On quantum adjacency algebras of Doob graphs and their irreducible modules

Abstract

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.