The universal Askey-Wilson algebra

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Linda Holmes
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1 University of Wisconsin-Madison

2 Overview This talk concerns an algebra called the Universal Askey-Wilson algebra. As we will see, is related to: Leonard pairs and Leonard triples of QRacah type Q-polynomial distance-regular graphs of QRacah type The modular group PSL 2 (Z) The equitable presentation of the quantum group U q (sl 2 ) The double affine Hecke algebra of type (C 1, C 1)

3 Motivation: Leonard pairs We recall the notion of a Leonard pair. To do this, we first recall what it means for a matrix to be tridiagonal. The following matrices are tridiagonal , Tridiagonal means each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. The tridiagonal matrix on the left is irreducible. This means each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero.

4 The Definition of a Leonard Pair We now define a Leonard pair. From now on F will denote a field. Definition (Terwilliger 1999) Let V denote a vector space over F with finite positive dimension. By a Leonard pair on V, we mean a pair of linear transformations A : V V and B : V V such that: 1 There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal. 2 There exists a basis for V with respect to which the matrix representing B is diagonal and the matrix representing A is irreducible tridiagonal.

5 Leonard pairs in summary In summary, for a Leonard pair A, B A B basis 1 diagonal irred. tridiagonal basis 2 irred. tridiagonal diagonal

6 Leonard pairs and orthogonal polynomials The term Leonard pair is motivated by a 1982 theorem of Doug Leonard concerning the QRacah polynomials and some related polynomials in the Askey scheme. For a detailed version of Leonard s theorem see the book E. Bannai and T. Ito. Algebraic Combinatorics I: Association Schemes. Benjamin Cummings. London 1984.

7 Leonard pair example Here is an example of a Leonard pair. Fix an integer d 3. Pick nonzero scalars a, b, c, q in F such that (i) q 2i 1 for 1 i d; (ii) Neither of a 2, b 2 is among q 2d 2, q 2d 4,..., q 2 2d ; (iii) None of abc, a 1 bc, ab 1 c, abc 1 is among q d 1, q d 3,..., q 1 d.

8 Leonard pair example, cont. Define for 0 i d and θ i = aq 2i d + a 1 q d 2i, θi = bq 2i d + b 1 q d 2i ϕ i = a 1 b 1 q d+1 (q i q i )(q i d 1 q d i+1 ) for 1 i d. (q i abcq i d 1 )(q i abc 1 q i d 1 )

9 Leonard pair example, cont. Define A = B = θ θ 1 1 θ θ d θ 0 ϕ 1 0 θ 1 ϕ 2 θ 2 ϕ d 0 θ d

10 Leonard pair example cont. Then the pair A, B is a Leonard pair on the vector space V = F d+1. A Leonard pair of this form is said to have QRacah type. This is the most general type of Leonard pair.

11 Leonard pairs of QRacah type Hau-wen Huang (former student of Chih-wen Weng in the Department of Applied Math, National Chiao Tung University, Taiwan) has proven the following beautiful theorem about the Leonard pairs of QRacah type.

12 The Z 3 -symmetric Askey-Wilson relations Theorem (Hau-wen Huang 2011) Referring to the above Leonard pair A, B of QRacah type, there exists an element C such that A + qbc q 1 CB q 2 q 2 = (a + a 1 )(q d+1 + q d 1 ) + (b + b 1 )(c + c 1 ) q + q 1 I, B + qca q 1 AC q 2 q 2 = (b + b 1 )(q d+1 + q d 1 ) + (c + c 1 )(a + a 1 ) q + q 1 I, C + qab q 1 BA q 2 q 2 = (c + c 1 )(q d+1 + q d 1 ) + (a + a 1 )(b + b 1 ) q + q 1 I. The above equations are called the Z 3 -symmetric Askey-Wilson relations.

13 Leonard triples In the previous example the Z 3 -symmetry involving A, B, C suggests that we should consider Leonard triples along with Leonard pairs. The notion of a Leonard triple is due to Brian Curtin and defined as follows. Definition (Brian Curtin 2007) By a Leonard triple on V we mean an ordered triple of linear transformations (A, B, C) in End(V ) such that for each φ {A, B, C} there exists a basis for V with respect to which the matrix representing φ is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal.

14 Leonard triples in summary In summary, for a Leonard triple A, B, C A B C basis 1 diagonal irred. tridiagonal irred. tridiagonal basis 2 irred. tridiagonal diagonal irred. tridiagonal basis 3 irred. tridiagonal irred. tridiagonal diagonal

15 Leonard pairs and Leonard triples For a moment let us return to our Leonard pair A, B of QRacah type. Consider the element C from the Z 3 -symmetric Askey-Wilson relations. Huang has found necessary and sufficient conditions on C for the triple A, B, C to be a Leonard triple. This is explained in the next two theorems.

16 Leonard pairs and Leonard triples, cont. Theorem (Hau-wen Huang 2011) The roots of the characteristic polynomial of C are {θ ε i }d i=0, where θ ε i = cq 2i d + c 1 q d 2i (0 i d)

17 Leonard pairs and Leonard triples, cont. Theorem (Hau-wen Huang 2011) The following (i) (iii) are equivalent. (i) The triple A, B, C is a Leonard triple; (ii) {θi ε}d i=0 are mutually distinct; (iii) c 2 is not among q 2d 2, q 2d 4,..., q 2 2d. The above Leonard triple is said to have QRacah type.

18 The Askey-Wilson algebra In 1992 Alexei Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. Essentially, AW is the algebra defined by three generators A, B, C subject to the Z 3 -symmetric Askey-Wilson relations (The original definition was somewhat different). The algebra AW is defined using four parameters a, b, c, q.

19 We now define a central extension of AW, called the universal Askey-Wilson algebra and denoted. The algebra involves just one parameter q. The algebra is defined as follows. For the rest of the talk, q denotes a nonzero scalar in F such that q 4 1.

20 Definition (Ter 2011) Define an F-algebra = q by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A + qbc q 1 CB q 2 q 2, B + qca q 1 AC q 2 q 2, C + qab q 1 BA q 2 q 2 is central in. We call the universal Askey-Wilson algebra.

21 By constuction, each Askey-Wilson algebra AW is a homomorphic image of. By construction, each Leonard pair or triple of QRacah type can be viewed as a -module.

22 and Q-polynomial DRGs We now briefly relate to Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter D 3 and distance matrices {A i } D i=0. Assume Γ has a Q-polynomial ordering {E i } D i=0 of its primitive idempotents. Assume that the Q-polynomial structure has QRacah type; this means (in the notation of Bannai/Ito) type I with each of s, s nonzero.

23 and Q-polynomial DRGs, cont. Fix a vertex x of Γ. For 0 i D let A i = A i (x) denote the dual distance matrix of Γ that corresponds to E i and x. Assume that each irreducible T -module is thin. Here T = T (x) is the subconstituent algebra of Γ with respect to x, generated by A 1 and A 1. Theorem (Arjana Zitnik, Ter, in preparation) With the above assumptions and notation, there exists a surjective algebra homomorphism T that sends the generator A to a linear combination of I, A 1 and the generator B to a linear combination of I, A 1.

24 Three central elements of We now describe from a ring theoretic point of view. Definition Define elements α, β, γ of such that A + qbc q 1 CB q 2 q 2 = B + qca q 1 AC q 2 q 2 = C + qab q 1 BA q 2 q 2 = α q + q 1, β q + q 1, γ q + q 1. Note that each of α, β, γ is central in.

25 A basis for Theorem (Ter 2011) The following is a basis for the F-vector space : A i B j C k α r β s γ t i, j, k, r, s, t N. We proved this using the Bergman Diamond Lemma.

26 An action of PSL 2 (Z) on Recall that the modular group PSL 2 (Z) has a presentation by generators p, s and relations p 3 = 1, s 2 = 1. Our next goal is to show that PSL 2 (Z) acts on as a group of automorphisms. Strategy: identify two automorphisms of that have orders 3 and 2. By construction has an automorphism that sends A B C A. This automorphism has order 3. To find an automorphism of that has order 2, we use another presentation for.

27 Alternate presentation for Theorem (Ter 2011) The algebra has a presentation by generators A, B, γ and relations A 3 B [3] q A 2 BA + [3] q ABA 2 BA 3 = (q 2 q 2 ) 2 (AB BA), B 3 A [3] q B 2 AB + [3] q BAB 2 AB 3 = (q 2 q 2 ) 2 (BA AB), A 2 B 2 B 2 A 2 + (q 2 + q 2 )(BABA ABAB) γa = Aγ, Here [n] q = (q n q n )/(q q 1 ). = (q q 1 ) 2 (AB BA)γ, γb = Bγ. The first two relations above are the tridiagonal relations.

28 An automorphism of that has order 2 By the alternate presentation has an automorphism that swaps A, B and fixes γ. This automorphism has order 2.

29 An action of PSL 2 (Z) on Theorem (Ter 2011) The group PSL 2 (Z) acts on as a group of automorphisms in the following way: This action is faithful. u A B C α β γ p(u) B C A β γ α s(u) B A C + AB BA q q 1 β α γ

30 The Casimir element Ω of Shortly we will describe the center Z( ). To do this we introduce a certain element Ω called the Casimir element. Definition Define Ω = q 1 ACB + q 2 A 2 + q 2 B 2 + q 2 C 2 q 1 Aα q 1 Bβ qcγ. We call Ω the Casimir element of.

31 The Casimir element Ω is central Theorem (Ter 2011) The Casimir element Ω is contained in Z( ). Moreover Ω is fixed by everything in PSL 2 (Z).

32 A basis for that involves Ω We are going to show that Z( ) is generated by Ω, α, β, γ provided that q is not a root of unity. To this end we display a basis for that involves Ω. Lemma (Ter 2011) The following is a basis for the F-vector space : A i B j C k Ω l α r β s γ t i, j, k, l, r, s, t N ijk = 0.

33 Algebraic independence Corollary (Ter 2011) The elements Ω, α, β, γ are algebraically independent over F.

34 The center Z( ) We now describe the center Z( ). Theorem (Ter 2011) Assume that q is not a root of unity. Then the algebra Z( ) is generated by Ω, α, β, γ. Moreover Z( ) is isomorphic to a polynomial algebra in four variables.

35 The algebra and U q (sl 2 ) Our next goal is to explain how is related to the quantum group U q (sl 2 ). Definition The F-algebra U = U q (sl 2 ) is defined by generators e, f, k ±1 and relations kk 1 = k 1 k = 1, ke = q 2 ek, kf = q 2 fk, ef fe = k k 1 q q 1. We call e, f, k ±1 the Chevalley generators for U.

36 Irreducible modules for U q (sl 2 ) We review the finite-dimensional irreducible modules for U q (sl 2 ). Lemma For all integers d 0 and ε {1, 1} there exists a U-module V d,ε with the following property: V d,ε has a basis {v i } d i=0 such that kv i = εq d 2i v i (0 i d), fv i = [i + 1] q v i+1 (0 i d 1), fv d = 0, ev i = ε[d i + 1] q v i 1 (1 i d), ev 0 = 0. The U-module V d,ε is irreducible provided that q is not a root of unity.

37 The Casimir element of U q (sl 2 ) Earlier we gave a Casimir element for. The algebra U also has a Casimir element, which we now recall. Definition Define Λ U as follows: Φ = ef (q q 1 ) 2 + q 1 k + qk 1. We call Λ the (normalized) Casimir element of U.

38 The Casimir element of U q (sl 2 ), cont. The following result is well known. Lemma The Casimir element Λ is in the center Z(U). Moreover on the U-module V d,ε Λ = ε(q d+1 + q d 1 )I.

39 The equitable presentation of U q (sl 2 ) When we defined U we used the Chevalley presentation. There is another presentation for U of interest, said to be equitable. Lemma (Tatsuro Ito, Chih-wen Weng, Ter 2000) The algebra U has a presentation by generators x, y ±1, z and relations yy 1 = y 1 y = 1, qxy q 1 yx q q 1 = 1, qyz q 1 zy q q 1 = 1, qzx q 1 xz q q 1 = 1. We call x, y ±1, z the equitable generators for U.

40 The U-module V d,ε from the equitable point of view In the equitable presentation the U-module V d,ε looks as follows. V d,ε has three bases such that: x y z basis 1 diagonal lower bidiagonal upper bidiagonal basis 2 upper bidiagonal diagonal lower bidiagonal basis 3 lower bidiagonal upper bidiagonal diagonal

41 The Casimir element Λ of U q (sl 2 ) In the equitable presentation of U the Casimir element looks as follows. Lemma (Ter 2011) The Casimir element Λ is equal to each of the following: qx + q 1 y + qz qxyz, qy + q 1 z + qx qyzx, qz + q 1 x + qy qzxy, q 1 x + qy + q 1 z q 1 zyx, q 1 y + qz + q 1 x q 1 xzy, q 1 z + qx + q 1 y q 1 yxz.

42 and U q (sl 2 ) We are now ready to describe how is related to U q (sl 2 ). Lemma (Ter 2011) Let a, b, c denote nonzero scalars in F. Then there exists an F-algebra homomorpism U q (sl 2 ) that sends A xa + ya 1 xy yx + q q 1 bc 1, B yb + zb 1 yz zy + q q 1 ca 1, C zc + xc 1 + zx xz q q 1 ab 1. The above homomorphism is not injective. To shrink the kernel we do the following.

43 and U q (sl 2 ) From now on let a, b, c denote mutually commuting indeterminates. Let F[a ±1, b ±1, c ±1 ] denote the F-algebra of Laurent polynomials in a, b, c that have all coefficients in F. Consider the F-algebra U F[a ±1, b ±1, c ±1 ], where U = U q (sl 2 ) and = F.

44 and U q (sl 2 ) Theorem (Ter 2011) There exists an injective F-algebra homomorphism : U F[a ±1, b ±1, c ±1 ] that sends A x a + y a 1 xy yx + q q 1 bc 1, B y b + z b 1 yz zy + q q 1 ca 1, C z c + x c 1 + zx xz q q 1 ab 1, where x, y, z denote the equitable generators for U.

45 and U q (sl 2 ) Theorem (Ter 2011) The homomorphism sends α Λ (a + a 1 ) + 1 (b + b 1 )(c + c 1 ), β Λ (b + b 1 ) + 1 (c + c 1 )(a + a 1 ), γ Λ (c + c 1 ) + 1 (a + a 1 )(b + b 1 ), where Λ denotes the Casimir element of U.

46 and U q (sl 2 ) Theorem (Ter 2011) Under the homomorphism the image of the Casimir element Ω is 1 (q + q 1 ) 2 Λ (a + a 1 ) 2 1 (b + b 1 ) 2 1 (c + c 1 ) 2 Λ (a + a 1 )(b + b 1 )(c + c 1 ) where Λ denotes the Casimir element of U.

47 and the universal DAHA of type (C 1, C 1). Our next goal is to describe how is related to the double affine Hecke algebra (DAHA) of type (C 1, C 1). This is the most general DAHA of rank 1. We will work with the universal version of DAHA. For notational convenience define a four element set I = {0, 1, 2, 3}.

48 The universal DAHA of type (C 1, C 1) Definition Let Ĥ q denote the F-algebra defined by generators {t ±1 i } i I and relations t i t 1 i = t 1 i t i = 1 i I, t i + t 1 i is central i I, t 0 t 1 t 2 t 3 = q 1. We call Ĥ q the universal DAHA of type (C 1, C 1). For notational convenience define T i = t i + t 1 i i I.

49 The elements X, Y of Ĥq We will describe how is related to Ĥq. To set the stage we first mention a few basic features of Ĥ q. Define X = t 3 t 0, Y = t 0 t 1. Note that X, Y are invertible.

50 A basis for Ĥq Theorem (Ter 2012) The following is a basis for the F-vector space Ĥ q : Y i X j t0 k T1 r T2 s T3 t i, j, k Z r, s, t N. This can be proven using the Bergman Diamond Lemma.

51 A basis for Ĥq Corollary (Ter 2012) The following are algebraically independent over F: t 0, T 1, T 2, T 3. Corollary (Ter 2012) The following are algebraically independent over F: T 0, T 1, T 2, T 3.

52 The center Z(Ĥq) We now describe the center Z(Ĥ q ). Theorem (Ter 2012) Assume that q is not a root of unity. Then the algebra Z(Ĥq) is generated by {T i } i I. Moreover Z(Ĥq) is isomorphic to a polynomial algebra in four variables.

53 Some automorphisms of Ĥq We mention some automorphisms of Ĥ q. We start with an obvious one. There exists an automorphism of Ĥq that sends t 0 t 1 t 2 t 3 t 0. We call this Z 4 -symmetry. This symmetry sends X Y q 1 X 1 q 1 Y 1 X.

54 The Artin braid group B 3 We will be discussing the Artin braid group B 3. Definition The group B 3 is defined by generators ρ, σ and relations ρ 3 = σ 2. For notational convenience define τ = ρ 3 = σ 2. There exists a group homomorphism B 3 PSL 2 (Z) that sends ρ p and σ s. Via this homomorphism we pull back the PSL 2 (Z) action on, to get a B 3 action on as a group of automorphisms. Next we explain how B 3 acts on Ĥ q as a group of automorphisms.

55 An action of B 3 on Ĥq Lemma The group B 3 acts on Ĥ q as a group of automorphisms such that τ(h) = t 1 0 ht 0 for all h Ĥ q and ρ, σ do the following: h t 0 t 1 t 2 t 3 ρ(h) t 0 t0 1 t 3t 0 t 1 t 2 σ(h) t 0 t0 1 t 3t 0 t 1 t 2 t1 1 t 1

56 An action of B 3 on Ĥq, cont. Lemma The B 3 action on Ĥ q does the following to the central elements {T i } i I. The generator τ fixes every central element. The generators ρ, σ satisfy the table below. h T 0 T 1 T 2 T 3 ρ(h) T 0 T 3 T 1 T 2 σ(h) T 0 T 3 T 2 T 1 We are now ready to describe how is related to Ĥ q.

57 and Ĥq Theorem (Ter 2012) There exists an injective F-algebra homomorphism ψ : Ĥ q that sends A t 1 t 0 + (t 1 t 0 ) 1, B t 3 t 0 + (t 3 t 0 ) 1, C t 2 t 0 + (t 2 t 0 ) 1.

58 and Ĥq Theorem (Ter 2012) The homomorphism ψ sends α (q 1 t 0 + qt t t t3 1 β (q 1 t 0 + qt t t t2 1 γ (q 1 t 0 + qt t t t1 1

59 and Ĥq Theorem (Ter 2012) Under the homomorphism ψ the image of the Casimir element Ω is (q + q 1 ) 2 (q 1 t 0 + qt0 1 )2 (t 1 + t1 1 )2 (t 2 + t2 1 )2 (t 3 + t3 1 )2 (q 1 t 0 + qt0 1 )(t 1 + t1 1 )(t 2 + t2 1 )(t 3 + t3 1 ).

60 and Ĥq Theorem (Ter 2012) For all g B 3 the following diagram commutes: ψ Ĥ q g ψ g Ĥ q

61 The image of under ψ Now consider the image of under ψ. As we will see, this image is related to the spherical subalgebra {h Ĥ q t 0 h = ht 0 }.

62 The image of under ψ Theorem (Ter 2012) Consider the image of under ψ. The spherical subalgebra {h Ĥq t 0 h = ht 0 } is generated by this image together with t ±1 0, T 1, T 2, T 3.

63 Notation For notational convenience, from now on identify with its image under the injection ψ : Ĥ q. From this point of view A = t 1 t 0 + (t 1 t 0 ) 1 = t 0 t 1 + (t 0 t 1 ) 1 = Y + Y 1, B = t 3 t 0 + (t 3 t 0 ) 1 = t 0 t 3 + (t 0 t 3 ) 1 = X + X 1, C = t 2 t 0 + (t 2 t 0 ) 1 = t 0 t 2 + (t 0 t 2 ) 1, α = (q 1 t 0 + qt T 2 T 3, β = (q 1 t 0 + qt T 1 T 2, γ = (q 1 t 0 + qt T 3 T 1, Ω = (q + q 1 ) 2 (q 1 t 0 + qt 1 0 )2 T 2 1 T 2 2 T 2 3 (q 1 t 0 + qt 1 0 )T 1T 2 T 3.

64 A presentation for the spherical subalgebra by generators and relations We now give a presentation of the spherical subalgebra {h Ĥ q t 0 h = ht 0 } by generators and relations. This will be our last result of the talk.

65 A presentation for the spherical subalgebra Theorem (Ter 2012) The spherical subalgebra {h Ĥ q t 0 h = ht 0 } is presented by generators and relations in the following way. The generators are A, B, C, t 0 ±1, {T i} 3 i=1. The relations assert that each of t±1 0, {T i } 3 i=1 is central and t 0t0 1 = 1, t0 1 t 0 = 1, A + qbc q 1 CB q 2 q 2 = B + qca q 1 AC q 2 q 2 = C + qab q 1 BA q 2 q 2 = α q + q 1, β q + q 1, γ q + q 1, q 1 ACB + q 2 A 2 + q 2 B 2 + q 2 C 2 q 1 Aα q 1 Bβ qcγ = (q + q 1 ) 2 (q 1 t 0 + qt 1 0 )2 T 2 1 T 2 2 T 2 3 (q 1 t 0 + qt 1 0 )T 1T 2 T 3,

66 A presentation for the spherical subalgebra, cont. Theorem where α = (q 1 t 0 + qt T 2 T 3, β = (q 1 t 0 + qt T 1 T 2, γ = (q 1 t 0 + qt T 3 T 1.

67 Summary In this talk we introduced the universal Askey-Wilson algebra. We showed how each Leonard pair and Leonard triple of QRacah type yields a -module. We discussed how is related to Q-polynomial distance-regular graphs of QRacah type. We gave several bases for, we described its center, and we showed how PSL 2 (Z) acts on as a group of automorphisms. We described how is related to U q (sl 2 ). Finally we described how is related to the universal DAHA of type (C 1, C 1). Thank you for your attention! THE END

68 References Hau-wen Huang. The classification of Leonard triples of QRacah type. Linear Algebra Appl. To appear. arxiv: T. Ito and P. Terwilliger. Double affine Hecke algebras of rank 1 and the Z 3 -symmetric Askey-Wilson relations. SIGMA 6 (2010) 065, 9 pages, arxiv: P. Terwilliger.. SIGMA 7 (2011) 069, 24 pages, arxiv: P. Terwilliger. and the equitable presentation of U q (sl 2 ). SIGMA 7 (2011) 099, 26 pages, arxiv: P. Terwilliger. and DAHA of type (C 1, C 1). Submitted for publication.

Leonard pairs and the q-tetrahedron algebra. Tatsuro Ito, Hjalmar Rosengren, Paul Terwilliger

Leonard pairs and the q-tetrahedron algebra. Tatsuro Ito, Hjalmar Rosengren, Paul Terwilliger Tatsuro Ito Hjalmar Rosengren Paul Terwilliger Overview Leonard pairs and the Askey-scheme of orthogonal polynomials Leonard pairs of q-racah type The LB-UB form and the compact form The q-tetrahedron