Algebraic Geometry Seminar

Algebras with a Negation Map

Speaker: Louis Rowen, Bar-Ilan University

Location: Warren Weaver Hall 705

Date: Thursday, September 8, 2016, 2 p.m.


Our objective in this talk is three-fold. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient, such as determinants, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras, one often is challenged by the lack of negation. Following an idea originating in work of Gaubert and the Max-Plus group and brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called systems, in the context of universal algebra, showing how these encompass the more viable (super)tropical versions. Special attention is paid to metatangible systems, whose algebraic theory is rich enough to provide a host of structural results. Some basic results also are obtained in linear algebra, linking determinants to linear independence. Formulating the structure axiomatically enables us to view the tropicalization functor as a morphism, thereby further explaining the mysterious link between classical algebraic results and their tropical analogs, as well as with hyperfields. Next, we use the tropicalization functor to analyze some tropical structures and propose tropical analogs of classical algebraic notions.