Algebra and Combinatorics Seminar

Date: April 12, 2019

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Michael Brannan, Texas A&M University

Title: Hopf algebras and non-local games

Abstract: A non-local game consists of two players, who are each provided questions from a referee and then supply answers. The game comes with rules which determine if the answers supplied by the players are correct or not. The players cooperate to win each round of the game, but the ``non-locality'' of the game means that the players cannot communicate by classical means during each round of the game. They can, however, agree upon a shared strategy for producing satisfactory answers. Non-local games are of interest in quantum information theory because quite often the only winning strategy is a so-called quantum strategy - i.e., one which utilizes some shared resource of quantum entanglement between the players. In this talk, I will focus on a particular class of non-local games, called synchronous games. For these games one can associate to a game an associative algebra A whose structure completely characterizes the existence of winning deterministic and probabilistic (quantum) strategies for these games. As a particular example, I will focus on the graph isomorphism game, which takes as inputs two graphs, and is devised so that a winning deterministic strategy requires that the two graphs be isomorphic. On the other hand, a probabilistic winning strategy relaxes this condition to the two graphs being what quantum information theorists call ``quantum isomorphic''. I will explain how the notion of quantum isomorphism mentioned above is intimately connected to the theory of Hopf bi-Galois objects: Two graphs are quantum isomorphic if and only if the game algebra A is a Hopf bi-Galois extension over the universal Hopf algebras coacting on the function algebras of the two graphs. I will explain how this Hopf-algebraic interpretation of the graph isomorphism game provides some fundamental new insights. **NOTE**: There will be a sequel to this talk given by Kari Eifler (TAMU) at 4pm in the Linear Analysis Seminar. Both talks will be complementary, yet self-contained.