论文标题
带有$ e_6 $对称和$ sl的Clebsch-Gordan系列的Calabi-yau代数（3）$
A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
论文作者
论文摘要
在经典不变理论的基础上，观察到偏光痕迹会产生$ u（sl（n））$的对角线嵌入$ u（sl（n））^{\ otimes l} $中的centraliser $ z_l（sl（n））$。然后，该论文专注于$ sl（3）$，案例$ l = 2 $。引入并明确证明具有三个发电机的Calabi - Yau代数$ \ Mathcal {a} $具有PBW基础和特定的中心元素。可以看出，$ z_2（sl（3））$通过单个显式关系固定中心元素的值，对代数$ \ Mathcal {a} $的商都是同构。当专注于Clebsch-Gordan系列$ U（SL（3））$中的三个最高权重表示后，出现了$ \ Mathcal {a} $的专业化，涉及表征三个最高权重的成对的数字。在$ u（sl（3））\ otimes u（sl（3））$中的这种实现中，定义关系和中心元素的值的系数具有与$ e_6 $类型的Weyl类的基本学位相对应的程度。通过在表示形式的六个参数与$ e_6 $的某些根之间的正确关联后，已清楚地表明了$ e_6 $的整个Weyl群体下的对称性。 $ u（sl（3））\ otimes u（sl（3））$在实现中的关系系数和中心元素的价值是根据与$ e_6 $相关的基本不变多项式表示的。还表明，代数$ \ Mathcal {a} $的关系可以与RACAH或HAHN代数的Heun类型运营商实现。
Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$. A Calabi--Yau algebra $\mathcal{A}$ with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that $Z_2(sl(3))$ is isomorphic to a quotient of the algebra $\mathcal{A}$ by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of $U(sl(3))$, a specialisation of $\mathcal{A}$ arises, involving the pairs of numbers characterising the three highest weights. In this realisation in $U(sl(3))\otimes U(sl(3))$, the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type $E_6$. With the correct association between the six parameters of the representations and some roots of $E_6$, the symmetry under the full Weyl group of type $E_6$ is made manifest. The coefficients of the relations and the value of the central element in the realisation in $U(sl(3))\otimes U(sl(3))$ are expressed in terms of the fundamental invariant polynomials associated to $E_6$. It is also shown that the relations of the algebra $\mathcal{A}$ can be realised with Heun type operators in the Racah or Hahn algebra.
