An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂

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⁻¹xᵢ₊₁xᵢ)/(q−q⁻¹) = 1, x³ᵢxᵢ₊₂ − [3]qx²ᵢxᵢ + ₂xᵢ + [3]qxᵢxᵢ₊₂x²ᵢ − xᵢ₊₂x³ᵢ=0, where...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2020
Автори:	Post, Sarah, Terwilliger, Paul
Формат:	Стаття
Мова:	Англійська
Опубліковано:	Інститут математики НАН України 2020
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/210713
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂. Sarah Post and Paul Terwilliger. SIGMA 16 (2020), 037, 35 pages

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citation_txt	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂. Sarah Post and Paul Terwilliger. SIGMA 16 (2020), 037, 35 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	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⁻¹xᵢ₊₁xᵢ)/(q−q⁻¹) = 1, x³ᵢxᵢ₊₂ − [3]qx²ᵢxᵢ + ₂xᵢ + [3]qxᵢxᵢ₊₂x²ᵢ − xᵢ₊₂x³ᵢ=0, where [3]q=(q³−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≠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.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 037, 35 pages An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 Sarah POST † and Paul TERWILLIGER ‡ † Department of Mathematics, University of Hawai‘i at Manoa, Honolulu, HI 96822, USA E-mail: spost@hawaii.edu URL: https://math.hawaii.edu/~sarah/ ‡ Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA E-mail: terwilli@math.wisc.edu Received August 18, 2019, in final form April 19, 2020; Published online May 04, 2020 https://doi.org/10.3842/SIGMA.2020.037 Abstract. 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−1xi+1xi q − q−1 = 1, x3ixi+2 − [3]qx 2 ixi+2xi + [3]qxixi+2x 2 i − xi+2x 3 i = 0, where [3]q = ( q3−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. Key words: quantum group; q-Serre relations; derivation; q-Onsager algebra 2020 Mathematics Subject Classification: 17B37 1 Introduction The algebra �q was introduced in [29]. It is associative, infinite-dimensional, and noncommu- tative. 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 sl2; these have degree 3 in one variable and degree 1 in the other variable. The cyclic group Z4 = Z/4Z acts on the oriented four-cycle as a group of rotational symmetries, and this induces a Z4-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 Uq ( ŝl2 ) . The algebra U+ q is defined by two generators, subject to the above q-Serre relations [20, Corol- lary 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 Oq. The algebra Oq originated in algebraic combinatorics [24, Lemma 5.4], [26] and statistical mechanics [1, Section 1], [2, Section 2]. Research on Oq is presently active in both areas; see [10, 15, 16, 17, 25, 27, 28, 29, mailto:spost@hawaii.edu https://math.hawaii.edu/~sarah/ mailto:terwilli@math.wisc.edu https://doi.org/10.3842/SIGMA.2020.037 2 S. Post and P. Terwilliger 30, 31, 32] and [1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12]. The algebraOq 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 Oq are related as follows. By [29, Proposition 5.6] there exists an injective algebra homomorphism Oq → �q that sends one Oq-generator to a linear combination of two adjacent �q-generators, and the other Oq-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 Oq 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 PV in one variable z that has constant coefficient 1; this PV is called the Drinfel’d polynomial of V . By [33, Proposition 1.4], the map V 7→ PV 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 sl2. 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 {xi}i∈Z4 and relations qxixi+1 − q−1xi+1xi q − q−1 = 1, (1.1) x3 ixi+2 − [3]qx 2 ixi+2xi + [3]qxixi+2x 2 i − xi+2x 3 i = 0, (1.2) where [3]q = ( q3 − q−3 ) / ( q − q−1 ) . 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 x1ξ = 0 and x3ξ = 0 and ξ 6= 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 {Un}n∈N such that (i) U0 6= 0; (ii) the sum U = ∑ n∈N Un is direct; (iii) for n ∈ N, x0Un ⊆ Un+1, x1Un ⊆ Un−1, x2Un ⊆ Un+1, x3Un ⊆ Un−1, where U−1 = 0. The sequence {Un}n∈N is described as follows. The subspace U0 has dimen- sion 1. The nonzero vectors in U0 are precisely the NIL vectors in U, and each of these vectors An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 3 generates U. Let ξ denote a NIL vector in U. Then for n ∈ N, the subspace Un is spanned by the vectors u1u2 · · ·unξ such that ui ∈ {x0, x2} 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 v1v2 · · · vn such that vi ∈ {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 7→ q2A and B 7→ q−2B. For a word v = v1v2 · · · vn in V, K(v) = vq〈v1,A〉+〈v2,A〉+···+〈vn,A〉, K−1(v) = vq〈v1,B〉+〈v2,B〉+···+〈vn,B〉 where 〈 , 〉 A B A 2 −2 B −2 2 We define four maps in End(V), denoted AL, BL, AR, BR. (1.3) For v ∈ V, AL(v) = Av, BL(v) = Bv, AR(v) = vA, BR(v) = vB. 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 1 ? v = v ? 1 = v for v ∈ V. For X ∈ {A,B} and a nontrivial word v = v1v2 · · · vn in V, X ? v = n∑ i=0 v1 · · · viXvi+1 · · · vnq〈v1,X〉+〈v2,X〉+···+〈vi,X〉, v ? X = n∑ i=0 v1 · · · viXvi+1 · · · vnq〈vn,X〉+〈vn−1,X〉+···+〈vi+1,X〉. It turns out that K is an automorphism of the q-shuffle algebra V. We define four maps in End(V), denoted A`, B`, Ar, Br. (1.4) For v ∈ V, A`(v) = A ? v, B`(v) = B ? v, Ar(v) = v ? A, Br(v) = v ? B. 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 A∗L, B∗L, A∗R, B∗R (1.5) 4 S. Post and P. Terwilliger and A∗` , B∗` , A∗r , B∗r . (1.6) We acknowledge that the maps (1.3), (1.4) and (1.5), (1.6) are well known in the literature on quantum groups and q-shuffle algebras. For instance, in [18, Section 3.4] the maps e′0 = K−1A∗r , e′1 = KB∗r , e′′0 = A∗` , e′′1 = B∗` give the Kashiwara operators for the negative part of Uq ( ŝl2 ) . In [18, Lemma 3.4.2] the above maps e′0, e′1 and f0 = AL, f1 = BL are used to obtain a module for the reduced q-analog Bq ( ŝl2 ) . 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 J+ = A3B − [3]qA 2BA+ [3]qABA 2 −BA3, J− = B3A− [3]qB 2AB + [3]qBAB 2 −AB3. 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 Uq ( ŝl2 ) ; 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− q2. Theorem 1.1. For each row in the tables below, the vector space V/J becomes a �q-module on which the generators {xi}i∈Z4 act as indicated. module label x0 x1 x2 x3 I AL Q(A∗` −B∗rK) BL Q ( B∗` −A∗rK−1 ) IS AR Q(A∗r −B∗`K) BR Q ( B∗r −A∗`K−1 ) IT BL Q ( B∗` −A∗rK−1 ) AL Q(A∗` −B∗rK) IST BR Q ( B∗r −A∗`K−1 ) AR Q(A∗r −B∗`K) module label x0 x1 x2 x3 II Q(AL −KBR) A∗` Q ( BL −K−1AR ) B∗` IIS Q(AR −KBL) A∗r Q ( BR −K−1AL ) B∗r IIT Q ( BL −K−1AR ) B∗` Q(AL −KBR) A∗` IIST Q ( BR −K−1AL ) B∗r Q(AR −KBL) A∗r 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 {xi}i∈Z4 act as indicated. module label x0 x1 x2 x3 III A` Q(A∗L −B∗RK) B` Q ( B∗L −A∗RK−1 ) IIIS Ar Q(A∗R −B∗LK) Br Q ( B∗R −A∗LK−1 ) IIIT B` Q ( B∗L −A∗RK−1 ) A` Q(A∗L −B∗RK) IIIST Br Q ( B∗R −A∗LK−1 ) Ar Q(A∗R −B∗LK) module label x0 x1 x2 x3 IV Q(A` −KBr) A∗L Q ( B` −K−1Ar ) B∗L IVS Q(Ar −KB`) A∗R Q ( Br −K−1A` ) B∗R IVT Q ( B` −K−1Ar ) B∗L Q(A` −KBr) A∗L IVST Q ( Br −K−1A` ) B∗R Q(Ar −KB`) A∗R Each �q-module in the tables is isomorphic to U. An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 5 We will state some additional results shortly. We recall the concept of a derivation. Let A denote an algebra, and let ϕ, φ denote auto- morphisms 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 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 x1 and x3 act on the alge- bra V/J as a derivation of the following sort: module label x1 x3 I, II (K, I)-derivation ( K−1, I ) -derivation IS, IIS (I,K)-derivation (I,K−1)-derivation IT, IIT ( K−1, I ) -derivation (K, I)-derivation IST, IIST ( I,K−1 ) -derivation (I,K)-derivation Theorem 1.4. For each �q-module in Theorem 1.2, the elements x1 and x3 act on the algebra U as a derivation of the following sort: module label x1 x3 III, IV (K, I)-derivation ( K−1, I ) -derivation IIIS, IVS (I,K)-derivation (I,K−1)-derivation IIIT, IVT ( K−1, I ) -derivation (K, I)-derivation IIIST, IVST ( I,K−1 ) -derivation (I,K)-derivation 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 �q- module. Appendix A contains some data on the q-shuffle product. Appendix B gives some matrix representations of the maps used in our main results. 2 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 6 S. Post and P. Terwilliger 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 6= 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 [n]q = qn − q−n q − q−1 , n ∈ Z. 3 The free algebra V with generators A, B Let A, B denote noncommuting indeterminates, and let V denote the free algebra with genera- tors A, B. For n ∈ N, a word of length n in V is a product v1v2 · · · vn such that vi ∈ {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. Definition 3.1. For n ∈ N, let Vn denote the subspace of V spanned by the words of length n. For notational convenience define V−1 = 0. Referring to Definition 3.1, the dimension of Vn is 2n. We have V = ∑ n∈N Vn (direct sum). (3.1) The vector 1 is a basis for V0. For r, s ∈ N we have VrVs ⊆ Vr+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 AL, BL, AR, BR. (3.2) For v ∈ V, AL(v) = Av, BL(v) = Bv, AR(v) = vA, BR(v) = vB. Lemma 3.3. For n ∈ N, ALVn ⊆ Vn+1, BLVn ⊆ Vn+1, ARVn ⊆ Vn+1, BRVn ⊆ Vn+1. The following lemma is about AL and BL; a similar result holds for AR and BR. Lemma 3.4. Let W denote a subspace of V that is closed under AL and BL. Assume that 1 ∈W . Then W = V. 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 J+ = A3B − [3]qA 2BA+ [3]qABA 2 −BA3, (3.3) J− = B3A− [3]qB 2AB + [3]qBAB 2 −AB3. (3.4) Definition 3.6. The quotient algebra V/J is often denoted by U+ q and called the positive part of Uq ( ŝl2 ) ; see for example [14, p. 40] or [20, Corollary 3.2.6]. An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 7 4 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). Definition 4.1. Define a bilinear form ( , ) : V × V → F as follows. Recall that the standard basis for V consists of the words in A, B. This basis is orthonormal with respect to ( , ). 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 7→ X∗ is an antiautomorphism of End(V). We now consider A∗L, B∗L, A∗R, B∗R. (4.1) Lemma 4.2. We have A∗L(1) = 0, B∗L(1) = 0, A∗R(1) = 0, B∗R(1) = 0. (4.2) Moreover for v ∈ V, A∗L(Av) = v, A∗L(Bv) = 0, B∗L(Av) = 0, B∗L(Bv) = v, A∗R(vA) = v, A∗R(vB) = 0, B∗R(vA) = 0, B∗R(vB) = v. 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)) = (AL(u), Av) = (Au,Av) = (u, v). Therefore A∗L(Av) = v since ( , ) is nondegenerate. � 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 δr,s is equal to 1 if r = s, and 0 if r 6= s. Lemma 4.3. For an integer n ≥ 1 and a word v = v1v2 · · · vn in V, A∗L(v) = v2 · · · vnδv1,A, B∗L(v) = v2 · · · vnδv1,B, A∗R(v) = v1 · · · vn−1δvn,A, B∗R(v) = v1 · · · vn−1δvn,B. Proof. Use Lemma 4.2. � Lemma 4.4. For n ∈ N, A∗LVn ⊆ Vn−1, B∗LVn ⊆ Vn−1, A∗RVn ⊆ Vn−1, B∗RVn ⊆ Vn−1. Proof. Use Lemma 4.3. � The next two lemmas are about A∗L and B∗L; similar results hold for A∗R and B∗R. Lemma 4.5. For v ∈ V the following are equivalent: 8 S. Post and P. Terwilliger (i) v ∈ V0; (ii) A∗Lv = 0 and B∗Lv = 0. Proof. (i)⇒ (ii): By (4.2). (ii)⇒ (i): Consider the orthogonal direct sum V = V0 +AV+BV. We have 0 = (A∗Lv,V) = (v,AV) and 0 = (B∗Lv,V) = (v,BV). The vector v is orthogonal to both AV and BV, and is therefore contained in V0. � Lemma 4.6. Let W denote a nonzero subspace of V that is closed under A∗L and B∗L. Then 1 ∈W . Proof. There exists 0 6= w ∈ W . Write w = n∑ i=0 wi with wi ∈ Vi for 0 ≤ i ≤ n and wn 6= 0. Call n the degree of w. Choose w such that this degree is minimal. By assumption A∗Lw ∈ W . By Lemma 4.4 the vector A∗Lw is either 0 or has degree less than n. So A∗Lw = 0 by the minimality of n. Similarly B∗Lw = 0. Now w ∈ V0 by Lemma 4.5. The result follows. � 5 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. Definition 5.1. Let K denote the automorphism of the free algebra V that sends A 7→ q2A and B 7→ q−2B. The automorphism K is described as follows. For a word v = v1v2 · · · vn in V, K(v) = vq〈v1,A〉+〈v2,A〉+···+〈vn,A〉, K−1(v) = vq〈v1,B〉+〈v2,B〉+···+〈vn,B〉, where 〈 , 〉 A B A 2 −2 B −2 2 We have KVn = Vn for n ∈ N, and also K∗ = K. Definition 5.2. Define S ∈ End(V) such that for each word v = v1v2 · · · vn in V, S(v) = vn · · · v2v1. The map S is described as follows. It is the unique antiautomorphism of the free algebra V that fixes A and B. We have SVn = Vn for n ∈ N. We have (S(u), S(v)) = (u, v) for u, v ∈ V. By this and S2 = I we obtain (S(u), v) = (u, S(v)) for all u, v ∈ V. Therefore S∗ = S. Lemma 5.3. We have KS = SK and ALS = SAR, ARS = SAL, BLS = SBR, BRS = SBL, (5.1) A∗LS = SA∗R, A∗RS = SA∗L, B∗LS = SB∗R, B∗RS = SB∗L. (5.2) Proof. The first five equations are readily verified by applying both sides to a word in V. We illustrate with a detailed proof of ALS = SAR. For a word v in V, ALS(v) = AS(v) = S(A)S(v) = S(vA) = SAR(v). Therefore ALS = SAR. To obtain the equations (5.2), apply the adjoint map to each equation in (5.1). � An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 9 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 TVn = Vn 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 = TS. Lemma 5.5. We have KT = TK−1 and ALT = TBL, ART = TBR, BLT = TAL, BRT = TAR, (5.3) A∗LT = TB∗L, A∗RT = TB∗R, B∗LT = TA∗L, B∗RT = TA∗R. (5.4) Proof. The first five equations are readily verified by applying both sides to a word in V. We illustrate with a detailed proof of ALT = TBL. For a word v in V, ALT (v) = AT (v) = T (B)T (v) = T (Bv) = TBL(v). Therefore ALT = TBL. To obtain the equations (5.4), apply the adjoint map to each equation in (5.3). � Recall J±, J from Definition 3.5. Lemma 5.6. We have K ( J± ) = q±4J±, S ( J± ) = −J±, T ( J± ) = J∓. (5.5) Moreover K(J) = J, S(J) = J, T (J) = J. (5.6) Proof. The relations (5.5) are routinely checked using (3.3) and (3.4). Line (5.6) follows from (5.5). � 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 1 ? v = v ? 1 = v for v ∈ V. For X ∈ {A,B} and a nontrivial word v = v1v2 · · · vn in V, X ? v = n∑ i=0 v1 · · · viXvi+1 · · · vnq〈v1,X〉+〈v2,X〉+···+〈vi,X〉, (6.1) v ? X = n∑ i=0 v1 · · · viXvi+1 · · · vnq〈vn,X〉+〈vn−1,X〉+···+〈vi+1,X〉. (6.2) For nontrivial words u = u1u2 · · ·ur and v = v1v2 · · · vs in V, u ? v = u1 ( (u2 · · ·ur) ? v ) + v1 ( u ? (v2 · · · vs) ) q〈u1,v1〉+〈u2,v1〉+···+〈ur,v1〉, (6.3) u ? v = ( u ? (v1 · · · vs−1) ) vs + ( (u1 · · ·ur−1) ? v ) urq 〈ur,v1〉+〈ur,v2〉+···+〈ur,vs〉. (6.4) For r, s ∈ N we have Vr ? Vs ⊆ Vr+s. Therefore the sum (3.1) is a grading for the q-shuffle algebra V. The following examples illustrate the shuffle product. 10 S. Post and P. Terwilliger Example 6.1. We have A ? A = ( 1 + q2 ) AA, A ? B = AB + q−2BA, B ? A = BA+ q−2AB, B ? B = ( 1 + q2 ) BB. Appendix A contains additional examples. Using these examples (or by [13, p. 10]) one obtains A ? A ? A ? B − [3]qA ? A ? B ? A+ [3]qA ? B ? A ? A−B ? A ? A ? A = 0, (6.5) B ? B ? B ? A− [3]qB ? B ? A ? B + [3]qB ? A ? B ? B −A ? B ? B ? B = 0. (6.6) 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 7→ A and B 7→ B. By construction θ(V) = U . Com- paring (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 Sn as the group of permutations of the set {1, 2, . . . , n}. For σ ∈ Sn, 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. Lemma 6.2. For an integer n ≥ 1 and a word v = v1v2 · · · vn in V, θ(v) = ∑ σ∈Sn vσ(1)vσ(2) · · · vσ(n) ∏ (i,j)∈Inv(σ) q〈vσ(i),vσ(j)〉. Proof. By induction on n, using (6.1) or (6.2). � By Lemma 6.2 we have θVn ⊆ Vn for n ∈ N. Lemma 6.3. For words u = u1u2 · · ·ur and v = v1v2 · · · vs in V, (θ(u), v) = (u, θ(v)). (6.7) For r 6= 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∑ σ ∏ (i,j)∈Inv(σ) q〈vi,vj〉, where the sum is over all σ ∈ Sr such that vk = uσ(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]. Corollary 6.4. We have θ∗ = θ. Proof. By Lemma 6.3 we have (θ(u), v) = (u, θ(v)) for u, v ∈ V. � Let the set U⊥ consist of the vectors in V that are orthogonal to U with respect to ( , ). Lemma 6.5. We have J = U⊥. An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 11 Proof. Use J = ker(θ) and Corollary 6.4, along with the fact that ( , ) is nondegenerate. Here are the details. For v ∈ V, v ∈ J ⇔ θ(v) = 0 ⇔ (θ(v),V) = 0 ⇔ (v, θ(V)) = 0 ⇔ (v, U) = 0 ⇔ v ∈ U⊥. � Note that θ(U) ⊆ U since U ⊆ V and θ(V) = U . 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 Kθ = θK, Sθ = θS, Tθ = θT. Proof. Apply each side to a word in V, and evaluate the result using Lemma 6.6. � Lemma 6.8. We have K(U) = U, S(U) = U, T (U) = U. 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 K(U) = Kθ(V) = θK(V) = θ(V) = U. � 7 The maps A`, B`, Ar, Br 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. Definition 7.1. We define four maps in End(V), denoted A`, B`, Ar, Br. (7.1) For v ∈ V, A`(v) = A ? v, B`(v) = B ? v, Ar(v) = v ? A, Br(v) = v ? B. Lemma 7.2. For n ∈ N, A`Vn ⊆ Vn+1, B`Vn ⊆ Vn+1, ArVn ⊆ Vn+1, BrVn ⊆ Vn+1. Proof. By (6.1), (6.2) and Definition 7.1. � The following lemma is about A` and B`; a similar result holds for Ar and Br. Lemma 7.3. Let W denote a subspace of U that is closed under A`, B`. Assume that 1 ∈ W . Then W = U . Proof. By Definition 7.1, and since U is the subalgebra of the q-shuffle algebra V generated by A, B. � 12 S. Post and P. Terwilliger We now consider A∗` , B∗` , A∗r , B∗r . (7.2) Lemma 7.4. For a word v = v1v2 · · · vn in V, A∗` (v) = n∑ i=0 v1 · · · vi−1vi+1 · · · vnδvi,Aq〈v1,A〉+〈v2,A〉+···+〈vi−1,A〉, B∗` (v) = n∑ i=0 v1 · · · vi−1vi+1 · · · vnδvi,Bq〈v1,B〉+〈v2,B〉+···+〈vi−1,B〉, A∗r(v) = n∑ i=0 v1 · · · vi−1vi+1 · · · vnδvi,Aq〈vn,A〉+〈vn−1,A〉+···+〈vi+1,A〉, B∗r (v) = n∑ i=0 v1 · · · vi−1vi+1 · · · vnδvi,Bq〈vn,B〉+〈vn−1,B〉+···+〈vi+1,B〉. 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. � Lemma 7.5. For n ∈ N, A∗`Vn ⊆ Vn−1, B∗`Vn ⊆ Vn−1, A∗rVn ⊆ Vn−1, B∗rVn ⊆ Vn−1. Proof. Use Lemma 7.4. � We now consider how the maps (7.1), (7.2) are related to S, T , θ. Lemma 7.6. We have SA` = ArS, SAr = A`S, SB` = BrS, SBr = B`S, (7.3) SA∗` = A∗rS, SA∗r = A∗`S, SB∗` = B∗rS, SB∗r = B∗`S. (7.4) Proof. The equations (7.3) follow from Lemma 6.6. We illustrate with a detailed proof of SA` = ArS. For v ∈ V, SA`(v) = S(A ? v) = S(v) ? S(A) = S(v) ? A = ArS(v). Therefore SA` = ArS. To obtain the equations (7.4), apply the adjoint map to each equation in (7.3). � Lemma 7.7. We have TA` = B`T, TAr = BrT, TB` = A`T, TBr = ArT, (7.5) TA∗` = B∗`T, TA∗r = B∗rT, TB∗` = A∗`T, TB∗r = A∗rT. (7.6) Proof. The equations (7.5) follow from Lemma 6.6. We illustrate with a detailed proof of TA` = B`T . For v ∈ V, TA`(v) = T (A ? v) = T (A) ? T (v) = B ? T (v) = B`T (v). Therefore TA` = B`T . To obtain the equations (7.6), apply the adjoint map to each equation in (7.5). � An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 13 Lemma 7.8. We have θAL = A`θ, θAR = Arθ, θBL = B`θ, θBR = Brθ, (7.7) θA∗` = A∗Lθ, θA∗r = A∗Rθ, θB∗` = B∗Lθ, θB∗r = B∗Rθ. (7.8) Proof. The equations (7.7) follow from the definition of θ below (6.6). We illustrate with a detailed proof of θAL = A`θ. For v ∈ V, θAL(v) = θ(Av) = θ(A) ? θ(v) = A ? θ(v) = A`θ(v). Therefore θAL = A`θ. To obtain the equations (7.8), apply the adjoint map to each equation in (7.7). � 8 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]. Lemma 8.1. For u, v ∈ V, A∗` (uv) = K(u)A∗` (v) +A∗` (u)v, B∗` (uv) = K−1(u)B∗` (v) +B∗` (u)v, A∗r(uv) = uA∗r(v) +A∗r(u)K(v), B∗r (uv) = uB∗r (v) +B∗r (u)K−1(v). Proof. 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. � Corollary 8.2. For the free algebra V, (i) A∗` and B∗rK are (K, I)-derivations; (ii) B∗` and A∗rK −1 are ( K−1, I ) -derivations; (iii) A∗r and B∗`K are (I,K)-derivations; (iv) B∗r and A∗`K −1 are ( I,K−1 ) -derivations. Proof. By Lemma 8.1 and the comments about derivations in Section 2. � Lemma 8.3. For u, v ∈ V, A∗L(u ? v) = K(u) ? A∗L(v) +A∗L(u) ? v, B∗L(u ? v) = K−1(u) ? B∗L(v) +B∗L(u) ? v, A∗R(u ? v) = u ? A∗R(v) +A∗R(u) ? K(v), B∗R(u ? v) = u ? B∗R(v) +B∗R(u) ? K−1(v). Proof. 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. � Corollary 8.4. For the q-shuffle algebra V, (i) A∗L and B∗RK are (K, I)-derivations; (ii) B∗L and A∗RK −1 are ( K−1, I ) -derivations; (iii) A∗R and B∗LK are (I,K)-derivations; (iv) B∗R and A∗LK −1 are ( I,K−1 ) -derivations. Proof. By Lemma 8.3 and the comments about derivations in Section 2. � 14 S. Post and P. Terwilliger 9 Some relations Consider the mapsK±1, (3.2), (4.1), (7.1), (7.2). In this section we describe howK±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 K, K−1, A∗L, B∗L, A∗R, B∗R, A`, B`, Ar, Br (9.1) satisfy the following relations: KK−1 = K−1K = I and KA∗L = q−2A∗LK, KB∗L = q2B∗LK, KA∗R = q−2A∗RK, KB∗R = q2B∗RK, KA` = q2A`K, KB` = q−2B`K, KAr = q2ArK, KBr = q−2BrK, A∗LA ∗ R = A∗RA ∗ L, B∗LB ∗ R = B∗RB ∗ L, A∗LB ∗ R = B∗RA ∗ L, B∗LA ∗ R = A∗RB ∗ L, A`Ar = ArA`, B`Br = BrB`, A`Br = BrA`, B`Ar = ArB`, A∗LBr = BrA ∗ L, B∗LAr = ArB ∗ L, A∗RB` = B`A ∗ R, B∗RA` = A`B ∗ R, A∗LB` = q−2B`A ∗ L, B∗LA` = q−2A`B ∗ L, A∗RBr = q−2BrA ∗ R, B∗RAr = q−2ArB ∗ R, A∗LA` − q2A`A ∗ L = I, A∗RAr − q2ArA ∗ R = I, B∗LB` − q2B`B ∗ L = I, B∗RBr − q2BrB ∗ R = I, A∗LAr −ArA∗L = K, B∗LBr −BrB∗L = K−1, A∗RA` −A`A∗R = K, B∗RB` −B`B∗R = K−1, A3 `B` − [3]qA 2 `B`A` + [3]qA`B`A 2 ` −B`A3 ` = 0, B3 `A` − [3]qB 2 `A`B` + [3]qB`A`B 2 ` −A`B3 ` = 0, A3 rBr − [3]qA 2 rBrAr + [3]qArBrA 2 r −BrA3 r = 0, B3 rAr − [3]qB 2 rArBr + [3]qBrArB 2 r −ArB3 r = 0. 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 K, K−1, AL, BL, AR, BR, A∗` , B∗` , A∗r , B∗r (9.2) satisfy the following relations: KK−1 = K−1K = I and KAL = q2ALK, KBL = q−2BLK, KAR = q2ARK, KBR = q−2BRK, KA∗` = q−2A∗`K, KB∗` = q2B∗`K, An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 15 KA∗r = q−2A∗rK, KB∗r = q2B∗rK, ALAR = ARAL, BLBR = BRBL, ALBR = BRAL, BLAR = ARBL, A∗`A ∗ r = A∗rA ∗ ` , B∗`B ∗ r = B∗rB ∗ ` , A∗`B ∗ r = B∗rA ∗ ` , B∗`A ∗ r = A∗rB ∗ ` , ALB ∗ r = B∗rAL, BLA ∗ r = A∗rBL, ARB ∗ ` = B∗`AR, BRA ∗ ` = A∗`BR, ALB ∗ ` = q2B∗`AL, BLA ∗ ` = q2A∗`BL, ARB ∗ r = q2B∗rAR, BRA ∗ r = q2A∗rBR, A∗`AL − q2ALA ∗ ` = I, A∗rAR − q2ARA ∗ r = I, B∗`BL − q2BLB ∗ ` = I, B∗rBR − q2BRB ∗ r = I, A∗rAL −ALA∗r = K, B∗rBL −BLB∗r = K−1, A∗`AR −ARA∗` = K, B∗`BR −BRB∗` = K−1, (A∗` ) 3B∗` − [3]q(A ∗ ` ) 2B∗`A ∗ ` + [3]qA ∗ `B ∗ ` (A∗` ) 2 −B∗` (A∗` ) 3 = 0, (B∗` )3A∗` − [3]q(B ∗ ` )2A∗`B ∗ ` + [3]qB ∗ `A ∗ ` (B ∗ ` )2 −A∗` (B∗` )3 = 0, (A∗r) 3B∗r − [3]q(A ∗ r) 2B∗rA ∗ r + [3]qA ∗ rB ∗ r (A∗r) 2 −B∗r (A∗r) 3 = 0, (B∗r )3A∗r − [3]q(B ∗ r )2A∗rB ∗ r + [3]qB ∗ rA ∗ r(B ∗ r )2 −A∗r(B∗r )3 = 0. 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 , (A∗L)3B∗L − [3]q(A ∗ L)2B∗LA ∗ L + [3]qA ∗ LB ∗ L(A∗L)2 −B∗L(A∗L)3 = 0, (9.3) (B∗L)3A∗L − [3]q(B ∗ L)2A∗LB ∗ L + [3]qB ∗ LA ∗ L(B∗L)2 −A∗L(B∗L)3 = 0, (9.4) (A∗R)3B∗R − [3]q(A ∗ R)2B∗RA ∗ R + [3]qA ∗ RB ∗ R(A∗R)2 −B∗R(A∗R)3 = 0, (9.5) (B∗R)3A∗R − [3]q(B ∗ R)2A∗RB ∗ R + [3]qB ∗ RA ∗ R(B∗R)2 −A∗R(B∗R)3 = 0. (9.6) 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 A∗L(U) = A∗Lθ(V) = θA∗` (V) ⊆ θ(V) = U. 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 quo- tient V/J , A3 LBL − [3]qA 2 LBLAL + [3]qALBLA 2 L −BLA3 L = 0, (9.7) B3 LAL − [3]qB 2 LALBL + [3]qBLALB 2 L −ALB3 L = 0, (9.8) A3 RBR − [3]qA 2 RBRAR + [3]qARBRA 2 R −BRA3 R = 0, (9.9) B3 RAR − [3]qB 2 RARBR + [3]qBRARB 2 R −ARB3 R = 0. (9.10) 16 S. Post and P. Terwilliger Proof. The subspace J is invariant under K±1 by Lemma 5.6. The subspace J is invariant under AL, BL, AR, BR 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 Z4 = Z/4Z of order 4. Definition 10.1. Define the algebra �∨q by generators {xi}i∈Z4 and relations qxixi+1 − q−1xi+1xi q − q−1 = 1, i ∈ Z4. (10.1) Note 10.2. The algebra �̃q from [29, Definition 6.1] is related to �∨q in the following way. There exists a surjective algebra homomorphism �̃q → �∨q that sends xi 7→ xi and c±1 i 7→ 1 for i ∈ Z4. The algebra �∨q is related to the free algebra V in the following way. Let ( �∨q )even (resp.( �∨q )odd ) denote the subalgebra of �∨q generated by x0, x2 (resp. x1, x3). Adapting the proof of [29, Proposition 6.17], we see that (i) there exists an algebra isomorphism V→ ( �∨q )even that sends A 7→ x0 and B 7→ x2; (ii) there exists an algebra isomorphism V→ ( �∨q )odd that sends A 7→ x1 and B 7→ x3; (iii) the multiplication map ( �∨q )even⊗ ( �∨q )odd → �∨q , u⊗v 7→ 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 ∈ { q2, q−2 } . Suppose we are given elements a, b, x, y, k, k−1 in any algebra such that kk−1 = k−1k = 1 and ax = xa, ay = ya, bx = xb, by = yb, (10.2) ka = rak, kb = r−1bk, kx = sxk, ky = s−1yk. (10.3) Then ( a− k−1x )3 (b− ky)− [3]q ( a− k−1x )2 (b− ky) ( a− k−1x ) + [3]q ( a− k−1x ) (b− ky) ( a− k−1x )2 − (b− ky) ( a− k−1x )3 = a3b− [3]qa 2ba+ [3]qaba 2 − ba3 + ( x3y − [3]qx 2yx+ [3]qxyx 2 − yx3 ) k−2s−4 and (b− ky)3 ( a− k−1x ) − [3]q(b− ky)2 ( a− k−1x ) (b− ky) + [3]q(b− ky) ( a− k−1x ) (b− ky)2 − ( a− k−1x ) (b− ky)3 = b3a− [3]qb 2ab+ [3]qbab 2 − ab3 + ( y3x− [3]qy 2xy + [3]qyxy 2 − xy3 ) k2s−4. An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 17 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: (i) a− k−1x, b− ky; (ii) a− xk−1, b− yk; (iii) a− kx, b− k−1y; (iv) a− xk, b− yk−1. Proof. (i) By Lemma 10.3. (ii) Apply (i) above with a, b replaced by s−1a, s−1b. (iii) Apply (i) above with k, r, s replaced by k−1, r−1, s−1. (iv) Apply (ii) above with k, r, s replaced by k−1, r−1, s−1. � 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− q2. Proposition 11.1. For each row in the tables below, the vector space V becomes a �∨q -module on which the generators {xi}i∈Z4 act as indicated. module label x0 x1 x2 x3 I AL Q(A∗` −B∗rK) BL Q ( B∗` −A∗rK−1 ) IS AR Q(A∗r −B∗`K) BR Q ( B∗r −A∗`K−1 ) IT BL Q ( B∗` −A∗rK−1 ) AL Q(A∗` −B∗rK) IST BR Q ( B∗r −A∗`K−1 ) AR Q(A∗r −B∗`K) module label x0 x1 x2 x3 II Q(AL −KBR) A∗` Q ( BL −K−1AR ) B∗` IIS Q(AR −KBL) A∗r Q ( BR −K−1AL ) B∗r IIT Q ( BL −K−1AR ) B∗` Q(AL −KBR) A∗` IIST Q ( BR −K−1AL ) B∗r Q(AR −KBL) A∗r On each �∨q -module in the tables, the actions of x1, x3 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 {xi}i∈Z4 act as indicated. module label x0 x1 x2 x3 III A` Q(A∗L −B∗RK) B` Q ( B∗L −A∗RK−1 ) IIIS Ar Q(A∗R −B∗LK) Br Q ( B∗R −A∗LK−1 ) IIIT B` Q ( B∗L −A∗RK−1 ) A` Q(A∗L −B∗RK) IIIST Br Q ( B∗R −A∗LK−1 ) Ar Q(A∗R −B∗LK) module label x0 x1 x2 x3 IV Q(A` −KBr) A∗L Q ( B` −K−1Ar ) B∗L IVS Q(Ar −KB`) A∗R Q ( Br −K−1A` ) B∗R IVT Q ( B` −K−1Ar ) B∗L Q(A` −KBr) A∗L IVST Q ( Br −K−1A` ) B∗R Q(Ar −KB`) A∗R On each �∨q -module in the tables, the actions of x0, x2 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 VI, and so on. 18 S. Post and P. Terwilliger 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 x0Vn ⊆ Vn+1, x1Vn ⊆ Vn−1, x2Vn ⊆ Vn+1, x3Vn ⊆ Vn−1 for n ∈ N. Proof. The result for x0 and x2 comes from Lemmas 3.3, 7.2. The result for x1 and x3 comes from Lemmas 4.4, 7.5. � Lemma 11.5. For each �∨q -module in Proposition 11.1, the elements x1 and x3 act on the free algebra V as a derivation of the following sort: module label x1 x3 I, II (K, I)-derivation ( K−1, I ) -derivation IS, IIS (I,K)-derivation ( I,K−1 ) -derivation IT, IIT ( K−1, I ) -derivation (K, I)-derivation IST, IIST ( I,K−1 ) -derivation (I,K)-derivation Proof. By Corollary 8.2. � Lemma 11.6. For each �∨q -module in Proposition 11.2, the elements x1 and x3 act on the q-shuffle algebra V as a derivation of the following sort: module label x1 x3 III, IV (K, I)-derivation ( K−1, I ) -derivation IIIS, IVS (I,K)-derivation ( I,K−1 ) -derivation IIIT, IVT ( K−1, I ) -derivation (K, I)-derivation IIIST, IVST ( I,K−1 ) -derivation (I,K)-derivation Proof. By Corollary 8.4. � 12 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 VI ↔ VIS, VIT ↔ VIST, VII ↔ VIIS, VIIT ↔ VIIST, VIII ↔ VIIIS, VIIIT ↔ VIIIST, VIV ↔ VIVS, VIVT ↔ VIVST. Proof. By Lemmas 5.3, 7.6 and since S is a bijection. � Lemma 12.2. The map T ∈ End(V) is an isomorphism of �∨q -modules from VI ↔ VIT, VIS ↔ VIST, VII ↔ VIIT, VIIS ↔ VIIST, VIII ↔ VIIIT, VIIIS ↔ VIIIST, VIV ↔ VIVT, VIVS ↔ VIVST. 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 VI → VIII, VIS → VIIIS, VIT → VIIIT, VIST → VIIIST, VII → VIV, VIIS → VIVS, VIIT → VIVT, VIIST → VIVST. An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 19 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 VI → VII and VIT → VIIT, and ϕ∗ is a �∨q -module isomorphism from VIII → VIV and VIIIT → VIVT. Definition 12.4. Define a map ϕ ∈ End(V) as follows. For a word v = v1v2 · · · vn in V we have ϕ(v) = v̂1v̂2 · · · v̂n(1), (12.1) where Â = Q(AL −KBR), B̂ = Q ( BL −K−1AR ) . (12.2) In particular ϕ(1) = 1. Lemma 12.5. We have ϕVn ⊆ Vn 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 = v1v2 · · · vn denote a word in V. For a subset Ω ⊆ {1, 2, . . . , n} we define a word vΩ as follows. Write Ω = {i1, i2, . . . , ik} with i1 < i2 < · · · < ik. Then vΩ = vi1vi2 · · · vik . Let Ω denote the complement of Ω in {1, 2, . . . , n}. The word vΩ is obtained from v1v2 · · · vn by deleting vj for each j ∈ Ω. Note that v∅ = 1. Lemma 12.6. For a word v = v1v2 · · · vn in V, ϕ(v) = Qn ∑ Ω vΩST (vΩ)(−1)\|Ω\|q−2\|Ω\| (∏ i,j∈Ω i<j q−〈vi,vj〉 )( ∏ i∈Ω, j∈Ω i<j q〈vi,vj〉 ) , 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. Proof. Expand the right-hand side of (12.1) using (12.2). � Lemma 12.7. We have Tϕ = ϕT and Tϕ∗ = ϕ∗T . Proof. We first show that Tϕ = ϕT . By Definition 5.4, T (A) = B and T (B) = A. By Lemma 5.5 and (12.2), TÂ = B̂T and TB̂ = ÂT . By these comments TÂ = T̂ (A)T and TB̂ = T̂ (B)T . We show that Tϕ(v) = ϕT (v) for all v ∈ V. Without loss of generality, we may assume that v is a word in V. Write v = v1v2 · · · vn. By Definition 12.4 and the above comments, Tϕ(v) = Tϕ(v1v2 · · · vn) = T v̂1v̂2 · · · v̂n(1) = T̂ (v1)T̂ (v2) · · · T̂ (vn)T (1) = T̂ (v1)T̂ (v2) · · · T̂ (vn)(1) = ϕ ( T (v1)T (v2) · · ·T (vn) ) = ϕT (v1v2 · · · vn) = ϕT (v). We have shown that Tϕ = ϕT . In this equation, apply the adjoint map to each side and use T ∗ = T to obtain Tϕ∗ = ϕ∗T . � Lemma 12.8. On V, ϕAL = Q(AL −KBR)ϕ, ϕBL = Q ( BL −K−1AR ) ϕ, (12.3) ϕ(A∗` −B∗rK)Q = A∗`ϕ, ϕ ( B∗` −A∗rK−1 ) Q = B∗`ϕ. (12.4) 20 S. Post and P. Terwilliger Proof. We first obtain (12.3). For x ∈ {A,B} we show that ϕxL = x̂ϕ, where x̂ is from (12.2). It suffices to show that ϕxL(v) = x̂ϕ(v) for all words v in V. Let the word v be given, and write v = v1v2 · · · vn. Using (12.1), ϕxL(v) = ϕxL(v1v2 · · · vn) = ϕ(xv1v2 · · · vn) = x̂v̂1v̂2 · · · v̂n(1) = x̂ϕ(v). 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 x = AL, y = Q(A∗` −B∗rK), z = BL, X = Q(AL −KBR), Y = A∗` , Z = Q ( BL −K−1AR ) . Note that ∆ = ϕy − Y ϕ. Referring to Proposition 11.1, from the VI data qxy − q−1yx q − q−1 = 1, qyz − q−1zy q − q−1 = 1, and from the VII data qXY − q−1Y X q − q−1 = 1, qY Z − q−1ZY q − q−1 = 1. By (12.3), ϕx = Xϕ, ϕz = Zϕ. We will show that X∆ = q−2∆x, Z∆ = q2∆z. (12.5) We have X∆ = Xϕy −XY ϕ = ϕxy −XY ϕ = q−2(ϕyx− Y Xϕ) = q−2(ϕyx− Y ϕx) = q−2∆x. Similarly Z∆ = Zϕy − ZY ϕ = ϕzy − ZY ϕ = q2(ϕyz − Y Zϕ) = q2(ϕyz − Y ϕz) = q2∆z. 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 (12.5), W is invariant under x = AL and z = BL. Note that ∆(1) = 0, since y(1) = 0, Y (1) = 0, ϕ(1) = 1. Therefore 1 ∈ W . By these comments and Lemma 3.4 we obtain W = V, so ∆ = 0. We have obtained the equation on the left in (12.4). In this equation, multiply each side on the left by T and on the right by T−1. Simplify the result using the first equation in Lemma 12.7, together with the equations TKT−1 = K−1, TA∗`T −1 = B∗` , TB∗rT −1 = A∗r from Lemmas 5.5, 7.7. This yields the equation on the right in (12.4). � Corollary 12.9. The map ϕ from Definition 12.4 is a homomorphism of �∨q -modules from VI → VII and VIT → VIIT. 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 ∈ Z4 the relation ϕxi = xiϕ holds on VI and VIT. The result follows. � An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 21 Next we consider the adjoint ϕ∗. Lemma 12.10. We have ϕ∗Vn ⊆ Vn for n ∈ N. Moreover ϕ∗(1) = 1. 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 V0. � Lemma 12.11. On V, A∗Lϕ ∗ = ϕ∗(A∗L −B∗RK)Q, B∗Lϕ ∗ = ϕ∗ ( B∗L −A∗RK−1 ) Q, Q(A` −KBr)ϕ∗ = ϕ∗A`, Q ( B` −K−1Ar ) ϕ∗ = ϕ∗B`. Proof. For each equation in Lemma 12.8, apply the adjoint map to each side. � Corollary 12.12. The map ϕ∗ is a homomorphism of �∨q -modules from VIII → VIV and VIIIT → VIVT. 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 ∈ Z4 the relation ϕ∗xi = xiϕ ∗ holds on VIII and VIIIT. The result follows. � Lemma 12.13. We have ϕ∗θ = θϕ. Proof. We define ∆ = ϕ∗θ − θϕ and show that ∆ = 0. By Lemma 12.3 and Corolla- ries 12.9, 12.12 we find that each of ϕ∗θ and θϕ is a �∨q -module homomorphism VI → VIV. So ∆ is a �∨q -module homomorphism VI → VIV. By Proposition 11.1, x0 and x2 act on VI as AL and BL, respectively. By Proposition 11.2, x0 and x2 act on VIV as Q(A` −KBr) and Q ( B` −K−1Ar ) , respectively. By these comments ∆AL = Q(A` −KBr)∆, ∆BL = Q ( B` −K−1Ar ) ∆. (12.6) Let W denote the kernel of ∆ on V. By (12.6), W is invariant under AL and BL. 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. Definition 12.14. Define ψ ∈ End(V) by ψ = KBRA ∗ L +K−1ARB ∗ L. (12.7) Lemma 12.15. We have ψV0 = 0, and ψVn ⊆ Vn for n ≥ 1. Proof. By Lemmas 3.3, 4.4. � Lemma 12.16. The map ψ acts on the standard basis for V as follows. We have ψ(1) = 0. For a nontrivial word v = v1v2 · · · vn in V, ψ(v) = v2v3 · · · vnT (v1)q〈v1,v2〉+〈v1,v3〉+···+〈v1,vn〉−2. Proof. Use Definition 5.1 and the comments below it, along with Lemma 4.3 and Defini- tion 12.14. � Lemma 12.17. The following (i)–(iv) hold for n ≥ 1. (i) The equation ψn = q−2nT holds on Vn. 22 S. Post and P. Terwilliger (ii) The equation ψ2n = q−4nI holds on Vn. (iii) The map I − ψ is invertible on Vn. (iv) On Vn, I − ψ = (AL −KBR)A∗L + ( BL −K−1AR ) B∗L. Proof. (i) Pick a word v = v1v2 · · · vn in V. Repeatedly applying Lemma 12.16 we obtain ψn(v) = T (v1)T (v2) · · ·T (vn)q−2n = T (v)q−2n. The result follows. (ii) By (i) above and since T 2 = I. (iii) Pick a vector v ∈ Vn such that (I − ψ)v = 0. We show that v = 0. By construction ψv = v. By this and (ii) we obtain v = ψ2nv = q−4nv. So ( 1− q−4n ) v = 0. In this equation the coefficient of v is nonzero, so v = 0. (iv) By (12.7) and since I = ALA ∗ L +BLB ∗ L on Vn. � Lemma 12.18. For n ≥ 1 we have Vn = (AL −KBR)Vn−1 + ( BL −K−1AR ) Vn−1. 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 Vn = (I − ψ)Vn = ( (AL −KBR)A∗L + ( BL −K−1AR ) B∗L ) Vn ⊆ (AL −KBR)A∗LVn + ( BL −K−1AR ) B∗LVn ⊆ (AL −KBR)Vn−1 + ( BL −K−1AR ) Vn−1. � Lemma 12.19. For n ∈ N we have (i) ϕVn = Vn; (ii) ϕ∗Vn = Vn. Proof. (i) By induction on n. For n = 0 the result holds, because 1 is a basis for V0 and ϕ(1) = 1. Next assume that n ≥ 1. The sum Vn = AVn−1 + BVn−1 is direct. In this equation apply ϕ to each side. By Lemma 12.8 and induction, ϕ(AVn−1) = ϕALVn−1 = (AL −KBR)ϕVn−1 = (AL −KBR)Vn−1, ϕ(BVn−1) = ϕBLVn−1 = ( BL −K−1AR ) ϕVn−1 = ( BL −K−1AR ) Vn−1. By these comments and Lemma 12.18, ϕVn = ϕ(AVn−1) + ϕ(BVn−1) = (AL −KBR)Vn−1 + ( BL −K−1AR ) Vn−1 = Vn. (ii) By Lemma 12.10 and (i) above, along with the fact that the dimension of Vn is finite. � Lemma 12.20. Each of ϕ, ϕ∗ is a bijection. Proof. By (3.1) and Lemma 12.19. � Proposition 12.21. The map ϕ is an isomorphism of �∨q -modules from VI → VII and VIT → VIIT. Moreover ϕ∗ is an isomorphism of �∨q -modules from VIII → VIV and VIIIT → VIVT. Proof. By Corollaries 12.9, 12.12 and Lemma 12.20. � An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 23 Lemma 12.22. We have ϕ(J) = J . Proof. We invoke Lemmas 12.13, 12.20 and J = ker(θ). For v ∈ V, v ∈ J ⇔ θ(v) = 0 ⇔ ϕ∗θ(v) = 0 ⇔ θϕ(v) = 0 ⇔ ϕ(v) ∈ J. The result follows. � Lemma 12.23. We have ϕ∗(U) = U . Proof. By Lemmas 12.13, 12.20 and θ(V) = U , ϕ∗(U) = ϕ∗θ(V) = θϕ(V) = 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. Lemma 13.1. The following (i), (ii) hold. (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 x0, x2 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 x1, x3 satisfy the q-Serre relations. Proof. (i) Use Propositions 9.4, 11.1 and Corollary 10.4. (ii) Use Propositions 9.3, 11.2 and Corollary 10.4. � Definition 13.2 (see [29, Definition 5.1]). Define the algebra �q by generators {xi}i∈Z4 and relations qxixi+1 − q−1xi+1xi q − q−1 = 1, x3 ixi+2 − [3]qx 2 ixi+2xi + [3]qxixi+2x 2 i − xi+2x 3 i = 0. Lemma 13.3. There exists an algebra homomorphism �∨q → �q that sends xi 7→ xi for i ∈ Z4. This homomorphism is surjective. Proof. Compare Definitions 10.1, 13.2. � 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 x0, x2 (resp. x1, x3). Then by [29, Proposition 5.5], (i) there exists an algebra isomorphism U+ q → �even q that sends A 7→ x0 and B 7→ x2; (ii) there exists an algebra isomorphism U+ q → �odd q that sends A 7→ x1 and B 7→ x3; (iii) the multiplication map �even q ⊗�odd q → �q, u⊗v 7→ uv is an isomorphism of vector spaces. 24 S. Post and P. Terwilliger Theorem 13.4. For each row in the tables below, the vector space V/J becomes a �q-module on which the generators {xi}i∈Z4 act as indicated. module label x0 x1 x2 x3 I AL Q(A∗` −B∗rK) BL Q ( B∗` −A∗rK−1 ) IS AR Q(A∗r −B∗`K) BR Q ( B∗r −A∗`K−1 ) IT BL Q ( B∗` −A∗rK−1 ) AL Q(A∗` −B∗rK) IST BR Q ( B∗r −A∗`K−1 ) AR Q(A∗r −B∗`K) module label x0 x1 x2 x3 II Q(AL −KBR) A∗` Q ( BL −K−1AR ) B∗` IIS Q(AR −KBL) A∗r Q ( BR −K−1AL ) B∗r IIT Q ( BL −K−1AR ) B∗` Q(AL −KBR) A∗` IIST Q ( BR −K−1AL ) B∗r Q(AR −KBL) A∗r Proof. By Proposition 11.1 and Lemma 13.1(i). � Theorem 13.5. For each row in the tables below, the vector space U becomes a �q-module on which the generators {xi}i∈Z4 act as indicated. module label x0 x1 x2 x3 III A` Q(A∗L −B∗RK) B` Q ( B∗L −A∗RK−1 ) IIIS Ar Q(A∗R −B∗LK) Br Q ( B∗R −A∗LK−1 ) IIIT B` Q ( B∗L −A∗RK−1 ) A` Q(A∗L −B∗RK) IIIST Br Q ( B∗R −A∗LK−1 ) Ar Q(A∗R −B∗LK) module label x0 x1 x2 x3 IV Q(A` −KBr) A∗L Q ( B` −K−1Ar ) B∗L IVS Q(Ar −KB`) A∗R Q ( Br −K−1A` ) B∗R IVT Q ( B` −K−1Ar ) B∗L Q(A` −KBr) A∗L IVST Q ( Br −K−1A` ) B∗R Q(Ar −KB`) A∗R Proof. By Proposition 11.2 and Lemma 13.1(ii). � Note 13.6. Going forward, the �q-module V/J with label I will be denoted (V/J)I, and so on. Proposition 13.7. For each �q-module in Theorem 13.4 the elements x1 and x3 act on the algebra V/J as a derivation of the following sort: module label x1 x3 I, II (K, I)-derivation ( K−1, I ) -derivation IS, IIS (I,K)-derivation ( I,K−1 ) -derivation IT, IIT ( K−1, I ) -derivation (K, I)-derivation IST, IIST ( I,K−1 ) -derivation (I,K)-derivation Proof. By Lemma 11.5 and the comments about derivations in Section 2. � Proposition 13.8. For each �q-module in Theorem 13.5 the elements x1 and x3 act on the algebra U as a derivation of the following sort: module label x1 x3 III, IV (K, I)-derivation ( K−1, I ) -derivation IIIS, IVS (I,K)-derivation ( I,K−1 ) -derivation IIIT, IVT ( K−1, I ) -derivation (K, I)-derivation IIIST, IVST ( I,K−1 ) -derivation (I,K)-derivation Proof. By Lemma 11.6 and the comments about derivations in Section 2. � An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 25 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. Proposition 14.1. The map V/J → V/J , x+J 7→ S(x) +J is an isomorphism of �q-modules from (V/J)I ↔ (V/J)IS, (V/J)IT ↔ (V/J)IST, (V/J)II ↔ (V/J)IIS, (V/J)IIT ↔ (V/J)IIST. The map U → U , x 7→ S(x) is an isomorphism of �q-modules from UIII ↔ UIIIS, UIIIT ↔ UIIIST, UIV ↔ UIVS, UIVT ↔ UIVST. Proof. By Lemma 12.1, together with the fact that S(J) = J by (5.6) and S(U) = U by Lemma 6.8. � Proposition 14.2. The map V/J → V/J , x+J 7→ T (x) +J is an isomorphism of �q-modules from (V/J)I ↔ (V/J)IT, (V/J)IS ↔ (V/J)IST, (V/J)II ↔ (V/J)IIT, (V/J)IIS ↔ (V/J)IIST. The map U → U , x 7→ T (x) is an isomorphism of �q-modules from UIII ↔ UIIIT, UIIIS ↔ UIIIST, UIV ↔ UIVT, UIVS ↔ UIVST. Proof. By Lemma 12.2, together with the fact that T (J) = J by (5.6) and T (U) = U by Lemma 6.8. � Proposition 14.3. The map V/J → U , x+ J 7→ θ(x) is an isomorphism of �q-modules from (V/J)I → UIII, (V/J)IS → UIIIS, (V/J)IT → UIIIT, (V/J)IST → UIIIST, (V/J)II → UIV, (V/J)IIS → UIVS, (V/J)IIT → UIVT, (V/J)IIST → UIVST. Proof. By Lemma 12.3 and the fact that J is the kernel of θ. � Proposition 14.4. The map V/J → V/J , x+J 7→ ϕ(x) + J is an isomorphsim of �q-modules from (V/J)I → (V/J)II, (V/J)IT → (V/J)IIT. The map U → U , x 7→ ϕ∗(x) is an isomorphism of �q-modules from UIII → UIV, UIIIT → UIVT. Proof. By Proposition 12.21 and Lemmas 12.22, 12.23. � Theorem 14.5. The following �q-modules are mutually isomorphic: (V/J)I, (V/J)IS, (V/J)IT, (V/J)IST, (V/J)II, (V/J)IIS, (V/J)IIT, (V/J)IIST, UIII, UIIIS, UIIIT, UIIIST, UIV, UIVS, UIVT, UIVST. 26 S. Post and P. Terwilliger Proof. By Propositions 14.1–14.4. � 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 UIV. Then 1 ∈W . Proof. By Proposition 11.2 the generators x1 and x3 act on UIV 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 UIII. Then 1 6∈W . Proof. We assume 1 ∈ W and get a contradiction. By Proposition 11.2 the generators x0 and x2 act on UIII as A` and B`, respectively. By Lemma 7.3, W is not proper. This contradicts our assumptions. � Theorem 14.8. The �q-modules in Theorem 14.5 are irreducible. Proof. By Theorem 14.5 along with Lemmas 14.6, 14.7 we find that none of the listed �q- modules contain a nonzero proper �q-submodule. The result follows. � 15 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), x1x0 = q2x0x1 + 1− q2, x1x2 = q−2x2x1 + 1− q−2, x3x2 = q2x2x3 + 1− q2, x3x0 = q−2x0x3 + 1− q−2. Next we express these relations in a uniform way. Lemma 15.1. For u ∈ {x0, x2} and v ∈ {x1, x3} the following holds in �∨q : vu = uvq〈u,v〉 + 1− q〈u,v〉, where 〈 , 〉 x1 x3 x0 2 −2 x2 −2 2 The following formula will be useful. Lemma 15.2. Pick n ∈ N. Referring to the algebra �∨q , pick ui ∈ {x0, x2} for 1 ≤ i ≤ n, and also v ∈ {x1, x3}. Then vu1u2 · · ·un = u1u2 · · ·unvq〈u1,v〉+〈u2,v〉+···+〈un,v〉 + n∑ i=1 u1 · · ·ui−1ui+1 · · ·unq〈u1,v〉+···+〈ui−1,v〉 ( 1− q〈ui,v〉 ) , where 〈 , 〉 is from Lemma 15.1. Proof. By Lemma 15.1 and induction on n. � An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 27 Corollary 15.3. Let V denote a �∨q -module. Pick ξ ∈ V such that x1ξ = 0 and x3ξ = 0. Then for n ∈ N and u1, u2, . . . , un ∈ {x0, x2} and v ∈ {x1, x3}, vu1u2 · · ·unξ = n∑ i=1 u1 · · ·ui−1ui+1 · · ·unξq〈u1,v〉+···+〈ui−1,v〉 ( 1− q〈ui,v〉 ) , where 〈 , 〉 is from Lemma 15.1. 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 x0, x2. 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 7→ x0 and B 7→ x2. Denote this isomorphism by κ. Recall the �∨q -module VI from Proposition 11.1. Lemma 15.4. Let V denote a �∨q -module. Pick ξ ∈ V such that x1ξ = 0 and x3ξ = 0. Then the map VI → V , v 7→ κ(v)ξ is a �∨q -module homomorphism. Proof. We show that for i ∈ Z4 the following diagram commutes: VI v 7→κ(v)ξ−−−−−→ V xi y yxi VI −−−−−→ v 7→κ(v)ξ V. Referring to the above diagram, the action xi : VI → VI is described in Proposition 11.1 along with Definition 5.1 and Lemma 7.4. The action xi : 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 x3 ixi+2 − [3]qx 2 ixi+2xi + [3]qxixi+2x 2 i − xi+2x 3 i = 0, i ∈ Z4. Lemma 15.5. Let V denote a �q-module that contains a nonzero vector ξ such that x1ξ = 0 and x3ξ = 0. Then the map in Lemma 15.4 has kernel J . Moreover the map (V/J)I → V , v + J 7→ κ(v)ξ is an injective �q-module homomorphism. Proof. Let L denote the kernel of the map in Lemma 15.4. This map sends 1 7→ ξ, and ξ is nonzero, so 1 6∈ L. Observe that L is a left ideal of the free algebra V. The set H = {v ∈ V\|κ(v)V = 0} is a 2-sided ideal of the free algebra V. By construction H ⊆ L. Recall the elements J± from (3.3), (3.4). We have κ(J+) = x3 0x2 − [3]qx 2 0x2x0 + [3]qx0x2x 2 0 − x2x 3 0, (15.1) κ(J−) = x3 2x0 − [3]qx 2 2x0x2 + [3]qx2x0x 2 2 − x0x 3 2. (15.2) 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 7→ κ(v)ξ is a �q-module homomorphism that has kernel L/J . This kernel L/J is a �q- submodule 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 6= V since 1 6∈ L, so L = J . Consequently the map (V/J)I → V , v + J 7→ κ(v)ξ is injective. The result follows. � 28 S. Post and P. Terwilliger 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 x1ξ = 0 and x3ξ = 0. Then the map (V/J)I → V , v + J 7→ κ(v)ξ is an isomorphism of �q-modules. Proof. By Lemma 15.5, the map (V/J)I → V , v + J 7→ 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)I → V , v + J 7→ κ(v)ξ is an isomorphism of �q-modules. � 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 x1ξ = 0 and x3ξ = 0. Proof. (i) ⇒ (ii): Without loss of generality, we may identify the �q-module V with the �q-module UIII listed in Theorem 14.5. The vector ξ = 1 has the desired properties. (ii)⇒ (i): By Theorem 14.5 and Proposition 15.6. � Definition 15.8. 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 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 infinite- dimensional; 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 7→ uξ is an iso- morphism of vector spaces. Proof. By Proposition 15.6. � Theorem 15.10. The �q-module U has a unique sequence of subspaces {Un}n∈N such that: (i) U0 6= 0; (ii) the sum U = ∑ n∈N Un is direct; (iii) for n ∈ N, x0Un ⊆ Un+1, x1Un ⊆ Un−1, x2Un ⊆ Un+1, x3Un ⊆ Un−1, where U−1 = 0. The sequence {Un}n∈N is described as follows. The subspace U0 has dimension 1. The nonzero vectors in U0 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, Un is spanned by the vectors u1u2 · · ·unξ, ui ∈ {x0, x2}, 1 ≤ i ≤ n. (15.3) An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 29 Proof. Concerning existence, without loss of generality we may identify the �q-module U with the �q-module UIII 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 Un = U ∩ Vn. One checks that the sequence {Un}n∈N satisfies the above conditions (i)–(iii). We have established existence. Going forward let {Un}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 ∈ U0. To prove the claim, note by condition (ii) that there exists n ∈ N and ξi ∈ Ui (0 ≤ i ≤ n) such that ξn 6= 0 and ξ = n∑ i=0 ξi. Since ξ is NIL, x1ξ = 0 and x3ξ = 0. In the sum ξ = n∑ i=0 ξi, apply x1 to each term and use condition (iii) to find x1ξn = 0. Similarly x3ξn = 0, so ξn 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 ξn. By Proposition 15.6 the map (V/J)I → U, v + J 7→ κ(v)ξn 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 Un+i. By these comments and (i), (ii) we obtain n = 0, so ξ = ξ0 ∈ U0. 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 = Un for n ∈ N. By (iii) we have U′n ⊆ Un 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 u1u2 · · ·unξ, n ∈ N, ui ∈ {x0, x2}, 1 ≤ i ≤ n. So U = ∑ n∈N U′n. By these comments and (ii) we obtain U′n = Un for n ∈ N. The claim is proven. By the claims, the sequence {Un}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. A ? A ? B A ? B ? A B ? A ? A AAB q[2]q q−1[2]q q−3[2]q ABA q−1[2]q q−1[2]q q−1[2]q BAA q−3[2]q q−1[2]q q[2]q B ? B ? A B ? A ? B A ? B ? B BBA q[2]q q−1[2]q q−3[2]q BAB q−1[2]q q−1[2]q q−1[2]q ABB q−3[2]q q−1[2]q q[2]q A ? A ? A ? B A ? A ? B ? A A ? B ? A ? A B ? A ? A ? A AAAB q3[3]q[2]q q[3]q[2]q q−1[3]q[2]q q−3[3]q[2]q AABA q[3]q[2]q ( 2q + q−1 ) [2]q ( q + 2q−1 ) [2]q q−1[3]q[2]q ABAA q−1[3]q[2]q ( q + 2q−1 ) [2]q ( 2q + q−1 ) [2]q q[3]q[2]q BAAA q−3[3]q[2]q q−1[3]q[2]q q[3]q[2]q q3[3]q[2]q B ? B ? B ? A B ? B ? A ? B B ? A ? B ? B A ? B ? B ? B BBBA q3[3]q[2]q q[3]q[2]q q−1[3]q[2]q q−3[3]q[2]q BBAB q[3]q[2]q ( 2q + q−1 ) [2]q ( q + 2q−1 ) [2]q q−1[3]q[2]q BABB q−1[3]q[2]q ( q + 2q−1 ) [2]q ( 2q + q−1 ) [2]q q[3]q[2]q ABBB q−3[3]q[2]q q−1[3]q[2]q q[3]q[2]q q3[3]q[2]q 30 S. Post and P. Terwilliger 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 2r × 2s. 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 AL : ( 1 0 ) ,  1 0 0 1 0 0 0 0  ,  1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0  , A∗L : ( 1 0 ) , ( 1 0 0 0 0 1 0 0 ) ,  1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0  , BL : ( 0 1 ) ,  0 0 0 0 1 0 0 1  ,  0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  , B∗L : ( 0 1 ) , ( 0 0 1 0 0 0 0 1 ) ,  0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1  , AR : ( 1 0 ) ,  1 0 0 0 0 1 0 0  ,  1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0  , An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 31 A∗R : ( 1 0 ) , ( 1 0 0 0 0 0 1 0 ) ,  1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0  , BR : ( 0 1 ) ,  0 0 1 0 0 0 0 1  ,  0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1  , B∗R : ( 0 1 ) , ( 0 1 0 0 0 0 0 1 ) ,  0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1  , A` : ( 1 0 ) ,  q[2]q 0 0 1 0 q−2 0 0  ,  q2[3]q 0 0 0 0 q[2]q 0 0 0 1 1 0 0 0 q−1[2]q 0 0 0 0 1 0 0 0 q−2 0 0 0 q−4 0 0 0 0  , A∗` : ( 1 0 ) , ( q[2]q 0 0 0 0 1 q−2 0 ) ,  q2[3]q 0 0 0 0 0 0 0 0 q[2]q 1 0 0 0 0 0 0 0 1 q−1[2]q 0 0 0 0 0 0 0 0 1 q−2 q−4 0  , B` : ( 0 1 ) ,  0 0 q−2 0 1 0 0 q[2]q  ,  0 0 0 0 q−4 0 0 0 q−2 0 0 0 1 0 0 0 0 q−1[2]q 0 0 0 1 1 0 0 0 q[2]q 0 0 0 0 q2[3]q  , B∗` : ( 0 1 ) , ( 0 q−2 1 0 0 0 0 q[2]q ) ,  0 q−4 q−2 1 0 0 0 0 0 0 0 0 q−1[2]q 1 0 0 0 0 0 0 0 1 q[2]q 0 0 0 0 0 0 0 0 q2[3]q  , Ar : ( 1 0 ) ,  q[2]q 0 0 q−2 0 1 0 0  ,  q2[3]q 0 0 0 0 q−1[2]q 0 0 0 1 1 0 0 0 q[2]q 0 0 0 0 q−4 0 0 0 q−2 0 0 0 1 0 0 0 0  , 32 S. Post and P. Terwilliger A∗r : ( 1 0 ) , ( q[2]q 0 0 0 0 q−2 1 0 ) ,  q2[3]q 0 0 0 0 0 0 0 q−1[2]q 1 0 0 0 0 0 0 0 1 q[2]q 0 0 0 0 0 0 0 0 q−4 q−2 1 0  , Br : ( 0 1 ) ,  0 0 1 0 q−2 0 0 q[2]q  ,  0 0 0 0 1 0 0 0 q−2 0 0 0 q−4 0 0 0 0 q[2]q 0 0 0 1 1 0 0 0 q−1[2]q 0 0 0 0 q2[3]q  , B∗r : ( 0 1 ) , ( 0 1 q−2 0 0 0 0 q[2]q ) ,  0 1 q−2 q−4 0 0 0 0 0 0 0 0 q[2]q 1 0 0 0 0 0 0 0 1 q−1[2]q 0 0 0 0 0 0 0 0 q2[3]q  , K : ( 1 ) , ( q2 0 0 q−2 ) ,  q4 0 0 0 0 1 0 0 0 0 1 0 0 0 0 q−4  ,  q6 0 0 0 0 0 0 0 0 q2 0 0 0 0 0 0 0 0 q2 0 0 0 0 0 0 0 0 q2 0 0 0 0 0 0 0 0 q−2 0 0 0 0 0 0 0 0 q−2 0 0 0 0 0 0 0 0 q−2 0 0 0 0 0 0 0 0 q−6  , S : ( 1 ) , ( 1 0 0 1 ) ,  1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1  ,  1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1  , T : ( 1 ) , ( 0 1 1 0 ) ,  0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0  ,  0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0  , θ : ( 1 ) , ( 1 0 0 1 ) ,  q[2]q 0 0 0 0 1 q−2 0 0 q−2 1 0 0 0 0 q[2]q  , An Infinite-Dimensional �q-Module Obtained from the q-Shuffle Algebra for Affine sl2 33 [2]q  q3[3]q 0 0 0 0 0 0 0 0 q q−1 q−3 0 0 0 0 0 q−1 q−1 q−1 0 0 0 0 0 q−3 q−1 q 0 0 0 0 0 0 0 0 q q−1 q−3 0 0 0 0 0 q−1 q−1 q−1 0 0 0 0 0 q−3 q−1 q 0 0 0 0 0 0 0 0 q3[3]q  , ϕ : ( 1 ) , Q ( 1 −q−2 −q−2 1 ) , Q2  1 −q−2 −q−4 q−6 −q−1[2]q q−1[2]q 0 0 0 0 q−1[2]q −q−1[2]q q−6 −q−4 −q−2 1  , Q3  1 −q−2 −q−4 −q−6 q−6 q−8 q−10 −q−12 −[3]q 2 + q−2 q−2 0 −q−4 0 0 0 0 0 q−1[2]q q−3[2]q −q−1[2]q −q−3[2]q 0 0 0 0 0 1 0 −q−2 −q−2 − 2q−4 q−4[3]q q−4[3]q −q−2 − 2q−4 −q−2 0 1 0 0 0 0 0 −q−3[2]q −q−1[2]q q−3[2]q q−1[2]q 0 0 0 0 0 −q−4 0 q−2 2 + q−2 −[3]q −q−12 q−10 q−8 q−6 −q−6 −q−4 −q−2 1  , ϕ∗ : ( 1 ) , Q ( 1 −q−2 −q−2 1 ) , Q2  1 −q−1[2]q 0 q−6 −q−2 q−1[2]q 0 −q−4 −q−4 0 q−1[2]q −q−2 q−6 0 −q−1[2]q 1  , Q3  1 −[3]q 0 0 q−4[3]q 0 0 −q−12 −q−2 2 + q−2 0 0 −q−2 − 2q−4 0 0 q−10 −q−4 q−2 q−1[2]q 0 −q−2 −q−3[2]q 0 q−8 −q−6 0 q−3[2]q 1 0 −q−1[2]q −q−4 q−6 q−6 −q−4 −q−1[2]q 0 1 q−3[2]q 0 −q−6 q−8 0 −q−3[2]q −q−2 0 q−1[2]q q−2 −q−4 q−10 0 0 −q−2 − 2q−4 0 0 2 + q−2 −q−2 −q−12 0 0 q−4[3]q 0 0 −[3]q 1  . 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Theory , to appear, arXiv:1612.02864. https://doi.org/10.3842/SIGMA.2018.044 https://arxiv.org/abs/1801.06083 https://doi.org/10.1142/S0219498804000940 https://arxiv.org/abs/math.QA/0305356 https://doi.org/10.1016/j.laa.2017.10.002 https://arxiv.org/abs/1706.00518 https://doi.org/10.1007/s10468-019-09862-y https://arxiv.org/abs/1612.02864 1 Introduction 2 Preliminaries 3 The free algebra V with generators A, B 4 A bilinear form on V 5 Some automorphisms and antiautomorphisms 6 The q-shuffle algebra V and the map 7 The maps A, B, Ar, Br 8 Derivations 9 Some relations 10 The algebra q 11 Sixteen q-module structures on V 12 Some homomorphisms between the sixteen q-modules 13 Sixteen q-module structures on V/J or U 14 The sixteen q-modules are mutually isomorphic and irreducible 15 The NIL modules for q A Data on the q-shuffle product B Some matrix representations References
id	nasplib_isofts_kiev_ua-123456789-210713
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2025-12-17T12:04:33Z
publishDate	2020
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spelling	Post, Sarah Terwilliger, Paul 2025-12-15T15:29:22Z 2020 An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂. Sarah Post and Paul Terwilliger. SIGMA 16 (2020), 037, 35 pages 1815-0659 2020 Mathematics Subject Classification: 17B37 arXiv:1806.10007 https://nasplib.isofts.kiev.ua/handle/123456789/210713 https://doi.org/10.3842/SIGMA.2020.037 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⁻¹xᵢ₊₁xᵢ)/(q−q⁻¹) = 1, x³ᵢxᵢ₊₂ − [3]qx²ᵢxᵢ + ₂xᵢ + [3]qxᵢxᵢ₊₂x²ᵢ − xᵢ₊₂x³ᵢ=0, where [3]q=(q³−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≠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. The first author acknowledges support by the Simons Foundation Collaboration Grant 3192112. The second author thanks Marc Rosso and Xin Fang for helpful comments about q-shuffle algebras. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ Article published earlier
spellingShingle	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂ Post, Sarah Terwilliger, Paul
title	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂
title_full	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂
title_fullStr	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂
title_full_unstemmed	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂
title_short	An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂
title_sort	infinite-dimensional □q-module obtained from the q-shuffle algebra for affine sl₂
url	https://nasplib.isofts.kiev.ua/handle/123456789/210713
work_keys_str_mv	AT postsarah aninfinitedimensionalqmoduleobtainedfromtheqshufflealgebraforaffinesl2 AT terwilligerpaul aninfinitedimensionalqmoduleobtainedfromtheqshufflealgebraforaffinesl2 AT postsarah infinitedimensionalqmoduleobtainedfromtheqshufflealgebraforaffinesl2 AT terwilligerpaul infinitedimensionalqmoduleobtainedfromtheqshufflealgebraforaffinesl2

An Infinite-Dimensional □q-Module Obtained from the q-Shuffle Algebra for Affine sl₂

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