On quantum adjacency algebras of Doob graphs and their irreducible modules

• Main Author: Palma, Tessie M.
• Format: text
• Language: English
• Published: Animo Repository 2020
• Subjects: Algebra; Quantum groups; Charts, diagrams, etc.; Graphic methods; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/1403; https://animorepository.dlsu.edu.ph/context/etd_doctoral/article/2437/viewcontent/Palma_Tessie_11698799_On_quantum_adjacency_algebras_of_Doob_graphs_and_their_irreducible_modules_Partial.pdf
• Institution: De La Salle University

For fixed integers n ≥ 1 and m ≥ 0, we consider the Doob graph D = D(n, m) which is formed by taking direct product of n copies of Shrikhande graph and m copies of complete graph K4. Fix a vertex x of D and let T = T(x) denote the Terwilliger algebra of D with respect to vertex x. Let A denote the adjacency matrix of D. There exists a decomposition of A into a sum A = L + F + R (1) of elements of T where L, F, and R are the lowering, flat, and raising matrices, re- spectively. We call (1) the quantum decomposition of A. In 2007, Hora and Obataintroduced a semi-simple matrix algebra based on the quantum decomposition of the adjacency matrix. This algebra is generated by the quantum components of the de- composition 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 show that there exists an algebra homomorphism U(so4) → Q where U(so4) is the universal enveloping algebra of the special orthogonal Lie algebra so4. We also show that Q is generated by the center and the homomorphic image of U(so4). Keywords. Terwilliger algebra, quantum adjacency algebra, Doob graphs, Q-polynomial distance-regular graph, special orthogonal Lie algebra