arXiv subjects

Hau-Wen Huang

Publications and source records attributed to Hau-Wen Huang.

At least 19 recordsLinked to original sources

arXiv2026

The Askey--Wilson algebras, the Lie algebra $\mathfrak{so}_{3}$, and their fermionic realizations

This paper establishes a comprehensive algebraic framework linking the Lie algebra $\mathfrak{so}_{3}$ to the Askey--Wilson algebras. First, we provide a manifestly symmetric reformulation of the algebra homomorphism from the universal Racah algebra $\Re$ to $U(\mathfrak{sl}_2)$ by exploiting a Lie algebra isomorphism between $\mathfrak{sl}_{2}$ and $\mathfrak{so}_{3}$. This perspective facilitates a natural extension to the quantum setting, where we construct an explicit algebra homomorphism from the universal Askey--Wilson algebra $\triangle_{q^4}$ to the nonstandard quantum algebra $U_{q}^{\prime}(\mathfrak{so}_{3})$. By viewing the finite-dimensional irreducible $U_{q}^{\prime}(\mathfrak{so}_{3})$-modules of classical type as $\triangle_{q^4}$-modules, we demonstrate that the decomposition patterns perfectly parallel the branching rules of $U(\mathfrak{so}_3)$ over $\Re$. Furthermore, we extend this correspondence to the fermionic setting by establishing algebra isomorphisms between the skew group rings over $U(\mathfrak{so}_3)$ and $U_q'(\mathfrak{so}_3)$ and their associated anticommutator spin algebras. Collectively, these results provide a unified correspondence that bridges the gap between integrable algebraic structures, quantum groups, and their fermionic analogues.

math.RA↗

arXiv2025

A skew group ring of $\mathbb Z/2\mathbb Z$ over $U(\mathfrak{sl}_2)$, Leonard triples and odd graphs

We employ a skew group ring of $\mathbb Z/2\mathbb Z$ over $U(\mathfrak{sl}_2)$ to construct modules over the universal Bannai--Ito algebra. In addition, we give the conditions under which the defining generators act as Leonard triples on the resulting modules. As a combinatorial realization, we establish an algebra homomorphism from the universal Bannai--Ito algebra onto the Terwilliger algebra of an odd graph. This homomorphism provides a unified description of Leonard triples on all irreducible modules over the Terwilliger algebra.

math.CO↗

arXiv2025

A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs

The dual Hahn polynomials $\{u_i(x)\}_{i=0}^d$ are a family of discrete orthogonal polynomials involving two real parameters $r$ and $s$. Let $L,L^*$ denote the corresponding Leonard pair. Assume that $r\not=0$ and $r+s=0$. We show that $L,(L^*+\frac{r-d}{2})^{2}$ is a Leonard pair. According to the theory of Leonard pairs, the polynomials $\{u_i(x)\}_{i=0}^d$ are not only the dual Hahn polynomials but also the Racah polynomials with respect to the same inner product.

math.CA↗

arXiv2025

An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs

The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism $\natural$ from the universal Hahn algebra $\mathcal H$ into $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$. Let $Ω$ denote a finite set of size $D$ and $2^Ω$ denote the power set of $Ω$. It is generally known that $\mathbb C^{2^Ω}$ supports a $U(\mathfrak{sl}_2)$-module. Let $k$ denote an integer with $0\leq k\leq D$ and fix a $k$-element subset $x_0$ of $Ω$. By identifying $\mathbb C^{2^Ω}$ with $\mathbb C^{2^{Ω\setminus x_0}}\otimes \mathbb C^{2^{x_0}}$ this induces a $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^Ω}$ denoted by $\mathbb C^{2^Ω}(x_0)$. Pulling back via $\natural$ the $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb C^{2^Ω}(x_0)$ forms an $\mathcal H$-module. When $1\leq k\leq D-1$ the $\mathcal H$-module $\mathbb C^{2^Ω}(x_0)$ enfolds the Terwilliger algebra of the Johnson graph $J(D,k)$ with respect to $x_0$. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs. Unfortunately some steps break down in the $q$-analog case. By making detours, the imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.

math.CO↗

arXiv2024

Verma modules and finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity

Assume that $\mathbb F$ is an algebraically closed field and fix a nonzero scalar $q\in \mathbb F$ with $q^4\not=1$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative algebra over $\mathbb F$ defined by generators and relations. The generators are $A,B,C$ and the relations assert that each of \begin{gather*} A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \qquad B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \qquad C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}} \end{gather*} commutes with $A,B,C$. The Verma $\triangle_q$-modules are a family of infinite-dimensional $\triangle_q$-modules with marginal weights. Under the condition that $q$ is not a root of unity, it was shown that every finite-dimensional irreducible $\triangle_q$-module has a marginal weight and is isomorphic to a quotient of a Verma $\triangle_q$-module. Assume that $q$ is a root of unity. We prove that every finite-dimensional irreducible $\triangle_q$-module with a marginal weight is isomorphic to a quotient of a Verma $\triangle_q$-module. Properly speaking, two natural families of finite-dimensional quotients of Verma $\triangle_q$-modules contain all finite-dimensional irreducible $\triangle_q$-modules with marginal weights up to isomorphism. Furthermore, we classify the finite-dimensional irreducible $\triangle_q$-modules with marginal weights up to isomorphism.

math.RT↗

arXiv2024

Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$

From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $\sharp$ from the universal Racah algebra $\Re$ into $U(\mathfrak{sl}_2)$. We use the Casimir elements of $\Re$ to describe the kernel of $\sharp$. By pulling back via $\sharp$ every $U(\mathfrak{sl}_2)$-module can be viewed as an $\Re$-module. We show that for any finite-dimensional $U(\mathfrak{sl}_2)$-module $V$, the $\Re$-module $V$ is completely reducible and three generators of $\Re$ act on every irreducible $\Re$-submodule of $V$ as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube $H(D,2)$ and a decomposition of the second distance operator of $H(D,2)$ induced by Johnson graphs.

math.CO↗

arXiv2023

The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs

The universal enveloping algebra $U(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ is a unital associative algebra over $\mathbb C$ generated by $E,F,H$ subject to the relations \begin{align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H. \end{align*} The element $$ Λ=EF+FE+\frac{H^2}{2} $$ is called the Casimir element of $U(\mathfrak{sl}_2)$. Let $Δ:U(\mathfrak{sl}_2)\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$ denote the comultiplication of $U(\mathfrak{sl}_2)$. The universal Hahn algebra $\mathcal H$ is a unital associative algebra over $\mathbb C$ generated by $A,B,C$ and the relations assert that $[A,B]=C$ and each of \begin{align*} [C,A]+2A^2+B, \qquad [B,C]+4BA+2C \end{align*} is central in $\mathcal H$. Inspired by the Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$, we discover an algebra homomorphism $\natural:\mathcal H\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$ that maps \begin{eqnarray*} A &\mapsto & \frac{H\otimes 1-1\otimes H}{4}, \\ B &\mapsto & \frac{Δ(Λ)}{2}, \\ C &\mapsto & E\otimes F-F\otimes E. \end{eqnarray*} By pulling back via $\natural$ any $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module can be considered as an $\mathcal H$-module. For any integer $n\geq 0$ there exists a unique $(n+1)$-dimensional irreducible $U(\mathfrak{sl}_2)$-module $L_n$ up to isomorphism. We study the decomposition of the $\mathcal H$-module $L_m\otimes L_n$ for any integers $m,n\geq 0$. We link these results to the Terwilliger algebras of Johnson graphs. We express the dimensions of the Terwilliger algebras of Johnson graphs in terms of binomial coefficients.

math.CO↗

arXiv2023

A Connection Behind the Terwilliger Algebras of $H(D,2)$ and $\frac{1}{2} H(D,2)$

The universal enveloping algebra $U(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ is a unital associative algebra over $\mathbb C$ generated by $E,F,H$ subject to the relations \begin{align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H. \end{align*} The distinguished central element $$ Λ=EF+FE+\frac{H^2}{2} $$ is called the Casimir element of $U(\mathfrak{sl}_2)$. The universal Hahn algebra $\mathcal H$ is a unital associative algebra over $\mathbb C$ with generators $A,B,C$ and the relations assert that $[A,B]=C$ and each of \begin{align*} α=[C,A]+2A^2+B, \qquad β=[B,C]+4BA+2C \end{align*} is central in $\mathcal H$. The distinguished central element $$ Ω=4ABA+B^2-C^2-2βA+2(1-α)B $$ is called the Casimir element of $\mathcal H$. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism $\natural:\mathcal H\rightarrow U(\mathfrak{sl}_2)$ that sends \begin{eqnarray*} A &\mapsto & \frac{H}{4}, \\ B & \mapsto & \frac{E^2+F^2+Λ-1}{4}-\frac{H^2}{8}, \\ C & \mapsto & \frac{E^2-F^2}{4}. \end{eqnarray*} We determine the image of $\natural$ and show that the kernel of $\natural$ is the two-sided ideal of $\mathcal H$ generated by $β$ and $16 Ω-24 α+3$. By pulling back via $\natural$ each $U(\mathfrak{sl}_2)$-module can be regarded as an $\mathcal H$-module. For each integer $n\geq 0$ there exists a unique $(n+1)$-dimensional irreducible $U(\mathfrak{sl}_2)$-module $L_n$ up to isomorphism. We show that the $\mathcal H$-module $L_n$ ($n\geq 1$) is a direct sum of two non-isomorphic irreducible $\mathcal H$-modules.

math.CO↗

arXiv2023

The Clebsch-Gordan Rule for $U(\mathfrak{sl}_2)$, the Krawtchouk Algebras and the Hamming Graphs

Let $D\geq 1$ and $q\geq 3$ be two integers. Let $H(D)=H(D,q)$ denote the $D$-dimensional Hamming graph over a $q$-element set. Let ${\mathcal T}(D)$ denote the Terwilliger algebra of $H(D)$. Let $V(D)$ denote the standard ${\mathcal T}(D)$-module. Let $ω$ denote a complex scalar. We consider a unital associative algebra $\mathfrak K_ω$ defined by generators and relations. The generators are $A$ and $B$. The relations are $A^2 B-2 ABA +B A^2 =B+ωA$, $B^2A-2 BAB+AB^2=A+ωB$. The algebra $\mathfrak K_ω$ is the case of the Askey-Wilson algebras corresponding to the Krawtchouk polynomials. The algebra $\mathfrak K_ω$ is isomorphic to ${\rm U}(\mathfrak{sl}_2)$ when $ω^2\not=1$. We view $V(D)$ as a $\mathfrak{K}_{1-\frac{2}{q}}$-module. We apply the Clebsch-Gordan rule for ${\rm U}(\mathfrak{sl}_2)$ to decompose $V(D)$ into a direct sum of irreducible ${\mathcal T}(D)$-modules.

math.CO↗

arXiv2022

The universal DAHA of type $(C_1^\vee,C_1)$ and Leonard triples

Assume that $\mathbb F$ is an algebraically closed field and $q$ is a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra generated by $A,B, C$ and the relations state that each of $$ A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $\triangle_q$. The universal DAHA $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations state that \begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \\ \hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \\ t_0t_1t_2t_3=q^{-1}. \end{gather*} It was given an $\mathbb F$-algebra homomorphism $\triangle_q\to \mathfrak H_q$ that sends \begin{eqnarray*} A &\mapsto & t_1 t_0+(t_1 t_0)^{-1}, \\ B &\mapsto & t_3 t_0+(t_3 t_0)^{-1}, \\ C &\mapsto & t_2 t_0+(t_2 t_0)^{-1}. \end{eqnarray*} Therefore any $\mathfrak H_q$-module can be considered as a $\triangle_q$-module. Let $V$ denote a finite-dimensional irreducible $\mathfrak H_q$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the $\triangle_q$-module $V$.

math.RT↗

arXiv2022

The space of Dunkl monogenics associated with $\mathbb Z_2^3$ and the universal Bannai--Ito algebra

Let $n\geq 0$ denote an integer. Let $\mathscr M_n$ denote the space of Dunkl monogenics of degree $n$ associated with the reflection group $\mathbb Z_2^3$. The universal Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative algebra over $\mathbb C$ generated by $X,Y,Z$ and the relations assert that each of \begin{gather*} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather*} commutes with $X,Y,Z$. When the multiplicity function $k$ is real-valued the space $\mathscr M_n$ supports a $\mathfrak{BI}$-module in terms of the symmetries of the spherical Dirac--Dunkl operator. Under the assumption that $k$ is nonnegative, it was shown that $\dim \mathscr M_n=2(n+1)$ and $\mathscr M_n$ is isomorphic to a direct sum of two copies of an $(n+1)$-dimensional irreducible $\mathfrak{BI}$-module. In this paper, we improve the aforementioned result.

math.RT↗

arXiv2022

Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero

Assume that $\mathbb F$ is an algebraically closed with characteristic $0$. The Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative $\mathbb F$-algebra generated by $X,Y,Z$ and the relations assert that each of \begin{gather*} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather*} is central in $\mathfrak{BI}$. In this paper we classify the finite-dimensional irreducible $\mathfrak{BI}$-modules up to isomorphism. As we will see the elements $X,Y,Z$ are not always diagonalizable on finite-dimensional irreducible $\mathfrak{BI}$-modules.

math.RT↗

arXiv2022

Finite-dimensional modules of the universal Askey--Wilson algebra and DAHA of type $(C_1^\vee,C_1)$

Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C$ and the relations state that each of $$ A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $\triangle_q$. The universal DAHA (double affine Hecke algebra) $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations state that \begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \\ \hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \\ t_0t_1t_2t_3=q^{-1}. \end{gather*} Each $\mathfrak H_q$-module is a $\triangle_q$-module by pulling back via the injection $\triangle_q\to \mathfrak H_q$ given by \begin{eqnarray*} A &\mapsto & t_1 t_0+(t_1 t_0)^{-1}, \\ B &\mapsto & t_3 t_0+(t_3 t_0)^{-1}, \\ C &\mapsto & t_2 t_0+(t_2 t_0)^{-1}. \end{eqnarray*} We classify the lattices of $\triangle_q$-submodules of finite-dimensional irreducible $\mathfrak H_q$-modules. As a consequence, for any finite-dimensional irreducible $\mathfrak H_q$-module $V$, the $\triangle_q$-module $V$ is completely reducible if and only if $t_0$ is diagonalizable on $V$.

math.RT↗

arXiv2022

The universal additive DAHA of type $(C_1^\vee,C_1)$ and Leonard triples

Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra generated by $A,B,C,D$ and the relations state that $[A,B]=[B,C]=[C,A]=2D$ and each of $$ [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC $$ is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak H$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i\}_{i=0}^3$ and the relations state that \begin{gather*} t_0+t_1+t_2+t_3 = -1, \\ \hbox{$t_i^2$ is central for all $i=0,1,2,3$}. \end{gather*} Any $\mathfrak H$-module can be considered as a $\Re$-module via the $\mathbb F$-algebra homomorphism $\Re\to \mathfrak H$ given by \begin{eqnarray*} A &\mapsto & \frac{(t_0+t_1-1)(t_0+t_1+1)}{4}, \\ B &\mapsto & \frac{(t_0+t_2-1)(t_0+t_2+1)}{4}, \\ C &\mapsto & \frac{(t_0+t_3-1)(t_0+t_3+1)}{4}. \end{eqnarray*} Let $V$ denote a finite-dimensional irreducible $\mathfrak H$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the $\Re$-module $V$.

math.RT↗

arXiv2021

The Terwilliger algebra of the halved cube

Let $D\geq 3$ denote an integer. For any $x\in \mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $\mathbb F_2^D$ consisting of all $x\in \mathbb F_2^D$ with even $w(x)$. The $D$-dimensional halved cube $\frac{1}{2}H(D,2)$ is a finite simple connected graph with vertex set $X$ and $x,y\in X$ are adjacent if and only if $w(x-y)=2$. Fix a vertex $x\in X$. The Terwilliger algebra $\mathcal T=\mathcal T(x)$ of $\frac{1}{2}H(D,2)$ with respect to $x$ is the subalgebra of ${\rm Mat}_X(\mathbb C)$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^*=A^*(x)$ where $A^*$ is a diagonal matrix with $$ A^*_{yy}=D-2w(x-y) \qquad \hbox{for all $y\in X$}. $$ In this paper we decompose the standard $\mathcal T$-module into a direct sum of irreducible $\mathcal T$-modules.

math.CO↗

arXiv2021

An algorithm to evaluate the spectral expansion

Assume that $X$ is a connected $(q+1)$-regular undirected graph of finite order $n$. Let $A$ denote the adjacency matrix of $X$. Let $λ_1=q+1>λ_2\geq λ_3\geq \ldots \geq λ_n$ denote the eigenvalues of $A$. The spectral expansion of $X$ is defined by $$ Δ(X)=λ_1-\max_{2\leq i\leq n}|λ_i|. $$ By the Alon--Boppana theorem, when $n$ is sufficiently large, $Δ(X)$ is quite high if $$ μ(X)=q^{-\frac{1}{2}} \max_{2\leq i\leq n}|λ_i| $$ is close to $2$. In this paper, with the inputs $A$ and a real number $\varepsilon>0$ we design an algorithm to estimate if $μ(X)\leq 2+\varepsilon$ in $O(n^ω\log \log_{1+\varepsilon} n )$ time, where $ω<2.3729$ is the exponent of matrix multiplication.

cs.DS↗

arXiv2020

Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity

Let $\mathbb F$ denote an algebraically closed field and assume that $q\in \mathbb F$ is a primitive $d^{\rm \, th}$ root of unity with $d\not=1,2,4$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B,C$ and the relations assert that each of \begin{gather*} A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \qquad B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \qquad C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}} \qquad \end{gather*} commutes with $A,B,C$. We show that every finite-dimensional irreducible $\triangle_q$-module is of dimension less than or equal to $$ \left\{ \begin{array}{ll} d \quad &\hbox{if $d$ is odd}; \newline d/2 \quad &\hbox{if $d$ is even}. \end{array} \right. $$ Moreover we provide an example to show that the bound is tight.

math.RT↗

arXiv2020

Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^\vee,C_1)$

Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C, D$ and the relations state that $$ [A,B]=[B,C]=[C,A]=2D $$ and each of \begin{gather*} [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC \end{gather*} is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak H$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $t_0,t_1,t_0^\vee,t_1^\vee$ and the relations state that $$ t_0+t_1+t_0^\vee+t_1^\vee=-1 $$ and each of $t_0^2, t_1^2, t_0^{\vee 2}, t_1^{\vee 2}$ is central in $\mathfrak H$. Each $\mathfrak H$-module is an $\Re$-module by pulling back via the algebra homomorphism $\Re\to \mathfrak H$ given by \begin{eqnarray*} A &\mapsto & \frac{(t_1^\vee+t_0^\vee)(t_1^\vee+t_0^\vee+2)}{4}, \\ B &\mapsto & \frac{(t_1+t_1^\vee)(t_1+t_1^\vee+2)}{4}, \\ C &\mapsto & \frac{(t_0^\vee+t_1)(t_0^\vee+t_1+2)}{4}. \end{eqnarray*} Let $V$ denote any finite-dimensional irreducible $\mathfrak H$-module. The set of $\Re$-submodules of $V$ forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the $\Re$-module $V$ is completely reducible if and only if $t_0$ is diagonalizable on $V$.

math.RT↗