Solving Racah problems through new algebraic structures

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Racahvraagstukken oplossen via nieuwe algebra¨ısche structuren

Solving Racah problems through new algebraic structures

Wouter van de Vijver

Promotoren:

Prof. Dr. Hendrik De Bie Prof. Dr. Luc Vinet

Proefschrift voorgelegd aan de Faculteit Wetenschappen van de Universiteit Gent tot het behalen van de graad van

Doctor in de wetenschappen: wiskunde Universiteit Gent

Faculteit Ingenieurswetenschappen en Architectuur

Vakgroep Electronica en Informatiesystemen - Clifford Research Group Academiejaar 2020-2021

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Deep in the human unconscious is a pervasive need for a logical universe that makes sense. But the real universe is always one step beyond logic

— Frank Herbert, Dune

There is in all things a pattern that is part of our universe. It has symmetry, elegance, and grace - those qualities you find always in that which the true artist captures. You can find it in the turning of the seasons, in the way sand trails along a ridge, in the branch clusters of the creosote bush or the pattern of its leaves. We try to copy these patterns in our lives and our society, seeking the rhythms, the dances, the forms that comfort. Yet, it is possible to see peril in the finding of ultimate perfection. It is clear that the ultimate pattern contains it own fixity. In such perfection, all things move toward death.

— Frank Herbert, Dune

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Woord Vooraf

Zes jaar geleden onving ik een bericht met de vraag of ik ge¨ınteresseerd zou zijn in een assistentenpositie aan de UGent. Dit bericht was het begin van een avontuur dat de volgende zes jaar van mijn leven zouden bepalen; zes jaar waarvoor ik eeuwig dankbaar ben. Ik zou een aantal mensen willen bedanken die de voorbije zes jaar zo mooi hebben gekleurd.

Ten eerst wil ik mijn promotor en tevens de auteur van dit bericht bedanken: Hendrik De Bie. Hij is mijn gids geweest zowel in de academische wereld als daarbuiten. Hij heeft mij intellectueel tot nieuwe hoogtes gebracht.

Hij heeft mij de kans gegeven om verschillende hoeken van de wereld te be- zoeken op zoek naar kennis en heeft mij zo besmet met een liefde voor reizen.

Ik heb altijd genoten van de vele discussies of het nu over wiskunde, lesgeven, reizen, wijn, de beurs of het leven in het algemeen ging.

I also would like to thank the two people who took up the mantle of copro- motor. I got to meet these two people on my first trip to the beautiful land of Canada: Vincent X. Genest and Luc Vinet. While Vincent left academia after two years, those two years were filled with fascinating mathematics.

I also enjoyed the knowledge and the infinite amount of energy Luc has to conduct research. I also appreciate the time he took to discuss where I want to go in life. I would also like to thank the man I met on my last trip to Canada: Nicolas Cramp´e. My collaboration with him propelled my research forward. I admire the way he thinks about mathematics and how he lives as a mathematical nomad.

Natuurlijk mag ik ook mijn collega’s niet vergeten. Het was een plezier om samen met jullie te mogen werken en lesopdrachten tot een goed einde te brengen. Dit zijn de zaken die ik niet zal vergeten: de goed raad van Tim waarmee ik mijn eerste bureau mocht delen als beginnend assistent, de humor van Michael en Lander, de discussies en gesprekken met Ali en Sigiswald, het sarcasme en de goede raad van Hilde, de bridgelessen van Fred, de filmavonden, de kalme uitstraling van Karel die de mijne evenaart, de onvermoeibare toewijding van Hennie voor haar studentjes, de positieve energie van Srdjan waarmee ik graag samen de architecten heb onderwezen,

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de weetjes van Denis, de interessante discussies over China met Ren, de sportiviteit van Astrid, de gevatte opmerkingen van Sam, de humor van Teppo, de werkethos van de briljante studente Hadewijch die ik heb mogen begeleiden en die ik nu collega mag noemen, en uiteraard de taart op het einde van de werkweek. Ook Roy, Nicolas, Alexis, Asmus, Yang, Brecht, Zo¨e en Frederick kleurden op hun eigen manier de voorbije zes jaar.

Mijn vrienden mag ik ook niet vergeten voor hun steun. Of het nu feestjes, bordspellen, naar de film of op caf´e gaan, de reizen, ... was, de avonturen die we beleefden hebben steeds voor de nodige (ont)spanning gezocht. Hierbij geef ik een speciale vermelding voor onze D&D campagne hebben die mij door de pandemie geholpen heeft.

Als laatste bedank ik ook mijn ouders. Zij hebben mij steeds de vrijheid gegeven mijn interesses na te streven en hebben mijn liefde voor wetenschap aangemoedigd.

Aan iedereen die er bij was de voorbije zes jaar, vanuit het diepste van mijn hart heel erg bedankt!

Wouter van de Vijver

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Chapter 1 Introduction

Racah problems appear in many fields ranging from physics to representation theory. The first Racah problem was the following: be given two representations on which the angular momentum operators in three dimensions act.

Both representations have a basis with nice properties with respect to the algebra of angular momentum operators. Now take the tensor product of the two representations. One can couple the two bases to find a basis for the tensor product but one can also construct a new a basis diagonalizing the total angular momentum operator which acts on both components of the tensor product. In this situation the Racah problem asks to give the overlap coefficients between these two bases. The solution to this particular Racah problem are the Clebsch-Gordan coefficients. These coefficients play an important role in understanding the angular momentum operators as well as the representation theory of the orthogonal group SO(3) as they appear in the components for the Wigner D-matrix. We will bring into play the 3j-coefficients which are more symmetric versions of the Clebsch-Gordan coefficients solving the same Racah problem. These 3j-coefficients have sis- ters called the 6j-coefficients. The 6j-coefficients appear when considering the tensor product of three representations. In fact there exists a whole family called 3nj-coefficients that appear when considering the tensor product of n representations. All these coefficients are examples of solutions to Racah problems . These coefficients have been shown to exhibit a plethora of interesting properties illuminating the representation theory of the algebra acting on these representation as well as tying different fields, physical and mathematical, together. The research presented in this thesis lies on that in- tersection of different fields. Its roots stem from a number of articles. One of these is the article [30]. This article studies the 6j-symbols. To uncover symmetries of the 6j-symbols a quadratic algebra was constructed from squares of angular momentum operators. This led to the introduction of the Racah

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algebra which is nowadays called the rank one Racah algebra. The symmetries of the Racah algebra obtained by permuting certain parameters of the algebra then give rise to symmetries of the 6j-symbols. The 6j-symbols also appear in the theory of orthogonal polynomials as they can be written as Racah polynomials. These Racah polynomials sit atop the discrete branch of the Askey-scheme. The Askey-scheme is the result of the work of R. Askey and J. Wilson [3, 4]. These mathematicians classified univariate orthogonal polynomials of hypergeometric type and the result of their labor can be summarized in the following scheme known as the Askey-scheme.

Wilson Racah

Continuous dual Hahn

Continuous

Hahn Hahn Dual Hahn

Meixner-

Pollaczek Jacobi Meixner Krawtchouk

Laguerre Charlier

Hermite

In [62], it was shown that the properties of each of these orthogonal polynomials can be encoded by a quadratic algebra. In the case of the Racah polynomials one recovers the Racah algebra generated by certain shift operators. The interaction between the quadratic algebra, physical models, orthogonal polynomials and the 3nj-coefficients found through the Racah problem can be summarized in the following diagram. This diagram was initially conceived in [28].

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Algebraic structure

Orthogonal polynomials Racah problem

Physical model

It will be our goal to explain the different arrows between the four subjects in the diagram. In this thesis three different algebraic structures will be put in the center of this diagram. These are the higher rank Racah algebra, the higher rank Bannai-Ito algebra and the Racah oscillator algebra. These higher rank algebras will help solve the Racah problem for the following three different Lie (super)algebras algebras: the simple linear Lie algebra sl2, the Lie superalgebraosp(1|2) and the oscillator algebrah.

Let us focus briefly on the physical models as it will be the topic of least concern in this thesis. Physical models are usually examined through the symmetries of their Hamiltonian. These symmetries generate an algebra from which many properties of the physical model can be derived. The three higher rank algebras introduced above will act as symmetry algebras for a Hamiltonian. We will show how we can view this Hamiltonian from a representation theory perspective. If one considers the tensor product of representation spaces of one of the three Lie (super)algebras, one can consider the action of this Lie (super)algebra on the whole n-fold tensor product.

This particular action commutes with the action of the symmetry algebra for the Hamiltonian. This mutual commutativity is akin to the Schur-Weyl duality. This action also involves well-known operators such as the Laplace or Dirac operator and the Euler operator. It follows that the higher rank algebras become symmetry algebras for these operators too. Moreover, the Hamiltonian can be constructed from the action of the Lie (super)algebra on the wholen-fold tensor product. This emphasizes the importance of the role of the three higher rank algebras as they are instrumental in understanding certain physical models and their Hamiltonians. In the next sections we will briefly discuss these three algebraic structures separately.

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1.1 The higher rank Racah algebra

We will study the Racah problem for the simple linear Lie algebrasl2through the Racah algebra. To be more precise, we will not be studying the Racah algebra itself but a higher rank generalisation. It is was observed in [25, 24]

that the Racah algebra could be obtained from a threefold tensor product of the universal enveloping algebra sl₂. From this, it is only natural to lift this construction beyond the threefold tensor product to the n-fold tensor product. This leads to the introduction of the higher rank Racah algebra.

This will be the first algebraic structure of interest in this thesis.

One of the main motivators of this construction comes from the study of superintegrable systems. A superintegrable system is a d-dimensional quantum system described by a hamiltonian that has 2d−1 algebraically independent constants of motion. In a number of papers second order superintegrable systems were classified in two and three dimensions [43, 42, 41, 48]. In particular in [43] the Racah algebra was realized as a symmetry algebra of a given Hamiltonian. This was achieved by considering a difference realization of the Racah algebra acting on the Wilson/Racah polynomials. In [28] this research was pushed further by showing that the given Hamiltonian can be realized as the total Casimir of sl2 in the threefold tensor product.

The first pursuits to higher rank were given in [42] and more importantly in [50]. In this last article a fourfold tensor product was considered together with difference operators acting on bivariate Racah polynomials as found in [55]. This led eventually to [9] where the higher rank Racah algebra was defined abstractly from a multifold tensor product of sl₂. The results of [9]

will be presented in this thesis.

The higher rank Racah algebra will be realized explicitly using Zⁿ2 Dunkl operators [19] and it will be shown to be the symmetry algebra of the Dunkl Laplacian. This is a different but equivalent system/model to the superintegrable systems that were previously known. The multivariate Racah polynomials as defined by Tratnik [55] are shown to be the corresponding Racah coefficients. In this thesis we will not only present these results but expand on them showing that Racah coefficients for sl₂ can be written as multivariate Racah polynomials beyond what Tratnik defined. We will call this a multivariate polynomials of Griffiths type. The choice of this term will be explained in the section of Racah oscillator algebra, Section 1.3.

The original Racah algebra has a well known realization by difference operators. In [29] a number of operators acting diagonally on the Racah polynomials were introduced. We call these operators Racah operators. In [37, 36] the connection between these operators and the higher rank Racah algebra were investigated. A full discrete realization of the higher rank Racah

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algebra was finally given in [15]. This result is also presented in this thesis.

In [23, 27] it was discovered that the Racah algebra can be embedded into a single copy of the universal enveloping algebra of sl₂ instead of a threefold tensor product. This raises a similar question of the higher rank Racah algebra. Can it be embedded into a generalization of the sl₂? In [16] it is shown that the higher rank Racah algebra can be embedded intosl_nand this result is presented in this thesis. A similar question will be answered for the Racah oscillator algebra in more depth.

1.2 The higher rank Bannai-Ito algebra

The second algebraic structure of interest is the higher rank Bannai-Ito algebra. Originally, the Bannai-Ito algebra was introduced to study the univariate Bannai-Ito polynomials and their bispectral behaviour [56]. In [12], the Bannai-Ito algebra was realized as the symmetry algebra of the three dimensional Dirac-Dunkl operator. This led to the introduction of the higher rank Bannai-Ito algebra as symmetry algebra of the n-dimensional Dirac- Dunkl operators. The higher rank Bannai-Ito algebra was then defined more abstractly as a subalgebra of the n-fold tensor product of osp(1|2) [10] and this will be revised in this thesis. The Racah problem ofosp(1|2) was solved in [26] for the rank one case and the general rank n case in [8]. While the Bannai-Ito algebra takes a less prominent place in this thesis, its role should not be underestimated. Much of the results obtained for the higher rank Racah algebra can be translated to similar results for the higher rank Bannai-Ito algebra and vice versa. One of the benefits of the higher rank Bannai-Ito algebra is the fact that its algebra relations are easier to manip- ulate. Moreover it is known in the rank one case that the Racah algebra can be embedded into the Bannai-Ito algebra [24]. This fact extends to the general ranknassl₂ can be embedded into osp(1|2). This makes the Bannai-Ito algebra in a sense more fundamental than the better known Racah algebra and a guide to further insights into the Racah algebra.

1.3 The Racah oscillator algebra

The last algebraic structure discussed in this thesis will be the Racah oscillator algebra. This algebra sits in the multifold tensor product of the enveloping algebra of the oscillator algebra h. It therefore aids in solving the Racah problem for the oscillator algebra h. The 3j- and 6j-coefficients for the oscillator algebra were already obtained in [59, 60]. These are the Krawtchouk

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polynomials appearing in the Askey-scheme. In [62] the 9j-coefficients were obtained by considering a fourfold coupling of the oscillator algebra. These coefficient depended on two variables but did not correspond to the bivariate Krawtchouk polynomials as defined by Tratnik. Nevertheless, they are orthogonal with respect to the trinomial distribution. Polynomials orthogonal with respect to the multinomials distribution where described in [31] by R.

C. Griffiths. Tratnik type Krawtchouk polynomials are orthogonal to the multinomial distribution and therefore form a subset to the Griffiths type polynomials. The bivariate Krawtchouk polynomials of Griffiths type did reappear in the work [33, 49] but were called Rahman polynomials. These Rahman polynomials reappeared in the work of P. Iliev and P. Terwilliger [38] as overlap coefficients between bases diagonalized by different Cartan algebras ofsl3. This suggests that the multivariate Krawtchouk polynomials of Griffiths type must play a role in the representation theory ofsl_n+1. In [35]

it was shown that overlap coefficients between bases diagonalized by two different Cartan subalgebras ofsln+1 gave the multivariate Krawtchouk polynomials of Griffiths type. This affirmed the role of the multivariate Krawtchouk polynomials of Griffiths. In this thesis this link will be explained more deeply.

The main theorem on the Racah oscillator algebra in this thesis says that the Lie algebra sl_n+1 can be embedded into the Racah oscillator algebra. This ties the representation theories for both algebras together and explains why multivariate Krawtchouk polynomials of Griffiths type show up in both theories. The multivariate Krawtchouk polynomials of Griffiths type also show up in the representation theory of the orthogonal group O(n+ 1) as shown in [57]. They appear in the matrix elements of the realization of O(n + 1) when acting on the energy eigenspaces of the isotropic (n+ 1)-dimensional harmonic oscillator. The orthogonal group O(n+ 1) acts as automorphism group on sln+1. Two Cartan subalgebras of sln+1 are related through such an automorphism. This action of O(n+ 1) as automorphism group can be made explicit and we will retrieve the same result for the matrix elements.

1.4 Organization of this thesis

This thesis is comprised of five chapters besides the introductory chapter.

In Chapter 2 we introduce a number of algebras, orthogonal polynomials and classical results pertaining to these subjects. This prepares us for the following chapters. Chapter 3 introduces the tensor algebra approach from which we will define the three different higher rank algebras. A number of lemmas will be proven. Most important is the embedding Lemma 3.12.

This last lemma is pivotal in discerning the algebraic relations of each of

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the higher rank algebras. A short section will be devoted to the Bannai-Ito algebra to show how the embedding Lemma 3.12 comes into play. A second big part of Chapter 3 is the introduction of labeling Abelian subalgebras which play an important role in the representation theory of higher rank algebras and in solving the Racah problem. Chapter 4 focuses on the higher rank Racah algebra. The algebra relations are found and a first realization is presented. This is the Dunkl model. This model is then used to solve the Racah problem for sl₂. Having found the Racah coefficients, we construct a second realization. This is the discrete model. In the final section of this chapter a link between the higher rank Racah algebra and the special linear Lie algebra sl_n is established. In Chapter 5 we focus on the Racah oscillator algebra. Once the algebra relations are found, the embedding ofsl_n is established. The Racah problem forh is solved. We then study the action of the orthogonal group O(n+ 1) as automorphism group on sl_n. A final short chapter 6 is included where topics for further research are discussed.

A large part of the text of this thesis has been taken from the following articles [6, 9, 10, 13, 15, 16]. The tensor algebra approach in Chapter 3 was developed in [9]. The relations of the Bannai-Ito algebra in Section 3.2.1 come from [10]. Section 3.3 on labeling Abelian algebras comes from both [6, 9].

The first article introduces labeling Abelian algebras for chains. The second article introduces labeling Abelian algebras for trees. Chapter 4 finds the relations of the higher rank Racah algebra and introduces the Dunkl model as in [9]. The sections on the overlap coefficients and discrete model mainly follow [15]. Section 4.5 on constructing bases related to labeling Abelian algebras related to trees is new and does not belong to any article. The final section 4.6 relating the higher rank Racah algebra to sl_n is the article [16].

Chapter 5 is the article [6].

The text has been adapted and notation has been harmonized to compose this thesis into a coherent work.

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Chapter 2 Preliminaries

2.1 Lie algebras and superalgebras

Let k be a field. A Lie algebra g is a k-vector space with a k-bilinear map called a commutator

[., .] :g×g→g satisfying for allx, y and z:

• [x, y] =−[y.x],

• the Jacobi identity

[x,[y, z]] + [y,[z, x]] + [z,[x, y]] = 0.

To every associative algebra A we can associate a Lie algebra Lie(A). The underlying set forLie(A) is the same as for the algebraAbut the Lie bracket is defined as

[X, Y] =X.Y −Y.X

The dot . is here the multiplication defined in A. We can go the other way around. For every Lie algebra g we define an associative algebra called the universal enveloping algebra U(g). Let V be the underlying vector space of g. Then the universal enveloping algebra is given by

U(g) =

∞

M

n=0

V^⊗n/S

whereS is the two-sided ideal generated by the set of relations{x⊗y−y⊗ x−[x, y]|x, y ∈ g}. These types of algebras play an important role in the representation theory of Lie algebras. We will be interested mainly in the

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Casimir elements of a given Lie algebra. Casimir elements are elements in U(g) that commute with g.

For our purposes we will need three Lie algebras. The special linear algebra of rank 1 denoted bysl2, its generalization the special linear algebra of rank n −1 denoted by sl_n, and the oscillator algebra h. We will also introduce the orthosymplectic Lie superalgebra osp(1|2).

2.1.1 The special linear algebra sl

2

The algebra sl₂ is a Lie algebra generated by 3 elements e, f and hwith the following relations:

[h, e] = 2e, [h, f] =−2f, [e, f] =h. (2.1) We will also need an alternative but isomorphic presentation of sl2 denoted bysu(1,1).

[J₀, J±] =±J±, [J−, J₊] = 2J₀. (2.2) We call J₊ the raising operator and J− the lowering operator. This algebra admits a Casimir element in the universal enveloping algebra U(su(1,1)). It is given by

C =J₀²−J₀−J₊J−. (2.3) This algebra also admits a coproductµ^∗. The coproduct is an algebra morphism:

µ^∗ :U(su(1,1)) → U(su(1,1))⊗ U(su(1,1)).

It is defined as follows on the generators

µ^∗(J_±) = J_±⊗1 + 1⊗J_± µ^∗(J0) = J0⊗1 + 1⊗J0

(2.4) and extended as an algebra morphism.

2.1.2 The special linear algebra sl

_n

The Lie algebra sln is generated by the following set of elements:

{e_kk+1, e_k+1k, h_k|1≤k ≤n−1}.

To give the relations ofsl_n we first introduce the Cartan matrix

A_ij =







2 if i=j

−1 if j =i±1 0 if |j−i|>1

with 1≤i, j ≤n−1.

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Here are the Chevalley-Serre relations that define sln:

[h_i, h_j] = 0, (2.5)

[e_ii+1, e_j+1j] =δ_ijh_i, (2.6)

[h_i, e_jj+1] =A_ije_jj+1, [h_i, e_j+1j] =−A_ije_j+1j, (2.7) ad(e_ii+1)^1−A^ij(e_jj+1) = 0, if i6=j (2.8) ad(e_i+1i)^1−A^ij(e_j+1j) = 0. if i6=j (2.9) The operator ad is the adjoint action: ad(x)(y) := [x, y]. The set {h_k|1 ≤ k ≤ n −1} generates the Cartan algebra of sl_n. When we consider the sl₂ case, there are three generators {e₁₂, e₂₁, h₁} with following relations:

[e₁₂, e₂₁] =h₁, [h₁, e₁₂] = 2e₁₂, [h₁, e₂₁] =−2e₂₁. (2.10) This coincides with the definition of sl₂ given by (2.1). From the Chevalley- Serre relations we see that every triple {e_ii+1, e_i+1i, h_i} generates a copy of sl2.

2.1.3 The oscillator algebra

The oscillator algebra or Heisenberg algebra h is the Lie algebra generated by four elements A±, A₀ and a central element a with following defining relations [53]:

[A−, A₊] =a, [A₀, A±] =±A±. (2.11) The operator A₊ is the raising operator and A₋ the lowering operator. It should be noted that this is not the classical definition of the Heisenberg algebra as we have added the operatorA₀. The Casimir elementQis contained in the universal enveloping algebra U(h) and is given by:

Q:=aA₀−A₊A−. (2.12)

The oscillator algebra also admits a coproductµ^∗. The coproduct in this case is a morphism:

µ^∗ :U(h)→ U(h)⊗ U(h).

µ^∗(A±) =A±⊗1 + 1⊗A±,

µ^∗(A₀) =A₀⊗1 + 1⊗A₀. (2.13) and extended as an algebra morphism.

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2.1.4 The Lie superalgebra osp(1|2)

Consider the algebra generated by the elements I± and I₀ with following relations:

[I₀, I_±] =±I_±, {I₊, I₋}= 2I₀ (2.14) where we have introduced the anticommutator {X, Y} = XY −Y X. This is the Lie superalgebraosp(1|2). A Lie superalgebra is aZ2-graded algebra.

A Z2-graded algebra is an algebra whose underlying vector space can be decomposed into a direct sum of two parts, an even part g0 and odd part g₁. We split the underlying vector space of osp(1|2) in its even part g₀ = Span {I₀, I_±²}

and its odd part g₁ = Span ({I±}). So g =g₀⊕g₁. As one can see from the relations (2.14), the bracket between two odd elements is calculated by the anticommutator. The even part on the other hand is a Lie algebra and a quick calculation reveals that g₀ ∼=sl₂:

[I₀, I_±²] =±2I_±², [I₋², I₊²] = 4I₀.

We will add to this algebra the grading operator P. It commutes with the even part and anticommutes with the odd part:

[I0, P] = 0, {I±, P}= 0, P² = 1. (2.15) The Casimir ofosp(1|2) is given by

Γ := 1

2([I−, I₊]−1)P. (2.16) This algebra osp(1|2) also admits a coproduct µ^∗. The coproduct is a morphism:

µ^∗ :U(osp(1|2)) → U(osp(1|2))⊗ U(osp(1|2)).

µ^∗(I±) =I±⊗P + 1⊗I±, µ^∗(I₀) =I₀⊗1 + 1⊗I₀, µ^∗(P) =P ⊗P.

(2.17)

and extended as an algebra morphism. This coproduct is not co-commutative as opposed to the the coproduct defined for sl₂ and h.

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2.2 The Racah and Bannai-Ito algebra

2.2.1 The Racah algebra

The Racah algebra R is a quadratic associative algebra introduced in [63]

and governs the properties of the Racah polynomials. Classically the Racah algebra is defined by two generatorsK₁ and K₂ and their commutatorK₃ :=

[K₁, K₂]. The defining relations are:

[K₂, K₃] =K₂²+{K₁, K₂}+dK₂+e₁,

[K₃, K₁] =K₁²+{K₁, K₂}+dK₁+e₂. (2.18) The numbers d, e₁ and e₂ are structure constants. Throughout this thesis we will use a different presentation of the Racah algebra which is called the equitable presentation [47, 54]. It is generated by three elements C₁₂, C₂₃ andC13 and four central elementsC1,C2,C3 andC123. These generators are linearly dependent:

C₁₂+C₁₃+C₂₃ =C₁₂₃+C₁+C₂+C₃. (2.19) If we take the commutator of both sides of equation (2.19) with each of the non-central generators we get the following:

[C₁₂, C₂₃] = [C₂₃, C₁₃] = [C₁₃, C₁₂]. (2.20) We set

F := 1

2[C₁₂, C₂₃].

The remaining relations are given by

[C₁₂, F] =C₂₃C₁₂−C₁₂C₁₃+ (C₂−C₁)(C₃−C₁₂₃), [C₂₃, F] =C₁₃C₂₃−C₂₃C₁₂+ (C₃−C₂)(C₁−C₁₂₃), [C₁₃, F] =C₁₂C₁₃−C₁₃C₂₃+ (C₁−C₃)(C₂−C₁₂₃).

(2.21)

To see the connection between the first realization (2.18) and the second (2.21), we set

K₁ = C₁₂

2 , K₂ = C₂₃ 2

and the structure constants are expressed in function of the central elements:

d=−(C₁₂₃+C₁+C₂+C₃), e₁ = (C₂−C₁)(C₃−C₁₂₃), e₂ = (C₂−C₃)(C₁−C₁₂₃).

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2.2.2 The Bannai-Ito algebra

The Bannai-Ito algebra BI is an algebra generated by three generators B_i, i= 1,2,3 and structure constantsω_i, i= 1,2,3.

{B₁, B₂}=B₃+ω₃, {B₁, B₃}=B₂+ω₂, {B₂, B₃}=B₁+ω₁.

(2.22)

Its Casimir is given by C =B₁²+B₂²+B₃².

2.3 Racah polynomials

2.3.1 Univariate Racah polynomials

The Racah polynomials have an intimate relation with the eponymous algebra and show up in many ways in the representation theory of the Racah algebra. They are defined as follows:

Definition 2.1. ([44]) Let r_n(α, β, γ, δ;x) be the classical univariate Racah polynomials with real parameters α,β,γ and δ, variable xand dgreen ∈N:

r_n(α, β, γ, δ;x) := (α+ 1)_n(β+δ+ 1)_n(γ+ 1)_n×

4F₃

−n, n+α+β+ 1,−x, x+γ+δ+ 1 α+ 1, β+δ+ 1, γ+ 1 ; 1

.

To see the connection between the Racah algebra and Racah polynomials consider a finite-dimensional irreducible representation V of R. Let the set {ψ_k} be a basis of V diagonalized by K₁ or equivalently C₁₂ and {ϕ_s} be a basis of V diagonalized by K₂ or equivalently C₂₃. What are the overlap coefficients between these two bases? In other words, find the numbers Rsk

such that

X

k

R_skψ_k=ϕ_s.

The overlap coefficients between these two bases will be Racah polynomials.

This is a well-known result. See for example [23, 25, 28, 30]. To make this statement more precise we will give the following proposition. We will use the equitable presentation of the Racah algebra. We also introduce the following polynomial:

κ(x, β) =

x+β+ 1

2 x+β−1 2

. (2.23)

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Proposition 2.2. LetV be an irreducible representation ofRwithdim(V) = N + 1. Assume that the central elements on this representation act as the following scalars:

C₁ =κ(0, β₀),

C₂ =κ(0, β₁−β₀−1), C3 =κ(0, β2−β1−1), C₁₂₃ =κ(N, β₂).

(2.24)

Assume that there exists a basis of V diagonalizing C₁₂ which we denote by {ψ_k} and a basis of V diagonalizing C₂₃ denoted by {ϕ_s} with the following eigenvalues:

C₁₂ψ_k=κ(k, β₁)ψ_k,

C₂₃ϕ_s=κ(s, β₂−β₀−1)ϕ_s.

The overlap coefficients are then up to a gauge constant equal to

Rˆ_sk =r_s(β₁−β₀−1, β₂−β₁−1,−N −1, β₁+N;k). (2.25) The constants β₀, β₁ and β₂ depend on the representation and can be calculated from the action of the central elements C1, C2, C3 and C123 by Formula (2.24). For this reason we will often denote the dependence on these central element:

Rˆsk = ˆRsk(C1, C2, C3, C123).

These polynomials ˆR_sk are orthogonal for the following weight:

ρ(`) = Γ(β₁+`)Γ(β₁−β₀+`)

Γ(β₀ + 1 +`)`! (β₁+ 2`)

× Γ(β₂+N +`)Γ(β₂−β₁+N −`) Γ(β₁+N + 1 +`)(N −`)! .

(2.26)

We have

N

X

`=0

ρ(`) λ(k)

Rˆ_k`Rˆ_k⁰_` =δ_k,k⁰ (2.27) with _λ(k)¹ equal to

1

λ(k) = Γ(k+β₂−β₀−1)

Γ(k+β₂ −β₁)Γ(k+β₁−β₀)k!(2k+β₂−β₀ −1)

× Γ(β₀ +N + 1−k)(N −k)!

Γ(k+β₂−β₀ +N)Γ(β₂+N +k).

(2.28)

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2.3.2 The shift operator L

A second way the Racah algebraRis connected with the Racah polynomials is through a discrete realization of the Racah algebra. This is a realization through shift operators. We have the following proposition

Proposition 2.3. Let Cˆ₁₂ =κ(x, β₁)and Cˆ₂₃ =−L₁+κ(0, β₂−β₀−1)with L₁ a shift operator explicitly given by

L₁ = (x+β₁−β₀)(x+β₁)(N +x+β₂)(N −x)

(2x+β₁)(2x+β₁+ 1) (E_x−1)+

(x+β₀)x(N −x−β₁+β₂)(N +x+β₁)

(2x+β1)(2x+β1−1) (E_x⁻¹−1)

with the operatorEx(f(x)) :=f(x+ 1). The central operatorsCˆ1,Cˆ2, Cˆ3 and Cˆ₁₂₃ are constants as given in Formula (2.24) and the operator Cˆ₁₃ is given by Formula (2.19). They generate a discrete realization of the Racah algebra R. The eigenvectors of Cˆ23 are exactly the univariate Racah polynomials

rk(β1−β0−1, β2−β1−1,−N −1, β1+N;x) with corresponding eigenvalues κ(k, β2−β0−1).

Proof. After explicit computation one concludes that this is indeed a discrete realization of the Racah algebra R. Note that ˆC₂₃ is in fact the three-point difference equation of the Racah polynomials (see also [28, 44]).

2.3.3 Multivariate Racah polynomials

M.V. Tratnik generalized the univariate Racah polynomials in [55] in search of a multivariate version of the Askey-scheme. These multivariate polynomials will appear in the representation theory of the higher rank Racah algebra in Chapter 4. We will use the definition given in [29].

Definition 2.4. Let 1 ≤ p ≤ n. The multivariate Racah polynomials are given by

R_p(~k;~x;β;~ N) =

p

Y

j=1

r_k_j(2|~k|j−1+β_j −β₀−1, β_j+1−β_j−1,

|~k|j−1−x_j+1−1,|~k|j−1+β_j+x_j+1;−|~k|j−1+x_j)

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with ~k := (k₁, . . . , k_n−2) ∈ Nⁿ⁻² which will set the degrees of each Racah polynomial in the product, the variables ~x= (x₁, . . . , xn−2), the parameters β~ := (β0, . . . , βn−1) ∈ Rⁿ and the parameter N := xn−1 ∈ N. We also introduced the notation |~k|_j =Pj

i=1k_i. We assume that|~k|n−2 ≤N.

In Proposition 2.3 we introduced the operator L₁. This operator has the univariate Racah polynomials as eigenvectors. In [29] J. S. Geronimo and P. Iliev defined a set of operators {L_i}. They proved that this set has the multivariate Racah polynomials as common eigenvectors. We introduce them here:

Definition 2.5. (See also [29]) Let j be a positive integer. The Racah operator is the shift operator:

Lj = X

~ν∈{−1,0,1}^j

~ν6=~0

G~ν(E~ν −1).

Here,E_~_ν is a shift operator defined as follows. LetE_x^νⁱ

i(f(x_j)) =f(x_j+δ_ijν_i).

Then we define E_~_ν = E_x^ν¹₁E_x^ν²₂. . . Ex^ν^jj. The G_~_ν are rational functions in the variables x₀, x₁, . . . , x_j+1 and β₀, . . . , β_j+1 and are defined as follows. We introduce the following functions:

B_i^0,0 :=x_i(x_i+β_i) +x_i+1(x_i+1+β_i+1) + (β_i+ 1)(β_i+1−1) 2

B_i^0,1 := (x_i+1+x_i +β_i+1)(x_i+1−x_i+β_i+1−β_i) B_i^1,0 := (x_i+1−x_i)(x_i+1+x_i+β_i+1)

B_i^1,1 := (x_i+1+x_i +β_i+1)(x_i+1+x_i+β_i+1+ 1).

Let I_if(x_i) := f(−x_i −β_i). We extend B^s,t such that s, t ∈ {−1,0,1} by defining:

B_i^−1,t :=I_i(B_i^1,t) B_i^s,−1 :=I_i+1(B_i^s,1) B_i^−1,−1 :=I_i(I_i+1(B_i^1,1)).

We also introduce

b⁰_i := (2x_i+β_i+ 1)(2x_i+β_i−1) b¹_i := (2xi+βi+ 1)(2xi+βi) b⁻¹_i :=I_i(b¹_i).

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Let|~ν|₀ be the number of zeroes appearing in ~ν. ThenG_~_ν is G_~_ν := 2^|~^ν|⁰

Qj

i=0B_i^νⁱ^,νⁱ⁺¹ Qj

i=1b^ν_iⁱ .

When we need to show explicitly the variables on which the Racah operator depends, we denote

L_j(x₀, x₁, . . . , x_j+1, β₀, . . . , β_j+1, E_x₁, . . . , E_x_j).

The following proposition is Theorem 3.9. in [29]

Proposition 2.6. For j ∈[p] the operators L_j act as follows on the multivariate Racah polynomials:

L_jR_p(~k;~x;β;~ N) =−|~k|j(|~k|j−1 +β_j−β₀)R_p(~k;~x;β;~ N).

The operators L_j commute with each other.

The eigenvalue can be rewritten in terms of the polynomial defined in 2.23:

−|~k|_j(|~k|_j −1 +β_j −β₀) =κ(0, β_j−β₀−1)−κ(|~k|_j, β_j−β₀−1).

2.4 Univariate Krawtchouk polynomials

The Krawtchouk polynomials are related to the representation theory of the algebrasl2. We give the definition from [44].

Definition 2.7. The Krawtchouk polynomials denoted byKn(x;p, N) where pis a parameter, x the variable, n the degree and N ∈N, are defined as

K_n(x;p, N) :=₂F₁

−n,−x

−N ;1 p

=

N

X

k=0

(−n)_k(−x)_k (−N)_kk!

1 p^k.

It is a well-known result that the overlap coefficients between two bases of an irreducible representation of sl2 diagonalized by two Cartan algebras related by an inner automorphism ofsl₂ are univariate Krawtchouk polynomials [57, 45]. We give the proof here.

LetV be a finite dimensional representation ofsl2 and ˜.an automorphism ofsl₂. The elementh is a Cartan generator ofsl₂. Let{ψ_k} be an eigenbasis

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of h and {φ_s} be an eigenbasis for ˜h. The indices k and s run from 0 to N with dim(V) = N + 1. We are interested in the overlap coefficients B_sk between these bases:

φ_s=

N

X

k=0

B_skψ_k. (2.29)

hψ_k =µ_kψ_k ˜hφ_s=ν_sφ_s.

The algebrasl2 has algebra relations [h, e] = 2e and [h, f] =−2f with e the raising operator andf the lowering operator on {ψ_k}:

eψ_k=a_kk+1ψ_k+1 f ψ_k=bkk−1ψk−1

and µ_k=µ₀+ 2k. From the algebra relation [e, f] =h it follows that bkk−1ak−1k−a_kk+1b_k+1k=µ_k.

LetA_k:=akk−1bk−1k. Then we have

A_k−A_k+1 = 2k+µ₀. From this we find

A_k =−k(k−1)−µ₀k−Ω

with Ω∈R. We express ˜h as a linear combination ofh,e and f.

˜h=R_hh+R_ee+R_ff

with R_eR_f+R²_h = 1. We have set up everything we need to find the overlap coefficients. Let the operator ˜h act on both sides of Equality (2.29).

˜hφ_s =

N

X

k=0

B_sk(R_hh+R_ee+R_ff)ψ_k. This gives

ν_sφ_s=

N

X

k=0

B_sk(R_hµ_kψ_k+R_ea_kk+1ψ_k+1+R_fbkk−1ψk−1).

We expand the left hand side into the basis ψ_k and we gather the terms on the right hand side:

N

X

k=0

ν_sB_skψ_k =

N

X

k=0

(B_skR_hµ_k+Bsk−1R_eak−1k+B_sk+1R_fb_k+1k)ψ_k.

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From this we find the recurrence relation

ν_sB_sk =B_sk+1R_fb_k+1k+B_skR_hµ_k+Bsk−1R_eak−1k.

We want to recognize this recurrence relation as one of the family of orthogonal polynomials. Let

B˜_sk =

k

Y

t=2

btt−1R_f

! B_sk to find

ν_sB˜_sk = ˜B_sk+1+R_hµ_kB˜_sk +R_eR_fak−1kbk−1kB˜sk−1. We write the coefficients as polynomials in x=ν_s:

xB˜k(x) = ˜Bk+1(x) +RhµkB˜k(x) +ReRfAkB˜k−1(x). (2.30) We want to compare this with the recurrence relation of the normalized Krawtchouk polynomials. The normalized Krawtchouk polynomials are given by

p_n(x) = (−N)_npⁿK_n(x;p, N), Its recurrence relation is given by

xp_n(x) =p_n+1+ (n(1−2r) +rN)p_n(x) +r(1−r)n(N + 1−n)pn−1(x) with n= 0,1, . . . , N. Let x=αy+β and introduce q_n(y) =p_n(αy+β)/αⁿ. The polynomial q_n(x) satisfies the following recurrence relation:

yq_n(y) = q_n+1+n(1−2r) +rN −β

α q_n(y) + r(1−r)n(N + 1−n)

α² qn−1(y).

We retrieve equation (2.30) if we set α= 1

2, r= 1−R_h

2 , β = N

2, k =n, Ω = 0, µ₀ =−N.

We can now explicitly write down the polynomials ˜B_k(x).

B˜_k(x) = 2^kp_k

x+N 2

= (−N)_k(1−R_h)^kK_k

x+N

2 ;1−R_h 2 , N

= (−N)_k(1−R_h)^k₂F₁

−k,−^x+N₂

−N ; 2 1−Rh

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The overlap coefficients are the Krawtchouk polynomials ˆK_k(x) (defined in the last line of the equation above) up to a normalization factor. We conclude that

B_sk =

k

Y

t=2

ftt−1R_f

!

(−N)_k(1−R_h)^kKˆ_k

νs+N

2 ;1−Rh

2 , N

.

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Chapter 3 Tensor algebra approach to higher rank algebras

3.1 Definition of higher rank algebras

Letgbe eithersu(1,1) or h. We will now construct operators inU(g)^⊗n. Let X be a generator ofg. Introduce the following element:

X_k = 1⊗ · · · ⊗1

| {z }

k−1 times

⊗X⊗1⊗ · · · ⊗1

| {z }

n−ktimes

. (3.1)

This gives an operator that lives in thekth component of the tensor product.

We will generalize this definition to operators living in many components of the tensor product. We first introduce some new notation: the set [n] :=

{1, . . . , n}. Let K be any non-empty subset of [n] and put X_K = X

k∈K

X_k. (3.2)

This operatorX_K lives in the components of the tensor product whose indices are in K. These operators generate subalgebras inside U(g)^⊗n

Lemma 3.1. Let K ⊂[n]. The operators J_0,K, J_+,K and J−,K generate an algebra isomorphic to su(1,1) inside U(su(1,1))^⊗n. We denote this algebra by su^K(1,1).

Likewise we have the following lemma for h

Lemma 3.2. LetK ⊂[n]. The operatorsA0,K, A+,K, A−,K andaK generate an algebra isomorphic to h inside U(h)^⊗n. We denote this algebra by h^K.

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Let C_K be the Casimir of su^K(1,1) andQ_K be the Casimir of h^K: C_K :=J_0,K² −J_0,K −J_+,KJ−,K

Q_K :=a_KA_0,K−A_+,KA−,K.

The elementsC_K are contained in the algebraU(su(1,1))^⊗n and generate an algebra.

Definition 3.3. The higher rank Racah algebra R_n of rank n −2 is the subalgebra of U(su(1,1))^⊗n generated by the following set of operators:

{C_K|K ⊂[n] and K 6=∅}.

Example 3.4. The simplest non-trivial case is given byR₃. This is the rank one case. It is generated by the set

{C₁, C₂, C₃, C₁₂, C₂₃, C₁₃, C₁₂₃}.

For ease of notation we abbreviate sets of the form{1,2} by 12 when in the index of a generator.

Similarly the elements Q_K fall in the algebra U(h)^⊗n and also generate an algebra:

Definition 3.5. We define the Racah oscillator algebra Rn(h) to be the subalgebra of U(h)^⊗n generated by the elements of the set

{Q_K|K ⊂[n] and K 6=∅}.

Example 3.6. The simplest non-trivial case is as before given by the rank one caseR₃(h). It is generated by the set

{Q₁, Q₂, Q₃, Q₁₂, Q₂₃, Q₁₃, Q₁₂₃}.

A similar construction exists when we consider the Lie superalgebraosp(1|2).

The way we need to construct operators in U(osp(1,1))^⊗n is a bit different between odd and even generators. We put

I±,k = 1⊗ · · · ⊗1

| {z }

k−1 times

⊗I±⊗P ⊗ · · · ⊗P

| {z }

n−ktimes

, I_0,k = 1⊗ · · · ⊗1

| {z }

k−1 times

⊗I₀⊗1⊗ · · · ⊗1

| {z }

n−ktimes

, P_k= 1⊗ · · · ⊗1

| {z }

k−1 times

⊗P ⊗1⊗ · · · ⊗1

| {z }

n−ktimes

.

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For a non-empty set K ⊂[n] we then define I±,K = X

k∈K

I±,k, I_0,K = X

k∈K

I_0,k, P_K = Y

k∈K

P_k. We have the following lemma:

Lemma 3.7. Let K ⊂ [n]. The operators I_0,K, I_+,K, I−,K and P_K generate an algebra isomorphic to osp(1|2) inside U(osp(1|2))^⊗n. We denote this algebra by osp^K(1|2).

The algebra osp^K(1|2) has the following Casimir:

Γ_K := 1

2([I_−,K, I_+,K]−1)P_K. This allows us to define an algebra inside U(osp(1|2))^⊗n:

Definition 3.8. The higher rank Bannai-Ito algebra BI_n of rank n −2 is the subalgebra of U(osp(1|2))^⊗n generated by the following set of operators:

{Γ_K|K ⊂[n] and K 6=∅}.

Example 3.9. As for the other algebras, the easiest non-trivial case is the rank one Bannai-Ito algebra BI₃. It is generated by the set

{Γ₁,Γ₂,Γ₃,Γ₁₂,Γ₂₃,Γ₁₃,Γ₁₂₃}.

Remark 3.10. Alternatively, each of the algebras R_n, R_n(h) and BI_n can be defined using the Hopf structure of the underlying Lie (super)algebra.

Let g be either su(1,1), h or osp(1|2). Let X be the Casimir of g. We act repeatedly on the Casimir X with the coproduct:

X1 :=X, Xm := (1⊗. . .⊗1

| {z }

n−2 times

⊗µ^∗)(Xm−1).

Each element Xm sits in a different algebra U(g)^⊗m. We will lift each of these elements intoU(g)^⊗n by the map τ_k

τ_k:

m−1

OU(g)→

m

OU(g).

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Whengis a Lie algebra, i.e. su(1,1) or h, we defineτ_k as follows on homoge- nous tensor products and extend by linearity:

τ_k(t₁⊗. . .⊗tm−1) := t₁⊗. . .⊗tk−1⊗1⊗t_k⊗. . .⊗tm−1.

τ_k(t₁⊗. . .⊗tm−1) :=t₁⊗. . .⊗tk−1⊗P^|t¹^t²^...t^k−1^|⊗t_k⊗. . .⊗tm−1. Using this mapτk, we have an alternative way of defining XA with A ⊂[n]:

X˜_A :=





−→

Y

k∈[n]\A

τ_k



 X^|A|

. (3.3)

The arrow represents the order in which the maps τ_k must be applied: τ_k must be applied before τ_m if k < m. The elements ˜C_A coincide with C_A, the elements ˜Q_A coincide with Q_A and the elements ˜Γ_A coincide with Γ_A. This leads to an alternative construction of the higher rank algebras. This approach has been generalized to the quantum algebraU_q(sl₂) in [7]. A more thorough analysis is given in [46].

We have the following lemma:

Lemma 3.11. The higher rank Racah algebra R_n sits in the centralizer of su^[n](1,1).

The higher rank Racah oscillator algebra R_n(h)sits in the centralizer of h^[n]. The higher rank Bannai-Ito algebraBI_n sits in the centralizer of osp^[n](1|2).

This is the centralizer defined for the regular commutator and not the super- commutator.

Proof. We will give the proof for Rn. The other cases are analogous. We calculate the commutator of a generator ofsu^[n](1,1) andC_K. Letbe either 0 or ±:

[CK, J,[n]] = [CK, J,K] + [CK, J,[n]\K] = 0.

The first commutator equals 0 as C_A is the Casimir of su^A(1,1) and the second commutator is 0 as both operators act on different components of U(su(1,1))^⊗n.

Solving Racah problems through new algebraic structures

Racahvraagstukken oplossen via nieuwe algebra¨ısche structuren