A Connection Behind the Terwilliger Algebras ofH(D,2)and½ H(D,2) - Huang, Hau-Wen; Wen, Chia-Yi (2022)
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 align* [H,E]=2E, [H,F]=-2F, [E,F]=H. align* The distinguished central element$$ \Lambda=EF+FE+\frac{H^2}{2} $$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 align* =[C,A]+2A^2+B, =[B,C]+4BA+2C align* is central in$\mathcal H$ . The distinguished central element$$ \Omega=4ABA+B^2-C^2-2\beta A+2(1-\alpha)B $$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 eqnarray* A & & H4, \\ B & & E^2+F^2+-14-H^28, \\ C & & E^2-F^24. eqnarray* We determine the image of$\natural$and show that the kernel of$\natural$is the two-sided ideal of$\mathcal H$generated by$\beta$and$16 \Omega-24 \alpha+3$ . 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.