Howe Duality and Algebras of the Askey–Wilson Type: An Overview

Abstract

The Askey–Wilson algebra and its relatives such as the Racah and Bannai–Ito algebras were initially introduced in connection with the eponym orthogonal polynomials. They have since proved ubiquitous. In particular they admit presentations as commutants that are related through Howe duality. This paper surveys these results.

1 Introduction

The quadratic algebras of Askey–Wilson type such as the Askey–Wilson algebra itself, the Racah and Bannai–Ito algebras and their specializations and contractions encode the bispectral properties of orthogonal polynomials that arise in recoupling coefficients such as the Clebsch–Gordan or Racah coefficients. It is therefore natural that these algebras be encountered in centralizers of the diagonal action of an algebra of interest such as , , or , on n-fold tensor products of representations of . Indeed, elements of these centralizers will be used as labeling operators to define bases whose overlaps will be expressed in terms of the corresponding orthogonal polynomials.

Often the algebra forms a reductive pair with another algebra in which case the Howe duality operates in certain modules. This leads to alternative characterizations of the quadratic algebras that are in correspondence: on the one hand commutants in representations of the universal enveloping algebra and on the other hand, realizations of the type mentioned above as centralizers in recoupling problems for . This is the topic of this brief review which is organized as follows. Section 2 presents the general framework. Section 3 describes as illustration the dual commutant picture for the Racah algebra; this will involve the reductive pair . Section 4 gives a summary of the different cases that have been analyzed and Sect. 5 provides a short outlook.

2 General Framework

We shall say following [1] that two algebras and have dual representations on a Hilbert space $\mathcal {H}$ if (1) this space carries fully reducible representations of both and , (2) the action of and commute, (3) the representation ρ of the direct sum defined by the actions of and on $\mathcal {H}$ is multiplicity-free, and (4) each irreducible representation of occurring in the decomposition of ρ is paired with a unique irreducible representation of and vice versa. This is the essence of Howe duality which can be proved in a number of situations. We shall consider such instances in this paper.

Consider now a setup with the representation of in $\mathcal {H} = V^{\otimes 2n}$ given by where is a representation of on the vector space V , is the coproduct, and Δ ⁽ⁿ⁾ is defined recursively by Δ ⁽ⁿ⁾ = (Δ ⊗ 1^⊗(n−1)) ∘ Δ ⁽ⁿ⁻¹⁾, with Δ ⁽⁰⁾ = 1. This symmetric situation makes it natural that there be an action of some other algebra on the carrier space $\mathcal {H}$ that commutes with the action of . Take the maximal Abelian subalgebra of to be with one-dimensional. The pairing under Howe duality with the representations of implies that decomposes into representations of the form with the σ _i’s being irreducible representations arising in the decomposition of $\bar {\sigma }^{\otimes 2}$. This quotienting by is a way of posing a generalized Racah problem for the recoupling of the n representations σ _i of .

We indicated in the introduction that the quadratic algebras $\mathcal {A}$ of Askey–Wilson type can be obtained as (subalgebras of) centralizers of diagonal actions in n-fold tensor products of representations. The intermediate Casimir elements in σ ₁ ⊗ σ ₂ ⊗⋯ ⊗ σ _n manifestly centralize the action of on . They are taken to generate the quadratic algebra of interest. This provides the first presentation of $\mathcal {A}$ as a commutant. The dual one is identified as follows in the present context. We know that is the commutant of in $\mathcal {H}$. Moreover from the application of Howe duality, the generators of the representation σ ₁ ⊗ σ ₂ ⊗⋯ ⊗ σ _n of are known to commute with those that represent the subalgebra . The non-trivial part of the centralizer of σ ₁ ⊗ σ ₂ ⊗⋯ ⊗ σ _n must therefore be obtained, in the given representation on , by those elements in the universal enveloping algebra of that commute with . In other words, $\mathcal {A}$ can also be identified as the commutant of in as represented on $\mathcal {H}$.

There is an equivalent way of looking at this. The pairing of the representations of and through Howe duality manifests itself in the fact that the Casimir elements of and are affinely related. Let $\mathcal {C}$ be a Casimir element of . Consider for example the intermediate Casimir element given by $\bar {\sigma }^{\otimes {4}}[((\varDelta \otimes \varDelta ) \circ \varDelta ) (\mathcal {C})] \otimes 1^{\otimes {(2n-4)}}$ corresponding to the embedding of in the first four factors of . There will be a subalgebra of that will be dually related to on the restriction of $\mathcal {H}$ to V ^⊗4 so that its Casimir element will be essentially the one of . Next, looking at the intermediate Casimir element of associated with a different embedding, for instance in the four last factors of , there will be a dual pairing with a different embedding in of the same subalgebra and again the two Casimir elements will basically coincide. These observations lead to the conclusion that the set of intermediate Casimir elements associated with the representation of is algebraically identical to the set of Casimir elements of the subalgebras of that form dual pairs with when intermediate representations of the latter are taken. It is not difficult to convince oneself that the set of invariants connected to the relevant subalgebras of consists in the commutant of the maximal Abelian subalgebra of as concluded differently before.

To summarize, in situations where Howe duality prevails with the pair of algebras that are dually represented on $\mathcal {H}$ and if the representation of is of the form , the quadratic algebras $\mathcal {A}$ of Askey–Wilson type can be viewed on one hand as the commutant of this action of on $\mathcal {H}$ and thus generated by the intermediate Casimir elements of , or on the other hand as the commutant of in the intervening representation of . We shall present next an example of how this can be concretely realized.

3 The Dual Presentations of the Racah Algebra

The Racah algebra $\mathcal {R}$ has three generators K ₁, K ₂, K ₃ that are subjected to the relations [2]:

$$\displaystyle \begin{aligned}{} [K_1,K_2]=K_3, \qquad \quad \ [K_2,K_3]&={K_2}^2+\{K_1,K_2\}+dK_2+e_1,\\ {} [K_3,K_1]&={K_1}^2+\{K_1,K_2\}+dK_1+e_2, \end{aligned} $$

(1)

where [A, B] = AB − BA, {A, B} = AB + BA and d, e ₁, e ₂ are central.

We shall explain how dual presentations of the algebra $\mathcal {R}$ as a commutant are obtained in the fashion described in Sect. 2. The dual pair will be and the representation space $\mathcal {H}$ will be that of the state space of six quantum harmonic oscillators with annihilation and creation operators verifying . The corresponding Hamiltonian is manifestly invariant under the rotations in six dimensions. These are encoded in the Lie algebra , realized by the generators and possessing the Casimir element $\mathcal {C}=\sum _{\mu <\nu }{L_{\mu \nu }}^2$.

The Lie algebra has generators J ₀, J _± that obey the following commutation relations: [J ₀, J _±] = ±J _±, [J ₊, J ₋] = −2J ₀, and its Casimir operator is given by C = J ₀ ² − J ₊ J ₋− J ₀. The six harmonic oscillators also provide a realization of this algebra through the addition of six copies of the metaplectic representation of , for which the generators are mapped to: , , ${J_-^{(\mu )}=\frac {1}{2}(a_{\mu })^2}$, μ = 1, …, 6. Note that the operators $\sum _{\mu = 1}^{6} J_\bullet ^{(\mu )}$ are invariant under rotations. The space of state vectors $\mathcal {H}$ thus carries commuting representations of and and Howe duality takes place.

The maximal Abelian algebra of is and is generated by the set {L ₁₂, L ₃₄, L ₅₆}. The non-abelian part of its commutant in the representation of on $\mathcal {H}$ is generated by the two invariants

$$\displaystyle \begin{aligned} K_1&=\frac{1}{8}\big({L_{12}}^2+{L_{34}}^2+{L_{13}}^2+{L_{23}}^2+{L_{14}}^2+{L_{24}}^2\big), \end{aligned} $$

(2)

$$\displaystyle \begin{aligned} K_2&=\frac{1}{8}\big({L_{34}}^2+{L_{56}}^2+{L_{35}}^2+{L_{36}}^2+{L_{45}}^2+{L_{46}}^2\big).{} \end{aligned} $$

(3)

Define K ₃ by [K ₁, K ₂] = K ₃. Working out the commutation relations of K ₃ with K ₁ and K ₂, it is found that they correspond to those (1) of the Racah algebra with the central parameters given by $d=-\frac {1}{8}\big (\mathcal {C}+{L_{12}}^2+{L_{34}}^2+{L_{56}}^2\big )$,$e_1=-\frac {1}{64}\big (\mathcal {C}-{L_{12}}^2-4\big )\big ({L_{34}}^2-{L_{56}}^2\big )$, and $e_2=-\frac {1}{64}\big (\mathcal {C}-{L_{56}}^2-4\big )\big (L_{34}^2-L_{12}^2\big )$. For details see [3]. By abuse of notation we designate the abstract generators and their realizations by the same letter.

Regarding the picture, let $J_\bullet ^{(\mu ,\nu , \rho , \lambda )} = J_\bullet ^{(\mu )} +J_\bullet ^{(\nu )} + J_\bullet ^{(\rho )} + J_\bullet ^{ (\lambda )}$ denote the addition of the four metaplectic representations labeled by the variables μ, ν, ρ, λ all assumed different. The corresponding Casimir operator is $C^{(\mu ,\nu , \rho , \lambda )}=(J_0^{(\mu ,\nu , \rho , \lambda )})^2$ $-J_+^{(\mu ,\nu , \rho , \lambda )}J_-^{(\mu ,\nu , \rho , \lambda )}-J_0^{(\mu ,\nu , \rho , \lambda )}$. Quite clearly, these actions of restricted to state vectors of four oscillators are paired with commuting actions of the Lie algebra of rotations in the four dimensions labeled by μ, ν, ρ, λ. It is hence not surprising to find, owing to Howe duality, that C ⁽¹²³⁴⁾ = −2K ₁ and C ⁽³⁴⁵⁶⁾ = −2K ₂, namely that the intermediate Casimir operators corresponding to the recouplings of the first four and last four of the six metaplectic representations are equal (up to a factor) to the Casimir elements of the two corresponding subalgebras of which together generate as we observed the non-trivial part of the commutant of in . This entails the description of the Racah algebra as the commutant in of the action of on $\mathcal {H}$. Alternatively, picking the representations associated with those of under Howe duality yields the sum of three irreducible representations of belonging to the discrete series; these are realized as dynamical algebras of three singular oscillators. Note that corresponding to the representation $J_\bullet ^{(\mu ,\nu )} = J_\bullet ^{(\mu )} +J_\bullet ^{(\nu )}$ is the Casimir $C^{(\mu \nu )} =-\frac {1}{4}\big ({L_{\mu \nu }}^2+1\big )$. With the dependence on the polar angles “rotated out”, the total Casimir element C ^(123456) becomes the Hamiltonian of the generic superintegrable system on the two-sphere; the constants of motion are the quotiented intermediate Casimir elements and the symmetry algebra that they generate is hence that of Racah.

4 More Dual Pictures: An Overview

The main algebras of Askey–Wilson type have been studied recently from the commutant and Howe duality viewpoints. We summarize in the following the main results and give in particular the dualities that are involved.

4.1 The Racah Family

The higher rank extension of the Racah algebra [4] defined as the algebra generated by all the intermediate Casimir elements of can be described in the framework of the preceding section with the help of the dual pair using in this case the module formed by the state vectors of 2n harmonic oscillators. It is then seen to be dually the commutant of in the oscillator representation of [5].

The case n = 2 is special and of particular interest since it pertains to the Clebsch–Gordan problem for , that is, the recoupling of the two irreducible representations σ ₁ and σ ₂. There are no intermediate Casimirs here; the relevant operators associated with the direct product basis and the recoupled one are respectively M ₁ = σ ₁(J ₀) − σ ₂(J ₀) and the total Casimir M ₂ = (σ ₁ ⊗ σ ₂)Δ(C). These are seen to obey the commutation relations of the Hahn algebra [6]:

$$\displaystyle \begin{aligned}{} [M_1,M_2] =M_3, \qquad [M_2,M_3]& =-2\{M_1,M_2\}+\delta_1,\\ {} [M_3,M_1] &=-2{M_1}^{2}-4M_2+\delta_2, \end{aligned} $$

(4)

where δ ₁ = 4(σ ₁(J ₀) + σ ₂(J ₀))(σ ₁(C) − σ ₂(C)) and δ ₂ = 2(σ ₁(J ₀) + σ ₂(J ₀))² + (σ ₁(C) + σ ₂(C)) are central. The name of the algebra comes from the fact that the 3j-coefficients involve dual Hahn polynomials. In the setup with four harmonic oscillators, with $\mathcal {H}$ carrying the product of four metaplectic representations, Howe duality will imply that the total Casimir element C ⁽¹²³⁴⁾ of coincides with the Casimir of —this is the same computation as the one described above. It is easily seen that σ ₁(J ₀) − σ ₂(J ₀) is derived from $\frac {1}{2}(N_1+N_2-N_3-N_4)$ under the quotient by with , i = 1, …, 4. It can in fact be checked directly, again abusing notation, that ${M_1=\frac {1}{2}(N_1+N_2-N_3-N_4)}$ and ${M_2 = -\frac {1}{4}\big ({L_{12}}^2+{L_{34}}^2+{L_{13}}^2+{L_{23}}^2+{L_{14}}^2+{L_{24}}^2\big )}$ satisfy the relations given in Eq. (4) with $\delta _1=-\frac {1}{2}(N_1 + N_2 + N_3 + N_4 + 2)( {L_{12}}^2 - {L_{34}}^2)$ and ${\delta _2 = \frac {1}{2}(N_1 + N_2 + N_3 + N_4 + 2)^2 - ( {L_{12}}^2 + {L_{34}}^2 + 2)}$, in correspondence with the preceding expressions for δ ₁ and δ ₂ in the realization $J_\bullet ^{(1234)}$ of . From the expressions of these last M ₁ and M ₂, we can claim that the Hahn algebra is the commutant of in represented on $\mathcal {H}$. Let us stress that it is the universal enveloping algebra of that intervenes here.

4.2 The Bannai–Ito Ensemble

The Bannai–Ito algebra [7] takes its name after the Bannai–Ito polynomials that enter in the Racah coefficients of the Lie superalgebra . This algebra has three generators K _i, i = 1, …, 3 that satisfy the relations

$$\displaystyle \begin{aligned}{} \{K_i, K_j\} = K_k + \omega_k, \qquad i \neq j \neq k \in \{1, 2, 3\} \end{aligned} $$

(5)

with ω _i central and {X, Y } = XY + Y X. The relevant reductive pair in this case is and the representation space $\mathcal {H}$ is that of Dirac spinors in six dimensions with the Clifford algebra generated by the elements γ _μ verifying {γ _μ, γ _ν} = −2δ _μν, μ, ν = 1, …, 6. That the pair is dually represented on $\mathcal {H}$ is seen as follows: The spinorial representation of with generators

$$\displaystyle \begin{aligned}{} J_{\mu \nu} = -iL_{\mu\nu}+\varSigma_{\mu\nu},\qquad L_{\mu\nu}=x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}, \qquad \varSigma_{\mu\nu} = \frac{i}{2}\gamma_{\mu} \gamma_{\nu} \end{aligned} $$

(6)

leaves invariant the following operators:

$$\displaystyle \begin{aligned}{} J_- = -i\sum_{1\leq\mu\leq6} \gamma_\mu \partial_\mu, \qquad J_+ = -i\sum_{1\leq\mu\leq6} \gamma_\mu x_\mu, \qquad J_0 = \sum_{1\leq\mu\leq6} x_\mu \partial_\mu, \end{aligned} $$

(7)

which in turn realize the commutation relations of the Lie superalgebra : [J ₀, J _±] = ±J _±, {J ₊, J ₋} = −2J ₀ with J ₀ even and J _± odd. Howe duality thus takes place. As a matter of fact, for any subset A ⊂{1, …, 6} of cardinality |A| the operators $J_-^{A}=-i\sum _{\mu \in A}\gamma _\mu \partial _\mu ,~ J_+^{A}=-i\sum _{\mu \in A}\gamma _\mu x_\mu$, and ${J_0^{A}={\textstyle {|A|\over 2}}+\sum _{\mu \in A}x_\mu \partial _\mu }$ realize . The Casimir element of is given by $C = \frac {1}{2} ([J_- , J_+] - 1)S$ with S the grade involution obeying S ² = 1, [S, J ₀] = 0, {S, J _±} = 0. In the realizations at hand, S ^A = i ^|A|∕2∏_{μ ∈ A} γ _μ with |A| even.

It can be checked that the operators

realize the relations (5) of the Bannai-Ito algebra upon taking the following: ω _ij = 2Γ _k Γ ₁₂₃ + 2Γ _i Γ _j, where Γ ₁ = J ₁₂, Γ ₂ = J ₃₄, Γ ₃ = J ₅₆, and ${\varGamma _{123} = \big (\frac {5}{2}-i\sum _{1\leq \mu <\nu \leq 6}L_{\mu \nu }\varSigma _{\mu \nu }\big )\varSigma _{12}\varSigma _{34}\varSigma _{56}}$. That these arise from dual pictures is explained as follows (see [8] for details). On the one hand, K ₁, K ₂, K ₃ are observed to belong to the commutant in of the subalgebra of spanned by {J ₁₂, J ₃₄, J ₅₆}. On the other hand, considering the Casimir elements C ^A of associated with the realization by the operators $\{ J_0^{A}, J_{\pm }^{A}, S^{A}\}$, we find that C ⁽¹²³⁴⁾ = K ₁, C ⁽³⁴⁵⁶⁾ = K ₂, and C ⁽¹²⁵⁶⁾ = K ₃. This confirms that the Bannai–Ito algebra can be dually presented either as the commutant of in the spinorial representation of or as the centralizer of the action of on $\mathcal {H}$. These considerations can be extended to higher dimensions [8] so as to obtain analogously dual commutant pictures for the Bannai–Ito algebras of higher ranks [9].

4.3 The Askey–Wilson Class

The Askey–Wilson algebra can be presented as follows:

(8)

with [A, B]_q = qAB − q ⁻¹ BA and α, β, γ central.

The algebra has three generators, J _± and J ₀, obeying [J ₀ , J _±] = ±J _± and ${J_-J_+ - q^2 J_+J_- = q^{2J_0}\,[ 2J_0 ]_q}$ with ${[x]_q=\frac {q^{x}-q^{-x}}{q-q^{-1}}}$. Its coproduct is defined by Δ(J ₀) = J ₀ ⊗ 1 + 1 ⊗ J ₀, $\varDelta (J_\pm ) = J_\pm \otimes q^{2J_0} + 1 \otimes J_\pm$. The Casimir operator C of is given by ${C = J_+J_- q^{-2J_0+1} - \frac {q}{(1-q^2)^2} \, \big ( q^{2J_0-1} + q^{-2J_0+1} \big ) + \frac {1+q^2}{(1-q^2)^2}}$.

The q-deformation of is defined as the algebra with generators L _i,i+1 (i = 1, …, N − 1) obeying the relations

$$\displaystyle \begin{aligned} & L_{i-1,i}\,L_{i,i+1}^2 - (q^{1/2}+q^{-1/2}) L_{i,i+1}\,L_{i-1,i}\,L_{i,i+1} + L_{i,i+1}^2\,L_{i-1,i} = -L_{i-1,i},\\ & L_{i,i+1}\,L_{i-1,i}^2 - (q^{1/2}+q^{-1/2}) L_{i-1,i}\,L_{i,i+1}\,L_{i-1,i} + L_{i-1,i}^2\,L_{i,i+1} = -L_{i,i+1},\\ & [ L_{i,i+1}, L_{j,j+1} ] = 0 \quad \mbox{for} \quad |i-j| > 1. \end{aligned} $$

We shall use the notation $L_{ik}^\pm = [L_{ij}^{\pm }\,,L_{jk}^{\pm }]_{q^{\pm 1/4}}$ for any i < j < k, and by definition ${L_{i,i+1}^\pm =L_{i,i+1}}$.

The reductive pair is the one which is of relevance for the Askey–Wilson algebra. Let us indicate how and are dually represented on the standard state space $\mathcal {H}$ of 2n independent q-oscillators described by operators $\{A_i^\pm$, $A_i^0\}$ such that ${[A_i^0, A_i^\pm ] = \pm A_i^\pm }$, ${[A_i^-, A_i^+] = q^{A_i^0}}$, ${A_i^- A_i^+ - q A_i^+ A_i^- = 1}$, i = 1, …, 2n. The algebra is represented on $\mathcal {H}$ by using the coproduct to embed it in the tensor product of 2n copies of the q-deformation of the metaplectic representation, this gives

$$\displaystyle \begin{aligned} J_0^{(2n)}&=\varDelta^{(2n-1)}\Big({\textstyle{1\over 2}}\left(A^{0}_i+{\textstyle{1\over 2}}\right)\Big) =\frac{1}{2}\sum_{i=1}^{2n} \left(A_i^{0}+\frac{1}{2}\right),\\ J_\pm^{(2n)}&=\varDelta^{(2n-1)}\left(\frac{1}{[2]_{q^{1/2}}}(A^{\pm}_i)^{2}\right) =\frac{1}{[2]_{q^{1/2}}}\sum_{i=1}^{2n} \Bigg((A_i^{\pm})^{2}\prod_{j=i+1}^{2n}q^{A_j^{0}+\frac{1}{2}}\Bigg). \end{aligned} $$

(9)

The algebra can also be realized in terms of 2n q-oscillators. The 2n − 1 generators take the form

$$\displaystyle \begin{aligned} L_{i,i+1}= q^{-\frac{1}{2}(A_i^0+\frac{1}{2})} \Big( q^{\frac{1}{4}}A_i^+A_{i+1}^- - q^{-\frac{1}{4}}A_i^-A_{i+1}^+ \Big), \qquad i=1,\dots,2n-1. \end{aligned} $$

It can be checked that $[J_0^{(2n)}, L_{i,i+1}]=[J_\pm ^{(2n)}, L_{i,i+1}]=0,~ i=1,\dots ,2n-1$, in other words, that and have commuting actions on the Hilbert space $\mathcal {H}$ of 2n q-oscillators. This sets the stage for Howe duality. In order to connect with the Askey–Wilson algebra we take n = 3. The expressions of the operators K _A and K _B acting on $\mathcal {H}$ that realize the relations (8) (together with the specific central elements) are rather involved and we shall refer the reader to [10] for the formulas. We shall only stress that these operators can be obtained in a dual way: They are affinely related to the generators of the commutant of in as well as to the intermediate Casimir elements C ⁽¹²³⁴⁾ = Δ ⁽³⁾(C) ⊗ 1 ⊗ 1 and C ⁽³⁴⁵⁶⁾ = 1 ⊗ 1 ⊗ Δ ⁽³⁾(C) of the q-metaplectic representation (see (9)). This can be extended to higher ranks by letting n be arbitrary [11]. For n = 2 we are looking at the Clebsch–Gordan problem for . The q-Hahn algebra that arises has two dual realizations [12]: one as the commutant of in and the other in terms of the following two operators, (Δ(J ₀) ⊗ 1 ⊗ 1) − (1 ⊗ 1 ⊗ Δ(J ₀)) and Δ ⁽²⁾(C) (the full Casimir element) in the q-metaplectic representation.

5 Conclusion

This paper has offered a summary of how the quadratic algebras of Racah, Hahn, Bannai–Ito, Askey–Wilson, and q-Hahn types can be given dual descriptions as commutant of Lie algebras, superalgebras, and quantum algebras. The connection between these dual pictures is rooted in Howe dualities whose various expressions have been stressed. The attentive reader will have noticed that the Clebsch–Gordan problem for has not been mentioned; this is because it has not been analyzed yet. We plan on adding this missing piece to complete the picture.