论文标题
$ h（d，2）$和$ \ frac {1} {2} {2} h（d，2）$的terwilliger代数背后的连接
A Connection Behind the Terwilliger Algebras of $H(D,2)$ and $\frac{1}{2} H(D,2)$
论文作者
论文摘要
$ \ mathfrak的通用代数$ U（\ Mathfrak {sl} _2）$的$ \ Mathfrak {Sl} _2 $是$ \ Mathbb c $的Unital Associative代数。 [E，f] = h。 \ end {align*}杰出的中心元素$$λ= ef+fe+\ frac {h^2} {2} $$称为$ u（\ mathfrak {sl} _2）$的casimir元素。通用的Hahn代数$ \ Mathcal H $是$ \ MATHBB C $与发电机$ a，b，c $的$ \ mathbb c $的一个联合缔合代数，并且关系断言，$ [a，b] = c $，\ begin \ begin {align*}α= [c，a]+2a+2a+2a^2+b，a]+2a^2+b，\ q quAD \ end {align*}在$ \ Mathcal H $中是中心。杰出的中央元素$$ω= 4aba+b^2-c^2-2βA+2（1-α）b $$称为$ \ Mathcal H $的Casimir元素。通过调查HyperCube的Terwilliger代数及其一半图之间的关系，我们发现了代数同构$ \自然：\ Mathcal H \ rightArrow U（\ Mathfrak {sl} _2 _2 _2）$发送\ begin \ begin {eqnArray*} a＆\ mapsto i} \ mapsto＆\ frac {e^2+f^2+λ-1} {4} - \ frac {h^2} {8} {8}，\\ c＆\ mapsto＆\ frac＆\ frac {e^2-f^2} {4} {4}。 \ end {eqnarray*}我们确定了$ \ natural $的图像，并表明$ \ natural $的内核是由$β$生成的$ \ Mathcal H $的双面理想，$β$和$16Ω-24α+3 $。通过$ \ natural $ $ $ u（\ mathfrak {sl} _2）$返回，可以将模块视为$ \ mathcal h $ -module。对于每个整数$ n \ geq 0 $，存在一个唯一的$（n+1）$ - 尺寸不可约$ u（\ mathfrak {sl} _2）$ - 模块$ l_n $ for isomorphism。我们表明，$ \ Mathcal H $ -MODULE $ L_N $（$ n \ geq 1 $）是两个非iSomorphic nordbhic nordribibible $ \ MATHCAL H $ MODULES的直接总和。
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.
