Date of Publication

2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics Department

Thesis Adviser

John Vincent S. Morales

Defense Panel Chair

Arlene A. Pascasio

Defense Panel Member

Jose Maria P. Balmaceda
Franis Joseph H. Campeña
Rafael Reno S. Cantuba
Leonor A. Ruivivar

Abstract/Summary

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 Obata
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 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

Abstract Format

html

Language

English

Format

Electronic

Physical Description

48 leaves

Keywords

Algebra; Quantum groups; Charts, diagrams, etc.; Graphic methods

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Embargo Period

5-23-2022

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