteen q -modules are mutually isomorphic and irreducible. In Section 15 we characterize these q -modules using the notion of a NIL qmodule. Appendix A contains some data on the q-shuffle product. Appendix B gives some matrix representations of the maps used in our main results.

Preliminaries
We now begin our formal argument. Recall the natural numbers N = {0, 1, 2, . . .} and integers Z = {0, ±1, ±2, . . .}. Let F denote a field. We will be discussing vector spaces, algebras, and tensor products. Every vector space discussed is over F. Every algebra discussed is over F, associative, and has a multiplicative identity. Every tensor product discussed is over F. Let A denote an algebra. By an automorphism (resp. antiautomorphism) of A we mean an F-linear bijection γ : A → A such that γ(uv) = γ(u)γ(v) (resp. γ(uv) = γ(v)γ(u)) for all u, v ∈ A. Let ϕ, φ denote automorphisms of A. By a (ϕ, φ)-derivation of A we mean an F-linear map δ : A → A such that δ(uv) = ϕ(u)δ(v) + δ(u)φ(v) for all u, v ∈ A. The set of all (ϕ, φ)derivations of A is closed under addition and scalar multiplication, and is therefore a vector space over F. Let δ denote a (ϕ, φ)-derivation of A. Then δ(1) = 0. For an automorphism σ of A, the composition δσ is a (ϕσ, φσ)-derivation of A, and σδ is a (σϕ, σφ)-derivation of A. Let J denote a 2-sided ideal of A with J = A, and consider the quotient algebra A = A/J. Assume that ϕ(J) = J and φ(J) = J. Then there exist automorphisms ϕ and φ of A such that ϕ(a + J) = ϕ(a) + J and φ(a + J) = φ(a) + J for all a ∈ A. Assume further that δ(J) ⊆ J. Then there exists a ϕ, φ -derivation δ of A such that δ(a + J) = δ(a) + J for all a ∈ A. In the main body of the paper we will suppress the overline notation.
Fix a nonzero q ∈ F that is not a root of unity. Recall the notation 3 The free algebra V with generators A, B Let A, B denote noncommuting indeterminates, and let V denote the free algebra with generators A, B. For n ∈ N, a word of length n in V is a product v 1 v 2 · · · v n such that v i ∈ {A, B} for 1 ≤ i ≤ n. We interpret the word of length zero to be the multiplicative identity in V; this word is called trivial and denoted by 1. The vector space V has a basis consisting of its words; this basis is called standard. Referring to Definition 3.1, the dimension of V n is 2 n . We have The vector 1 is a basis for V 0 . For r, s ∈ N we have V r V s ⊆ V r+s . By these comments the sum (3.1) is a grading of V in the sense of [23, p. 704]. Let End(V) denote the algebra consisting of the F-linear maps V → V with the following property: the matrix that represents the map with respect to the standard basis for V has finitely many nonzero entries in each row. Definition 3.2. We define four maps in End(V), denoted For v ∈ V, The following lemma is about A L and B L ; a similar result holds for A R and B R . Proof . The free algebra V is generated by A, B.
Definition 3.5. Let J denote the 2-sided ideal of the free algebra V generated by 3) Definition 3.6. The quotient algebra V/J is often denoted by U + q and called the positive part of U q sl 2 ; see for example [14, p. 40]

A bilinear form on V
We continue to discuss the free algebra V with generators A, B. In this section we endow V with a symmetric nondegenerate bilinear form. We describe the corresponding adjoints of the four maps listed in (3.2). The bilinear form ( , ) is symmetric and nondegenerate. The summands in (3.1) are mutually orthogonal with respect to ( , ).
We now recall the adjoint map. By linear algebra, for X ∈ End(V) there exists a unique X * ∈ End(V) such that (Xu, v) = (u, X * v) for all u, v ∈ V. With respect to the standard basis for V, the matrices representing X and X * are transposes. The element X * is called the adjoint of X with respect to ( , ). The adjoint map End(V) → End(V), X → X * is an antiautomorphism of End(V). We now consider Proof . This follows from Definition 4.1 and the meaning of the adjoint. We illustrate with a detailed proof of A * L (Av) = v. For u ∈ V, (u, A * L (Av)) = (A L (u), Av) = (Au, Av) = (u, v).
We now describe how the maps (4.1) act on the standard basis for V. In view of (4.2), we focus on the nontrivial basis elements. Recall that the Kronecker delta δ r,s is equal to 1 if r = s, and 0 if r = s.
The next two lemmas are about A * L and B * L ; similar results hold for A * R and B * R .
For v ∈ V the following are equivalent: The vector v is orthogonal to both AV and BV, and is therefore contained in V 0 .
Call n the degree of w. Choose w such that this degree is minimal. By assumption A * L w ∈ W . By Lemma 4.4 the vector A * L w is either 0 or has degree less than n. So A * L w = 0 by the minimality of n. Similarly B * L w = 0. Now w ∈ V 0 by Lemma 4.5. The result follows.

Some automorphisms and antiautomorphisms
We continue to discuss the free algebra V with generators A, B. In this section we introduce the automorphisms K, T of V and the antiautomorphism S of V.
We have KV n = V n for n ∈ N, and also K * = K.
The map S is described as follows. It is the unique antiautomorphism of the free algebra V that fixes A and B. We have SV n = V n for n ∈ N. We have (S(u), Proof . The first five equations are readily verified by applying both sides to a word in V. We illustrate with a detailed proof of A L S = SA R . For a word v in V, Therefore A L S = SA R . To obtain the equations (5.2), apply the adjoint map to each equation in (5.1).
Definition 5.4. Let T denote the automorphism of the free algebra V that swaps A and B.
The map T is described as follows. We have T V n = V n for n ∈ N. We have (T (u), T (v)) = (u, v) for u, v ∈ V. By this and T 2 = I we obtain (T (u), v) = (u, T (v)) for u, v ∈ V. Therefore T * = T . We have ST = T S.
Proof . The first five equations are readily verified by applying both sides to a word in V. We illustrate with a detailed proof of Therefore To obtain the equations (5.4), apply the adjoint map to each equation in (5.3).
Lemma 5.6. We have 6 The q-shuffle algebra V and the map θ We have been discussing the free algebra V. There is another algebra structure on V, called the q-shuffle algebra [13,21,22]. For this algebra the product is denoted by . To describe this product, we start with some special cases. We have For r, s ∈ N we have V r V s ⊆ V r+s . Therefore the sum (3.1) is a grading for the q-shuffle algebra V.
The following examples illustrate the shuffle product.
Example 6.1. We have

Appendix A contains additional examples. Using these examples (or by [13, p. 10]) one obtains
For the q-shuffle algebra V let U denote the subalgebra generated by A, B. The algebra U is described as follows. There exists an algebra homomorphism θ from the free algebra V to the q-shuffle algebra V, that sends A → A and B → B. By construction θ(V) = U . Comparing (3.3), (3.4) with (6.5), (6.6) we obtain θ(J ± ) = 0. Recall that J ± generate the 2-sided ideal J of the free algebra V. Consequently θ(J) = 0, so the kernel ker(θ) contains J. By [22,Theorem 15] we have ker(θ) = J, so θ induces an algebra isomorphism V/J → U . By this and V/J = U + q , we get an algebra isomorphism U + q → U . This isomorphism is discussed around [22,Theorem 15] and also [19, p. 696].
Our next goal is to describe how θ acts on the standard basis for V. By construction θ(1) = 1. Pick an integer n ≥ 1. We view the symmetric group S n as the group of permutations of the set {1, 2, . . . , n}. For σ ∈ S n , by an inversion for σ we mean an ordered pair (i, j) of integers such that 1 ≤ i < j ≤ n and σ(i) > σ(j). Let Inv(σ) denote the set of inversions for σ.
Proof . By induction on n, using (6.1) or (6.2). By Lemma 6.2 we have θV n ⊆ V n for n ∈ N.
For r = s the common value (6.7) is 0. For r = s = 0 the common value (6.7) is 1. For r = s ≥ 1 the common value (6.7) is equal to where the sum is over all σ ∈ S r such that v k = u σ(k) for 1 ≤ k ≤ r.
Proof . By Lemma 6.2 and since the standard basis for V is orthonormal with respect to ( , ).
The following result is a reformulation of [13, Theorem 4.5].
Let the set U ⊥ consist of the vectors in V that are orthogonal to U with respect to ( , ).
Proof . Use J = ker(θ) and Corollary 6.4, along with the fact that ( , ) is nondegenerate. Here are the details.
Lemma 6.6. For the q-shuffle algebra V, the maps K and T are automorphisms and S is an antiautomorphism.
Proof . Use the definition of the q-shuffle product.
Lemma 6.7. We have Proof . Apply each side to a word in V, and evaluate the result using Lemma 6.6.
Lemma 6.8. We have Proof . By Lemma 6.7 and since each of K, S, T is invertible. We give the details for the equation involving K. We have In Definition 3.2 we used the free algebra V to obtain the four maps listed in (3.2). In this section we use the q-shuffle algebra V to obtain four analogous maps. We investigate how these maps and their adjoints are related to S, T , θ.
The following lemma is about A and B ; a similar result holds for A r and B r .
Proof . By Definition 7.1, and since U is the subalgebra of the q-shuffle algebra V generated by A, B.
We now consider Proof . For each map X in (7.1), use (6.1), (6.2) to compute the matrix representing X with respect to the standard basis for V. The transpose of this matrix represents X * with respect to the standard basis for V. The result follows.
We now consider how the maps (7.1), (7.2) are related to S, T , θ.
Lemma 7.6. We have Proof . The equations (7.3) follow from Lemma 6.6. We illustrate with a detailed proof of Therefore SA = A r S. To obtain the equations (7.4), apply the adjoint map to each equation in (7.3).
Lemma 7.7. We have Proof . The equations (7.5) follow from Lemma 6.6. We illustrate with a detailed proof of Therefore T A = B T . To obtain the equations (7.6), apply the adjoint map to each equation in (7.5).
Lemma 7.8. We have Proof . The equations (7.7) follow from the definition of θ below (6.6). We illustrate with a detailed proof of Therefore θA L = A θ. To obtain the equations (7.8), apply the adjoint map to each equation in (7.7).

Derivations
In this section we interpret the maps (4.1), (7.2) using the derivation concept from Section 2.
We acknowledge that most if not all of the results in this section are well known to the experts; see for example [13,Sections 3 and 4].
Without loss of generality, we may assume that u and v are words in V. In this case the result is readily checked using Lemma 7.4.
Without loss, we may assume that u and v are words in V. In this case the result is readily checked using (6.3), (6.4) and Lemma 4.3.

Some relations
Consider the maps K ±1 , (3.2), (4.1), (7.1), (7.2). In this section we describe how K ±1 , (4.1), (7.1) are related, and how K ±1 , (3.2), (7.2) are related. We also discuss the subspaces J and U . We acknowledge that most if not all of the relations in this section are well known to the experts, see for example [18,Sections 3.3

and 3.4].
Proposition 9.1. The maps satisfy the following relations: KK −1 = K −1 K = I and Proof . The first 16 relations above are checked by applying each side to a word in V. The next 16 relations are obtained by setting u = A or u = B or v = A or v = B in Lemma 8.3. The last four relations follow from (6.5), (6.6) and the fact that the q-shuffle product is associative.
Proposition 9.2. The maps satisfy the following relations: KK −1 = K −1 K = I and Proof . Apply the adjoint map to each relation in Proposition 9.1.
Proposition 9.3. The subspace U is invariant under each of the maps (9.1). On U , Proof . The subspace U is invariant under K ±1 by Lemma 6.8. The subspace U is invariant under the last eight maps in (9.1), in view of Lemma 7.8. We illustrate with a detailed proof of A * L (U ) ⊆ U . We have Next we show that the relation (9.3) holds on U . Let X denote the map on the left in (9.3), and note that X * (v) = −J + v for all v ∈ V. We show that XU = 0. To do this, it suffices to show that (XU, V) = 0. We have (XU, V) = (U, X * V) = (U, J + V) and J + V ⊆ J = U ⊥ . Therefore (XU, V) = 0. We have shown that the relation (9.3) holds on U . One similarly shows that the relations (9.4)-(9.6) hold on U .
Proposition 9.4. The subspace J is invariant under each of the maps (9.2). On the quotient V/J,

8)
Proof . The subspace J is invariant under K ±1 by Lemma 5.6. The subspace J is invariant under A L , B L , A R , B R since J is a 2-sided ideal of the free algebra V. The subspace J is invariant under A * , B * , A * r , B * r by (7.8) and J = ker(θ). We verify that the relation (9.7) holds on V/J. Let Y denote the map on the left in (9.7). To show that Y is zero on V/J, it suffices to show that Y V ⊆ J. This is the case, since Y V = J + V ⊆ J. We have verified that (9.7) holds on V/J. One similarly verifies that the relations (9.8)-(9.10) hold on V/J. 10 The algebra ∨ q In this section we introduce an algebra ∨ q and describe how it is related to the free algebra V. We also discuss the q-Serre relations. In the next section we will obtain sixteen ∨ q -module structures on V. Recall the cyclic group Z 4 = Z/4Z of order 4.
The algebra ∨ q is related to the free algebra V in the following way. Let (iii) the multiplication map ∨ q even ⊗ ∨ q odd → ∨ q , u ⊗ v → uv is an isomorphism of vector spaces.
We need a fact about the q-Serre relations. We will take a moment to establish this fact, and then return to the main topic. Lemma 10.3. Pick scalars r, s ∈ q 2 , q −2 . Suppose we are given elements a, b, x, y, k, k −1 in any algebra such that kk −1 = k −1 k = 1 and ax = xa, ay = ya, bx = xb, by = yb, Proof . To verify each equation, expand the left-hand side and evaluate the result using (10.2), (10.3).
Corollary 10.4. With the notation and assumptions of Lemma 10.3, if a, b and x, y satisfy the q-Serre relations, then so do any of the following pairs: Proof . (i) By Lemma 10.3.
(ii) Apply (i) above with a, b replaced by s −1 a, s −1 b.
11 Sixteen ∨ q -module structures on V In Definition 10.1 we defined the algebra ∨ q . In this section we describe sixteen ∨ q -module structures on V. The first eight involve the maps from (9.2), and the rest involve the maps from (9.1). For notational convenience define Q = 1 − q 2 .
Proposition 11.1. For each row in the tables below, the vector space V becomes a ∨ q -module on which the generators {x i } i∈Z 4 act as indicated.
module label x 0 On each ∨ q -module in the tables, the actions of x 1 , x 3 satisfy the q-Serre relations.
Proof . By the relations in Proposition 9.2 along with Corollary 10.4.
Proposition 11.2. For each row in the tables below, the vector space V becomes a ∨ q -module on which the generators {x i } i∈Z 4 act as indicated.
module label x 0 On each ∨ q -module in the tables, the actions of x 0 , x 2 satisfy the q-Serre relations.
Proof . By the relations in Proposition 9.1, along with Corollary 10.4. Note 11.3. Going forward, the ∨ q -module V with label I will be denoted V I , and so on.
We now describe the sixteen ∨ q -modules in more detail.
Lemma 11.4. For each ∨ q -module V in Propositions 11.1, 11.2 we have Proof . The result for x 0 and x 2 comes from Lemmas 3.3, 7.2. The result for x 1 and x 3 comes from Lemmas 4.4, 7.5.
Lemma 11.5. For each ∨ q -module in Proposition 11.1, the elements x 1 and x 3 act on the free algebra V as a derivation of the following sort: module label Lemma 11.6. For each ∨ q -module in Proposition 11.2, the elements x 1 and x 3 act on the q-shuffle algebra V as a derivation of the following sort: module label

Some homomorphisms between the sixteen ∨ q -modules
In the previous section we gave sixteen ∨ q -module structures on V. In this section we describe some homomorphisms between them.
Lemma 12.1. The map S ∈ End(V) is an isomorphism of ∨ q -modules from Proof . By Lemmas 5.3, 7.6 and since S is a bijection.
Proof . By Lemmas 5.5, 7.7 and since T is a bijection.
Lemma 12.3. The map θ ∈ End(V) is a homomorphism of ∨ q -modules from Proof . By Lemma 7.8 and the first equation in Lemma 6.7.
Our next goal is to display a map ϕ ∈ End(V) such that ϕ is a ∨ q -module isomorphism from V I → V II and V IT → V IIT , and ϕ * is a ∨ q -module isomorphism from V III → V IV and V IIIT → V IVT .
Lemma 12.5. We have ϕV n ⊆ V n for n ∈ N.
Proof . By Lemma 3.3 and Definition 12.4.
Shortly we will describe ϕ from an another point of view. In this description we use the following notation. Let v = v 1 v 2 · · · v n denote a word in V. For a subset Ω ⊆ {1, 2, . . . , n} we define a word v Ω as follows.
where the sum is over all subsets Ω of {1, 2, . . . , n}. The maps S and T are from Definitions 5.2 and 5.4, respectively.
Without loss of generality, we may assume that v is a word in V. Write v = v 1 v 2 · · · v n . By Definition 12.4 and the above comments, We have shown that T ϕ = ϕT . In this equation, apply the adjoint map to each side and use T * = T to obtain T ϕ * = ϕ * T .
Proof . We first obtain (12.3). For x ∈ {A, B} we show that ϕx L = xϕ, where x is from (12.2). It suffices to show that ϕx L (v) = xϕ(v) for all words v in V. Let the word v be given, and write v = v 1 v 2 · · · v n . Using (12.1), We have obtained (12.3). Next we obtain the equation on the left in (12.4). For that equation let ∆ denote the left-hand side minus the right-hand side. We show that ∆ = 0. For notational convenience define Note that ∆ = ϕy − Y ϕ. Referring to Proposition 11.1, from the V I data and from the V II data We will show that X∆ = q −2 ∆x, Z∆ = q 2 ∆z. (12.5) We have We have shown (12.5). We can now easily show that ∆ = 0. We define W = {v ∈ V | ∆v = 0} and show that W = V. By Proof . The ∨ q -modules in the corollary statement are described in Proposition 11.1. Using these descriptions and Lemma 12.8, we find that for i ∈ Z 4 the relation ϕx i = x i ϕ holds on V I and V IT . The result follows.
Proof . The first assertion follows from Lemma 12.5 and the fact that the summands in (3.1) are mutually orthogonal. To obtain the last assertion, use ϕ(1) = 1 and the fact that 1 is a basis for V 0 .
Lemma 12.11. On V, Proof . For each equation in Lemma 12.8, apply the adjoint map to each side.
Proof . The ∨ q -modules in the corollary statement are described in Proposition 11.2. Using these descriptions and Lemma 12.11, we find that for i ∈ Z 4 the relation ϕ * x i = x i ϕ * holds on V III and V IIIT . The result follows.
Proof . We define ∆ = ϕ * θ − θϕ and show that ∆ = 0. By Lemma 12.3 and Corollaries 12.9, 12.12 we find that each of ϕ * θ and θϕ is a ∨ q -module homomorphism V I → V IV . So ∆ is a ∨ q -module homomorphism V I → V IV . By Proposition 11.1, x 0 and x 2 act on V I as A L and B L , respectively. By Proposition 11.2, x 0 and x 2 act on V IV as Q(A − KB r ) and Q B − K −1 A r , respectively. By these comments Let W denote the kernel of ∆ on V. By (12.6), W is invariant under A L and B L . We have ∆(1) = 0 since θ(1) = 1, ϕ(1) = 1, ϕ * (1) = 1. Therefore 1 ∈ W . By these comments and Lemma 3.4 we obtain W = V, so ∆ = 0.
Next we show that ϕ and ϕ * are bijections.
Proof . Use Definition 5.1 and the comments below it, along with Lemma 4.3 and Definition 12.14.
(i) The equation ψ n = q −2n T holds on V n .
(ii) The equation ψ 2n = q −4n I holds on V n .
(iii) The map I − ψ is invertible on V n .
(iv) On V n , Repeatedly applying Lemma 12.16 we obtain The result follows.
(ii) By (i) above and since T 2 = I.
(iii) Pick a vector v ∈ V n such that (I − ψ)v = 0. We show that v = 0. By construction ψv = v. By this and (ii) (iv) By (12.7) and since Lemma 12.18. For n ≥ 1 we have Proof . The inclusion ⊇ follows from Lemma 3.3. Concerning the inclusion ⊆, use parts (iii), (iv) of Lemma 12.17 along with Lemma 4.4 to obtain Lemma 12.19. For n ∈ N we have (i) ϕV n = V n ; (ii) ϕ * V n = V n .
Proof . (i) By induction on n. For n = 0 the result holds, because 1 is a basis for V 0 and ϕ(1) = 1. Next assume that n ≥ 1. The sum V n = AV n−1 + BV n−1 is direct. In this equation apply ϕ to each side. By Lemma 12.8 and induction, By these comments and Lemma 12.18, (ii) By Lemma 12.10 and (i) above, along with the fact that the dimension of V n is finite.
Proposition 12.21. The map ϕ is an isomorphism of ∨ q -modules from V I → V II and V IT → V IIT . Moreover ϕ * is an isomorphism of ∨ q -modules from V III → V IV and V IIIT → V IVT .
The result follows.
Proof . By Lemmas 12.13, 12.20 and θ(V) = U , 13 Sixteen q -module structures on V/J or U In Section 1 we informally discussed the algebra q . In this section we formally bring in q , and review its basic properties. We then display sixteen q -module structures on V/J or U . In order to motivate things, we mention a result about the ∨ q -modules from Propositions 11.1 and 11.2.
(i) For each ∨ q -module V in Proposition 11.1, the subspace J is a ∨ q -submodule. On the quotient ∨ q -module V/J the actions of x 0 , x 2 satisfy the q-Serre relations. (ii) For each ∨ q -module V from Proposition 11.2, the subspace U is a ∨ q -submodule on which the actions of x 1 , x 3 satisfy the q-Serre relations.
Lemma 13.3. There exists an algebra homomorphism ∨ q → q that sends x i → x i for i ∈ Z 4 . This homomorphism is surjective.
Recall the algebra U + q = V/J from Definition 3.6. This algebra is related to q in the following way. Let even q (resp. odd q ) denote the subalgebra of q generated by x 0 , x 2 (resp. x 1 , x 3 ). Then by [29,Proposition 5.5], (i) there exists an algebra isomorphism U + q → even q that sends A → x 0 and B → x 2 ; (ii) there exists an algebra isomorphism U + q → odd q that sends A → x 1 and B → x 3 ; (iii) the multiplication map even Theorem 13.4. For each row in the tables below, the vector space V/J becomes a q -module on which the generators {x i } i∈Z 4 act as indicated.
Theorem 13.5. For each row in the tables below, the vector space U becomes a q -module on which the generators {x i } i∈Z 4 act as indicated.
Note 13.6. Going forward, the q -module V/J with label I will be denoted (V/J) I , and so on. 14 The sixteen q -modules are mutually isomorphic and irreducible In the previous section we gave sixteen q -module structures on V/J or U . In this section we show that these q -modules are mutually isomorphic and irreducible.
Proof . By Lemma 12.1, together with the fact that S(J) = J by (5.6) and S(U ) = U by Lemma 6.8.
Proof . By Lemma 12.2, together with the fact that T (J) = J by (5.6) and T (U ) = U by Lemma 6.8.
is an isomorphism of q -modules from Proof . By Lemma 12.3 and the fact that J is the kernel of θ.
Our next goal is to show that the q -modules in Theorem 14.5 are irreducible.
Lemma 14.6. Let W denote a nonzero q -submodule of U IV . Then 1 ∈ W .
Proof . By Proposition 11.2 the generators x 1 and x 3 act on U IV as A * L and B * L , respectively. Now 1 ∈ W by Lemma 4.6.
Lemma 14.7. Let W denote a proper q -submodule of U III . Then 1 ∈ W .
Proof . We assume 1 ∈ W and get a contradiction. By Proposition 11.2 the generators x 0 and x 2 act on U III as A and B , respectively. By Lemma 7.3, W is not proper. This contradicts our assumptions.

The NIL modules for q
In this section we characterize the q -modules in Theorem 14.5, using the notion of a NIL q -module. We start with some comments about ∨ q . Reformulating the relations (10.1), Next we express these relations in a uniform way.
Lemma 15.1. For u ∈ {x 0 , x 2 } and v ∈ {x 1 , x 3 } the following holds in ∨ q : where , The following formula will be useful.
Proof . Referring to the equation displayed in Lemma 15.2, apply each side to ξ and note that vξ = 0.
Recall that ∨ q even is the subalgebra of ∨ q generated by x 0 , x 2 . Recall the free algebra V with generators A, B. By our comments below Note 10.2, there exists an algebra isomorphism V → ∨ q even that sends A → x 0 and B → x 2 . Denote this isomorphism by κ. Recall the ∨ q -module V I from Proposition 11.1.
We show that for i ∈ Z 4 the following diagram commutes: Referring to the above diagram, the action x i : V I → V I is described in Proposition 11.1 along with Definition 5.1 and Lemma 7.4. The action x i : V → V is clear for i even, and described in Corollary 15.3 for i odd. Using these descriptions we chase each word in V around the diagram, and confirm that the diagram commutes.
Let V denote a q -module. We view V as a ∨ q -module on which Since V is a q -module, the elements (15.1), (15.2) are zero on V . Therefore J ± ∈ H. Recall that J is the 2-sided ideal of the free algebra V generated by J ± . Consequently J ⊆ H. We mentioned earlier that H ⊆ L, so J ⊆ L. By this and Lemma 15.4, the map (V/J) I → V , v + J → κ(v)ξ is a q -module homomorphism that has kernel L/J. This kernel L/J is a qsubmodule of the q -module (V/J) I , and the q -module (V/J) I is irreducible by Theorem 14.8, so L/J = 0 or L/J = V/J. Thus L = J or L = V. We have L = V since 1 ∈ L, so L = J.
Let V denote a nonzero q -module. For ξ ∈ V , we say that V is generated by ξ whenever V does not have a proper q -submodule that contains ξ.
Proposition 15.6. Let V denote a q -module that is generated by a nonzero vector ξ such that x 1 ξ = 0 and x 3 ξ = 0. Then the map (V/J) I → V , v + J → κ(v)ξ is an isomorphism of q -modules.
Proof . By Lemma 15.5, the map (V/J) I → V , v + J → κ(v)ξ is an injective q -module homomorphism. For this map the image is a q -submodule of V that contains ξ, so this image is equal to V . By these comments the map (V/J) Theorem 15.7. For a q -module V the following are equivalent: (i) V is isomorphic to the q -modules in Theorem 14.5; (ii) v is generated by a nonzero vector ξ such that x 1 ξ = 0 and x 3 ξ = 0.
Proof . (i) ⇒ (ii): Without loss of generality, we may identify the q -module V with the q -module U III listed in Theorem 14.5. The vector ξ = 1 has the desired properties.
Definition 15.8. Let V denote a q -module. A vector ξ ∈ V is called NIL whenever x 1 ξ = 0 and x 3 ξ = 0 and ξ = 0. The q -module V is called NIL whenever it is generated by a NIL vector.
By Theorem 15.7, up to isomorphism there exists a unique NIL q -module, which we denote by U. Also by Theorem 15.7, the q -module U is isomorphic to each of the q -modules from Theorem 14.5. By Theorem 14.8 the q -module U is irreducible. The q -module U is infinitedimensional; indeed it is isomorphic to U + q as a vector space, as we now clarify. Recall the algebra isomorphism U + q → even q from below Lemma 13.3.
Lemma 15.9. Identify the algebra U + q with even q via the algebra isomorphism from below Lemma 13.3. Let ξ denote a NIL vector in U. Then the map U + q → U, u → uξ is an isomorphism of vector spaces.
Theorem 15.10. The q -module U has a unique sequence of subspaces {U n } n∈N such that: where U −1 = 0.
The sequence {U n } n∈N is described as follows. The subspace U 0 has dimension 1. The nonzero vectors in U 0 are precisely the NIL vectors in U, and each of these vectors generates U. Let ξ denote a NIL vector in U. Then for n ∈ N, U n is spanned by the vectors Proof . Concerning existence, without loss of generality we may identify the q -module U with the q -module U III listed in Theorem 14.5. Recall that U is the subalgebra of the q-shuffle algebra V generated by A, B. For n ∈ N define U n = U ∩ V n . One checks that the sequence {U n } n∈N satisfies the above conditions (i)-(iii). We have established existence. Going forward let {U n } n∈N denote any sequence of subspaces that satisfies the above conditions (i)-(iii). Let ξ denote a NIL vector in U. We claim that ξ ∈ U 0 . To prove the claim, note by condition (ii) that there exists n ∈ N and ξ i ∈ U i (0 ≤ i ≤ n) such that ξ n = 0 and ξ = n i=0 ξ i . Since ξ is NIL, x 1 ξ = 0 and x 3 ξ = 0. In the sum ξ = n i=0 ξ i , apply x 1 to each term and use condition (iii) to find x 1 ξ n = 0. Similarly x 3 ξ n = 0, so ξ n is NIL. Since the q -module U is irreducible, it is generated by any nonzero vector in U. In particular the q -module U is generated by ξ n . By Proposition 15.6 the map (V/J) I → U, v + J → κ(v)ξ n is an isomorphism of q -modules. Consider the image of this map. On one hand, the image is equal to U. On the other hand, by (iii) and the definition of κ above Lemma 15.4, the image is contained in i∈N U n+i . By these comments and (i), (ii) we obtain n = 0, so ξ = ξ 0 ∈ U 0 . We have proven the claim. For n ∈ N let U n denote the subspace of U spanned by the vectors (15.3). We claim that U n = U n for n ∈ N. By (iii) we have U n ⊆ U n for n ∈ N. Earlier we mentioned an isomorphism (V/J) I → U; its existence shows that the vector space U is spanned by the vectors So U = n∈N U n . By these comments and (ii) we obtain U n = U n for n ∈ N. The claim is proven. By the claims, the sequence {U n } n∈N satisfying conditions (i)-(iii) is unique, and it fits the description given in the last paragraph of the theorem statement.
A Data on the q-shuffle product Recall the q-shuffle algebra V from Section 6. In the following tables we express some q-shuffle products in terms of the standard basis for V.

B Some matrix representations
Consider the free algebra V generated by A, B. Earlier in the paper we described many maps in End(V). For such a map X, consider the matrix that represents X with respect to the standard basis for V. The rows and columns are indexed by the words in V. For words u, v the (u, v)entry is equal to (u, Xv). We will display some of these entries shortly. For the above matrix and r, s ∈ N the submatrix X(r, s) has rows and columns indexed by the words of length r and s, respectively. The matrix X(r, s) has dimensions 2 r × 2 s . We are going to display the nonzero X(r, s) such that 0 ≤ r, s ≤ 3. For this display we use the following word order: word length word order 0 1 1 A, B 2 AA, AB, BA, BB 3 AAA, AAB, ABA, BAA, ABB, BAB, BBA, BBB For each X we now display the nonzero X(r, s) such that 0 ≤ r, s ≤ 3. From the dimensions of X(r, s) it is clear what is r and s, so we do not state this explicitly. We have