For fixed integers \(n \ge 1\) and \(m \ge 0\), we consider the Doob graph \(D=D(n,m)\) formed by taking direct product of n copies of Shrikhande graph and m copies of complete graph \(K_4\). Fix a vertex x of D, and let \(T=T(x)\) denote the Terwilliger algebra of D with respect to x. Let A denote the adjacency matrix of D. There exists a decomposition of A into a sum \(A = L + F + R\) of elements in T where L, F, and R are the lowering, flat, and raising matrices, respectively. We call \(A = L + F + R\) the quantum decomposition of A. Hora and Obata (Quantum Probability and Spectral Analysis of Graphs. Theoretical and Mathematical Physics, Springer, Berlin, 2007) introduced a semi-simple matrix algebra based on the quantum decomposition of the adjacency matrix. This algebra is generated by the quantum components of the decomposition and is called the quantum adjacency algebra of the graph. Let \(Q=Q(x)\) denote the quantum adjacency algebra of D with respect to x. In this paper, we display an action of the special orthogonal Lie algebra \(\mathfrak {so}_4\) on the standard module for D. We also prove Q is generated by the center and the homomorphic image of the universal enveloping algebra \(U(\mathfrak {so}_4)\). To do these, we exploit the work of Tanabe (JAC 6: 173–195, 1997) on irreducible T-modules of D.

对于固定整数\（n \ ge 1 \）和\（m \ ge 0 \），我们考虑Doob图\（D = D（n，m）\）通过取Shrikhande图的n个副本的直接积和完整图形\（K_4 \）的m个副本。修复顶点X的d，让\（T = T（x）的\）表示的代数特威利格d相对于X。令A表示D的邻接矩阵。存在A分解为T中元素的和\（A = L + F + R \）的情况，其中L，F和R分别是下降矩阵，平坦矩阵和上升矩阵。我们称\（A = L + F + R \）为A的量子分解。Hora和Obata（图的量子概率和频谱分析。理论和数学物理，柏林，Springer，2007年）基于邻接矩阵的量子分解引入了半简单矩阵代数。该代数由分解的量子成分生成，称为图的量子邻接代数。令\（Q = Q（x）\）表示D相对于x的量子邻接代数。在本文中，我们展示了特殊正交李代数的作用D的标准模块上的\（\ mathfrak {so} _4 \）。我们还证明Q是由通用包络代数\（U（\ mathfrak {so} _4）\）的中心和同构图像生成的。为此，我们利用了田边（JAC 6：173–195，1997）关于D的不可约T-模块的工作。