The $q$-Higgs and Askey-Wilson algebras

TL;DR

This paper introduces a $q$-analogue of the Higgs algebra related to harmonic oscillators on the 2-sphere, connecting it to the Askey--Wilson algebra through Howe duality in quantum algebra representations.

Contribution

It constructs a $q$-Higgs algebra as a commutant in quantum groups and links it to the Askey--Wilson algebra via Howe duality, revealing new algebraic structures.

Findings

$q$-Higgs algebra as a commutant in $U_q(rak u(4))$

Connection established between $q$-Higgs and Askey--Wilson algebras

Representation theory insights into quantum symmetry algebras

Abstract

A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ in the $q$-oscillator representation of the quantized universal enveloping algebra $U_q(\mathfrak{u}(4))$. This $q$-Higgs algebra is also found as a specialization of the Askey--Wilson algebra embedded in the tensor product $U_q(\mathfrak{su}(1,1))\otimes U_q(\mathfrak{su}(1,1))$. The connection between these two approaches is established on the basis of the Howe duality of the pair $\big(\mathfrak{o}_{q^{1/2}}(4),U_q(\mathfrak{su}(1,1))\big)$.

Equations

$$(x{)}_q:=\frac{1-q^x}{1-q}\text{and}[x{]}_q:=\frac{q^x-q^{-x}}{q-q^{-1}}.$$

$$\big{[}j_{0}\,,j_{\pm}\big{]}=\pm j_{\pm},\qquad\big{[}j_{+}\,,j_{-}\big{]}=[2j_{0}]_{q}.$$

$$\Delta (j_0)=j_0\otimes 1+1\otimes j_0,\Delta (j_{+})=j_{+}\otimes q^{2j_0}+1\otimes j_{+},\Delta (j_{-})=j_{-}\otimes 1+q^{-2j_0}\otimes j_{-}.$$

$$\big{[}J_{0}\,,J_{\pm}\big{]}=\pm J_{\pm},\qquad J_{-}J_{+}-q^{2}J_{+}J_{-}=q^{2J_{0}}\,\big{[}2J_{0}\big{]}_{q}.$$

$$\Delta (J_0)=J_0\otimes 1+1\otimes J_0,\Delta (J_{\pm })=J_{\pm }\otimes q^{2J_0}+1\otimes J_{\pm }.$$

$$C=J_{+}J_{-}q^{-2J_{0}+1}-\frac{q}{(q^{2}-1)^{2}}\,\big{(}q^{2J_{0}-1}+q^{-2J_{0}+1}\big{)}+\frac{q^{2}+1}{(q^{2}-1)^{2}}.$$

$$L_{i-1,i}{L}_{i,i+1}^2-(q+q^{-1})L_{i,i+1}{L}_{i-1,i}{L}_{i,i+1}+L_{i,i+1}^2{L}_{i-1,i}=-L_{i-1,i},$$

$$L_{i,i+1}{L}_{i-1,i}^2-(q+q^{-1})L_{i-1,i}{L}_{i,i+1}{L}_{i-1,i}+L_{i-1,i}^2{L}_{i,i+1}=-L_{i,i+1},$$

$$\displaystyle\big{[}L_{i,i+1}\,,L_{j,j+1}\big{]}=0\quad\text{for}\quad|i-j|>1.$$

$$\displaystyle\begin{aligned} j_{1}&=ig\big{\{}q^{\frac{1}{2}j_{0}}\,,\,j_{+}+j_{-}\big{\}},\\ j_{2}&=g\big{\{}q^{-\frac{1}{2}j_{0}}\,,\,j_{+}-j_{-}\big{\}},\end{aligned}\qquad\quad g=\frac{1}{(q^{\frac{1}{4}}+q^{-\frac{1}{4}})(q^{\frac{1}{2}}+q^{-\frac{1}{2}})},$$

$$[j_1,j_2{]}_q=j_3,[j_2,j_3{]}_q=j_1,[j_3,j_1{]}_q=j_2.$$

$$L_{12}{L}_{23}^2-(q+q^{-1})L_{23}{L}_{12}{L}_{23}+L_{23}^2{L}_{12}$$

$$L_{23}{L}_{12}^2-(q+q^{-1})L_{12}{L}_{23}{L}_{12}+L_{12}^2{L}_{23}$$

$$L_{23}{L}_{34}^2-(q+q^{-1})L_{34}{L}_{23}{L}_{34}+L_{34}^2{L}_{23}$$

$$L_{34}{L}_{23}^2-(q+q^{-1})L_{23}{L}_{34}{L}_{23}+L_{23}^2{L}_{34}$$

$$\displaystyle\big{[}L_{12}\,,L_{34}\big{]}=0.$$

$$L_{13}^{\pm}=\big{[}L_{12}\,,L_{23}\big{]}_{q^{\pm 1}},\qquad L_{24}^{\pm}=\big{[}L_{23}\,,L_{34}\big{]}_{q^{\pm 1}},\qquad L_{14}^{\pm}=\big{[}L_{13}^{\pm}\,,L_{34}\big{]}_{q^{\pm 1}},$$

$$C_4$$

$$C_4^{\prime }$$

$$\big{[}A_{i}^{0}\,,A_{i}^{\pm}\big{]}=\pm A_{i}^{\pm},\qquad\big{[}A_{i}^{-}\,,A_{i}^{+}\big{]}=q^{A_{i}^{0}},\qquad A_{i}^{-}A_{i}^{+}-qA_{i}^{+}A_{i}^{-}=1,\qquad i=1,\dots,n\,,$$

$$N_i=A_i^{+}{A}_i^{-}=\frac{1-q^{A_i^0}}{1-q}=(A_i^0{)}_q.$$

$$A_i^0|n_i⟩=n_i|n_i⟩,A_i^{+}|n_i⟩=\sqrt{\frac{1-q^{n_i+1}}{1-q}}|n_i+1⟩,A_i^{-}|n_i⟩=\sqrt{\frac{1-q^{n_i}}{1-q}}|n_i-1⟩.$$

$$\chi (j_0)=\frac{1}{2}(A_1^0-A_2^0),\chi (j_{+})=q^{-\frac{1}{4}(A_1^0+A_2^0-1)}{A}_1^{+}{A}_2^{-},\chi (j_{-})=q^{-\frac{1}{4}(A_1^0+A_2^0-1)}{A}_1^{-}{A}_2^{+},$$

$$\displaystyle\chi(j_{1})=\frac{iq^{\frac{1}{4}}}{q^{\frac{1}{2}}+q^{-\frac{1}{2}}}\,q^{-\frac{1}{2}A_{2}^{0}}\big{(}q^{\frac{1}{4}}A_{1}^{-}A_{2}^{+}+q^{-\frac{1}{4}}A_{1}^{+}A_{2}^{-}\big{)},$$

$$\displaystyle\chi(j_{2})=\frac{q^{\frac{1}{4}}}{q^{\frac{1}{2}}+q^{-\frac{1}{2}}}\,q^{-\frac{1}{2}A_{1}^{0}}\big{(}q^{\frac{1}{4}}A_{1}^{+}A_{2}^{-}-q^{-\frac{1}{4}}A_{1}^{-}A_{2}^{+}\big{)}.$$

$$\mu (J_0)={\mathcal{J}}_0=\frac{1}{2}\left(A^0+\frac{1}{2}\right),\mu (J_{\pm })={\mathcal{J}}_{\pm }=\frac{1}{[2{]}_{q^{1/2}}}(A^{\pm }{)}^2.$$

$$\displaystyle\mathscr{L}_{i,i+1}=q^{-\frac{1}{2}A_{i}^{0}+\frac{1}{4}}\big{(}q^{\frac{1}{4}}A_{i}^{+}A_{i+1}^{-}-q^{-\frac{1}{4}}A_{i}^{-}A_{i+1}^{+}\big{)},\qquad i=1,2,3.$$

$$M^{+}$$

$$M^{-}$$

$$L$$

$$H=\sum _{i=1}^4(A_i^0+\frac{1}{2}).$$

$$T^{\pm}=q^{\alpha(A_{1}^{0}+A_{2}^{0})}\big{(}q^{A_{2}^{0}+\frac{1}{2}}(A_{1}^{\pm})^{2}+(A_{2}^{\pm})^{2}\big{)},\qquad\alpha\in\mathbb{C}.$$

$$\displaystyle\begin{aligned} \big{[}L,M^{\pm}\big{]}&=\pm 4M^{\pm},\\ \big{[}M^{+},M^{-}\big{]}&=\frac{(1+q)}{q(1-q)^{3}}\;q^{H}\Big{(}(q+q^{-1})(q^{L}-q^{-L})-2\big{(}q^{\frac{1}{2}H}+q^{-\frac{1}{2}H}\big{)}(q^{\frac{1}{2}L}-q^{-\frac{1}{2}L})\Big{)}\\ &+\frac{(1+q)}{q^{2}(1-q)}\;q^{H}\Big{(}\big{(}q^{-\frac{1}{2}H}{\mathscr{L}_{12}}^{2}+q^{\frac{1}{2}H}{\mathscr{L}_{34}}^{2}\big{)}q^{\frac{1}{2}L}-\big{(}q^{\frac{1}{2}H}{\mathscr{L}_{12}}^{2}+q^{-\frac{1}{2}H}{\mathscr{L}_{34}}^{2}\big{)}q^{-\frac{1}{2}L}\Big{)}.\end{aligned}$$

Full text

Introduction

The Higgs algebra was first obtained by Higgs Higgs1979 as the algebra of the conserved quantities of the Coulomb problem and harmonic oscillator on the $2$-sphere. Shown to be isomorphic to the Hahn algebra Vinet2018, it was also identified as the symmetry algebra of the Hartmann potential Granovskii1991, of certain ring-shaped potentials Granovskii1992 and of the singular oscillator in two dimensions Letourneau1995, Genest2014. The Higgs algebra stands between Lie algebras and quantized universal enveloping algebras, as it can be viewed both as a deformation of the $\mathfrak{su}(2)$ Lie algebra Bonatsos1995 and a truncation of the $U_{q}(\mathfrak{sl}_{2})$ quantum algebra Zhedanov1992. It has been obtained as the quantum finite W-algebra $W(\mathfrak{sp}(4),2\,\mathfrak{sl}(2))$ Bowcock1994, Barbarin1995 and has also appeared in the context of Heisenberg quantization of identical particles Leinaas1993.

The Higgs algebra can be presented in the following form

[TABLE]

where $\alpha_{1}$, $\alpha_{2}$ are central elements.

We here aim to construct a $q$-deformation of (Introduction) that preserves the general algebraic underpinnings of this structure. This will lead to an algebra that differs from the one in Chung2014 where a certain $q$-extension of the Higgs algebra was defined by simply replacing the cubic expression in $D$ by one involving $q$-numbers (see (1.1)).

We propose to obtain a $q$-analogue of the Higgs algebra by following a commmutant approach similar to Frappat2018 (see also Gaboriaud2018a, Gaboriaud2018b), where the ordinary Higgs algebra was obtained as the commutant of the $\mathfrak{o}(2)\oplus\mathfrak{o}(2)$ subalgebra of $\mathfrak{o}(4)$ in the oscillator representation of $U(\mathfrak{u}(4))$. This characterization was shown to be in duality in the sense of Howe Howe1987, Howe1989, Howe1989a, Rowe2012 with the well-established embedding of the Hahn algebra in $U(\mathfrak{su}(1,1))\otimes U(\mathfrak{su}(1,1))$ Granovskii1988, Zhedanov1993. While Howe duality, sometimes called “complementarity”, has not been thoroughly studied in the context of $q$-algebras (see for instance Quesne1992, Smirnov1992, Green1999, Lehrer2011, Futorny2017), the results in Noumi1996 will provide appropriate background for our purposes. The merit of the approach we propose is that the $q$-Higgs algebra obtained as a commutant also appears in a dual fashion as a specialization of the Askey–Wilson algebra Zhedanov1991, Terwilliger2011, Huang2017 in the tensor product $U_{q}(\mathfrak{su}(1,1))^{\otimes 2}$.

Let us now briefly present the contents of the paper. In Section 1, the $q$-deformations of $\mathfrak{su}(1,1)$ and $\mathfrak{o}(n)$ (respectively denoted $U_{q}(\mathfrak{su}(1,1))$ and $\mathfrak{o}_{q}(n)$) will be introduced along with their $q$-oscillator realizations. In Section 2, a $q$-deformation of the Higgs algebra will be obtained as a commutant of $\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)\subset\mathfrak{o}_{q^{1/2}}(4)$ in the $q$-oscillator realization of $U_{q}(\mathfrak{u}(4))$. The embedding of a special case of the Askey–Wilson algebra into $U_{q}(\mathfrak{su}(1,1))^{\otimes 2}$ will be presented in Section LABEL:sec:aw. As will be shown in Section LABEL:sec:dualpair, the $q$-Higgs algebra proves to be isomorphic to that specialization of the Askey–Wilson algebra, and this result will be explained by invoking the fact that the pair \big{(}\mathfrak{o}_{q^{1/2}}(4),U_{q}(\mathfrak{su}(1,1))\big{)} behaves as a Howe dual pair in this context. Concluding remarks and perspectives will form the last section.

1 The $U_{q}(\mathfrak{su}(1,1))$, $\mathfrak{o}_{q}(n)$ algebras and their $q$-oscillators realizations

The duality connection that we shall invoke in our discussion involves the algebras $U_{q}(\mathfrak{su}(1,1))$ and $\mathfrak{o}_{q}(n)$. We shall thus begin by introducing these algebras and their $q$-oscillator realizations.

Let $q$ be a complex number such that $|q|<1$. One defines for any number $x$ the following $q$-numbers:

[TABLE]

The same notation will be used for operators.

1.1 The $U_{q}(\mathfrak{su}(1,1))$ and $\mathfrak{o}_{q}(n)$ quantum algebras

$U_{q}(\mathfrak{sl}_{2})$ Drinfeld1985, Jimbo1986 is the quantized universal enveloping algebra with three generators $j_{0}$ and $j_{\pm}$ subjected to the relations

[TABLE]

It is endowed with a Hopf structure with coproduct $\Delta:U_{q}(\mathfrak{sl}_{2})\to U_{q}(\mathfrak{sl}_{2})\otimes U_{q}(\mathfrak{sl}_{2})$

[TABLE]

We shall denote by $U_{q}(\mathfrak{su}(1,1))$ the non-compact real form of $U_{q}(\mathfrak{sl}_{2})$ that has the three generators $J_{\pm}$ and $J_{0}$ obeying

[TABLE]

The coproduct $\Delta:U_{q}(\mathfrak{su}(1,1))\to U_{q}(\mathfrak{su}(1,1))\otimes U_{q}(\mathfrak{su}(1,1))$ will read

[TABLE]

The Casimir operator $C$ of this algebra has the following expression

[TABLE]

The coproduct being an algebra morphism, the relations (1.5) define an embedding of $U_{q}(\mathfrak{su}(1,1))$ into $U_{q}(\mathfrak{su}(1,1))\otimes U_{q}(\mathfrak{su}(1,1))$.

Remark 1.1

In the limit $q\to 1$, one recovers the usual $\mathfrak{su}(1,1)$ Lie algebra with Casimir operator $C=J_{+}J_{-}-{J_{0}}^{2}+J_{0}$. Moreover, the standard presentation of $U_{q}(\mathfrak{su}(1,1))$ Klimyk1997 is recovered if one considers instead the generators $\widetilde{J}_{0}=J_{0}$,  $\widetilde{J}_{+}=J_{+}q^{-J_{0}}$ and $\widetilde{J}_{-}=q^{-J_{0}}J_{-}$, which satisfy the commutation relations \big{[}\widetilde{J}_{0}\,,\widetilde{J}_{\pm}\big{]}=\pm\widetilde{J}_{\pm} and \big{[}\widetilde{J}_{-}\,,\widetilde{J}_{+}\big{]}=\big{[}2\widetilde{J}_{0}\big{]}_{q} and have co-commutative coproduct.

We introduce next the non-standard $q$-deformation $\mathfrak{o}_{q}(n)$ of $\mathfrak{o}(n)$ which is defined as the associative unital algebra with generators $L_{i,i+1}$ ($i=1,\dots,n-1$) and relations

[TABLE]

In the litterature, this non-standard deformation is often denoted $U^{\prime}_{q}(\mathfrak{so}_{n})$, see for instance Gavrilik2000, Klimyk2001, Klimyk2002, Iorgov2005. It has been shown in Noumi1996a that $\mathfrak{o}_{q}(n)$ can be viewed as a $q$-analogue of the symmetric space based on the pair $(\mathfrak{gl}(n),\mathfrak{o}(n))$. Although it has no Hopf structure on its own, it is a coideal subalgebra of $U_{q}(\mathfrak{sl}(n))$ Noumi1996a and appears in many areas of mathematical physics Klimyk2002.

The two cases where $n=3$ and $n=4$ are especially of interest to us.

Let us first note that it is possible to consider a so-called “Cartesian” presentation Zhedanov1992a, Havlicek1999, Havlicek2001 of $U_{q}(\mathfrak{sl}_{2})$, in which the three generators play an “equitable” role, and which corresponds to the non-standard deformation $\mathfrak{o}_{q}(3)$ (equivalently $U^{\prime}_{q}(\mathfrak{so}_{3})$ in refs. Havlicek1999, Havlicek2001) of the universal enveloping algebra $U(\mathfrak{so}(3))$, obtained by modifying the defining relations for the skew-symmetric generators of $\mathfrak{so}(3)$.

It goes like this. With $j_{0}$, $j_{\pm}$, the $U_{q}(\mathfrak{sl}_{2})$ generators, form the following elements:

[TABLE]

where $\{a,b\}=ab+ba$  is the anticommutator and $g$ is a normalization factor. Defining $j_{3}\equiv[j_{1},j_{2}]_{q}$, where $[a,b]_{q}:=q^{\frac{1}{2}}ab-q^{-\frac{1}{2}}ba$  is the $q$-commutator, $j_{1}$, $j_{2}$ and $j_{3}$ then satisfy the “Cartesian” relations

[TABLE]

Upon identifying $L_{12}=j_{1}$, $L_{23}=j_{2}$, one finds that this corresponds precisely to the relations (1.7) for the algebra $\mathfrak{o}_{q}(3)$. Note that the relations (1.7c) do not exist in this case.

For what follows, it will also be useful to have the formulas for $\mathfrak{o}_{q}(4)$ in full. These relations read Klimyk1994

[TABLE]

It is immediate to see that $L_{12},L_{23}$ and $L_{23},L_{34}$ respectively generate two $\mathfrak{o}_{q}(3)$ subalgebras of $\mathfrak{o}_{q}(4)$, however they do not appear within a direct sum, in contrast to what happens with $\mathfrak{o}(4)$.

If one introduces the following elements:

[TABLE]

where $[a,b]_{q}$ is defined as above and $[a,b]_{q^{-1}}:=q^{-\frac{1}{2}}ab-q^{\frac{1}{2}}ba$, the two independent Casimir operators of the algebra $\mathfrak{o}_{q}(4)$ are then given by Noumi1996, Gavrilik2000, Havlicek2001

[TABLE]

1.2 The $q$-oscillator algebras, Schwinger and metaplectic realizations

Let us now recall the properties of the $q$-oscillator operators that will be used to realize the algebras presented above. The $q$-oscillator algebra $\mathcal{A}_{q}(n)$ Macfarlane1989, Biedenharn1989, Floreanini1991 is defined as the unital associative algebra over ${\mathbb{C}}$ generated by $n$ independent sets of $q$-oscillators $\{A_{i}^{\pm}$, $A_{i}^{0}\}$ verifying

[TABLE]

and such that the commutators between elements with different indices $i$ are equal to zero. The last two relations allow one to express $N_{i}=A_{i}^{+}A_{i}^{-}$ in terms of $A_{i}^{0}$:

[TABLE]

In the limit $q\to 1$, $A_{i}^{0}$ coincides with the usual number operator ${N}_{i}$.

The $q$-oscillator algebra has the following representation on the space spanned by the standard occupancy number states $|n_{1},\cdots,n_{n}\rangle=|n_{1}\rangle\otimes\cdots\otimes|n_{n}\rangle$ ($n_{i}\in{\mathbb{N}}$):

[TABLE]

These commuting $q$-oscillators can now be used to build realizations of the algebras considered above.

Firstly, the algebra $\mathfrak{o}_{q}(3)$ can be realized à la Schwinger in terms of two $q$-oscillators. More precisely, using the homomorphism $\chi:U_{q}(\mathfrak{sl}_{2})\to\mathcal{A}_{q}(2)$ given by

[TABLE]

and the identification (1.8), the following realization of $\mathfrak{o}_{q}(3)$ is obtained:

[TABLE]

Another key ingredient is the metaplectic realization of $U_{q}(\mathfrak{su}(1,1))$, which is given by the homomorphism $\mu:U_{q}(\mathfrak{su}(1,1))\to\mathcal{A}_{q}(1)$:

[TABLE]

One sees immediately that it is a $q$-deformation of the usual metaplectic representation of $\mathfrak{su}(1,1)$.

Finally, we shall also use the realization of $\mathfrak{o}_{q^{1/2}}(4)$ in terms of $4$ $q$-oscillators which is provided by:

[TABLE]

One checks that the $\mathscr{L}_{i,i+1}$ indeed verify relations of the form (1.10) but whose $q$’s have been replaced by $q^{1/2}$’s. Furthermore, $\mathscr{L}_{12}$, $\mathscr{L}_{34}$ commute and hence generate a $\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$.

2 The commutant of $\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)$ in the $q$-oscillator realization of $U_{q}(\mathfrak{u}(4))$ and the $q$-Higgs algebra

It was shown in Frappat2018 that the Higgs algebra appears as the commutant of $\mathfrak{o}(2)\oplus\mathfrak{o}(2)$ in the universal enveloping algebra $U(\mathfrak{u}(4))$. This section aims to define the $q$-Higgs algebra through a $q$-analogue of this commutant picture.

We consider first the $\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ generated by $\mathscr{L}_{12}$ and $\mathscr{L}_{34}$, and look for its commutant in $U_{q}(\mathfrak{u}(4))$.

Introduce the following three operators

[TABLE]

which commute with the generators $\mathscr{L}_{12}$ and $\mathscr{L}_{34}$ (in the limit $q\to 1$, $\mathscr{L}_{12}$ and $\mathscr{L}_{34}$ correspond to rotations in the $(1,2)$ and $(3,4)$ planes).

One notes that each big parenthesis in the expression of the $M^{\pm}$ operators can actually be obtained by applying the coproduct of $U_{q}(\mathfrak{su}(1,1))$ to the $\mathscr{J}_{\pm}$ generators. Recalling that the bilinears of the form $E_{ij}=A_{i}^{+}A_{j}^{-}$,  $i,j=1,2,3,4$  realize the $U_{q}(\mathfrak{u}(4))$ algebra Hayashi1990, it can be observed that $M^{\pm}$, $L$ generate the non-trivial part of the commutant of $\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)$ in the $q$-oscillator realization of $U_{q}(\mathfrak{u}(4))$.

It is immediate to see that $M^{\pm}$ and $L$ also commute with the central element

[TABLE]

One could ask how were the expressions for $L$, $M^{\pm}$ obtained. First, the operator $L$ obviously commutes with $\mathscr{L}_{12}$ and $\mathscr{L}_{34}$. Second, instead of obtaining the factors in $M^{\pm}$ from the coproduct one can look for elements $T^{\pm}$ in $\mathcal{A}_{q}(2)$ that commute with $\mathscr{L}_{12}$; this is most easily done “on-shell”, that is, by solving $[\mathscr{L}_{12},T^{\pm}]|n_{1},n_{2}\rangle=0$  for any two $q$-oscillator states. One thus arrives at

[TABLE]

Since $A_{1}^{0}+A_{2}^{0}={\textstyle{\frac{1}{2}}}(L+H)$, only the second factor of $T^{\pm}$ is relevant. The same is done with $\mathscr{L}_{34}$ on the direct product states $|n_{3},n_{4}\rangle$. It is then clear that the only combinations of the operators (2.3) and their $(3,4)$ analogues that will belong to $U_{q}(\mathfrak{u}(4))$ are those occurring in $M^{\pm}$.

It now remains to determine the algebra formed by the three generators $M^{\pm}$ and $L$.

[TABLE]

Proposition 2.1

The operators $M^{\pm}$ and $L$ have the following commutators:

[TABLE]

The elements ${\mathscr{L}_{12}}$, ${\mathscr{L}_{34}}$ and $H$, which are central, play the role of structure constants. We shall take these relations to define abstractly the (universal) $q$-Higgs algebra.