论文标题
$ u（\ mathfrak {sl} _2）$和约翰逊图的Terwilliger代数
The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs
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*}元素$$λ= ef+fe+\ frac {h^2} {2} $$称为$ u（\ mathfrak {sl} _2）$的casimir元素。令$δ：u（\ mathfrak {sl} _2）\ to u（\ mathfrak {sl} _2）\ otimes u（\ mathfrak {sl} _2）$表示$ u（\ mathfrak {sl} _2）_通用的Hahn代数$ \ MATHCAL H $是$ a，b，c $生成的$ \ mathbb c $超过$ \ mathbb c $的一个联合联想代数，并且关系断言$ [a，b] = c $，并且每个\ begin {align*} [align*} [c，a]+2a^2 a]+2a^2+b，\ q quad nis in in Central Internal Internal Internal Internal Internal Internal Internal in Cintral Internal Internal Internal Intral {b，b，b，b，b，b，b，b，b，crign* $ \ MATHCAL H $。受$ U（\ Mathfrak {sl} _2）$的clebsch-Gordan系数的启发，我们发现了一个代数同构$ \ natural：\ mathcal h \ to u（\ mathfrak {\ mathfrak {sl} _2 _2 _2） &\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）$ - 模块可以被视为$ \ Mathcal H $ -Module。对于任何整数$ n \ geq 0 $，都有一个唯一的$（n+1）$ - 尺寸不可约$ u（\ mathfrak {sl} _2）$ - 模块$ l_n $ to isomorphism to insomorphism。我们研究$ \ MATHCAL H $ -MODULE $ L_M \ otimes l_n $的分解，任何整数$ m，n \ geq 0 $。我们将这些结果与约翰逊图的Terwilliger代数联系起来。我们表达了约翰逊图的Terwilliger代数的尺寸。
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.
