Face Algebras of Ribbon Graphs
Faculty Mentor Name
Alex Dugas
Research or Creativity Area
Other
Abstract
Given a surface such as a sphere or torus, we may embed on it a system of arrows called a ribbon graph such that each arrow is a part of a face cycle. Then from such a configuration, we define a "face algebra" as being an algebraic structure constructed from sequences or paths of arrows that are part of a cycle. We show how they are related to other well-studied class of algebras called Jacobian algebras of dimer models, which are studied in the fields of representation theory and string theory. We investigate abstract properties of face algebras, and prove that they are finite-dimensional and symmetric. This project also included a programming component, wherein the computer algebra system GAP was used to compute examples of face algebras starting from the permutations that define the underlying ribbon graph.
Face Algebras of Ribbon Graphs
Given a surface such as a sphere or torus, we may embed on it a system of arrows called a ribbon graph such that each arrow is a part of a face cycle. Then from such a configuration, we define a "face algebra" as being an algebraic structure constructed from sequences or paths of arrows that are part of a cycle. We show how they are related to other well-studied class of algebras called Jacobian algebras of dimer models, which are studied in the fields of representation theory and string theory. We investigate abstract properties of face algebras, and prove that they are finite-dimensional and symmetric. This project also included a programming component, wherein the computer algebra system GAP was used to compute examples of face algebras starting from the permutations that define the underlying ribbon graph.
