Algebraic Approach to Quantum Isomorphisms

Abstract

In very brief, this thesis is a study of quantum isomorphisms. We have started with two pairs of quantum isomorphic graphs and looked for generalizations of those.

We have learned that those two pairs of graphs are related by Godsil-McKay switching and one of the graphs is an orthogonality graph of lines in a root system. These two observations lead to research in two directions.

First, since quantum isomorphisms preserve coherent algebras, we studied a question of when Godsil-McKay switching preserved coherent algebras. In this way, non isomorphic graphs related by Godsil-McKay switching with isomorphic coherent algebras are candidates to being quantum isomorphic and non isomorphic.

Second, while it was known that one of the graphs in a pair was an orthogonality graphs of the lines in a root system $E_8,$ we showed that a graph from another pair is also an orthogonality graph of the lines in a root system $F_4.$ We have studied orthogonality graphs of lines in root systems $B_{2^d},C_{2^d},D_{2^d}$ and showed that they have quantum symmetry.

Finally, we have touched upon structures of quantum permutations, relationships between fractional and quantum isomorphisms as well as connection to quantum independence and chromatic numbers.