Conference abstracts
Session S02 - Commutative Algebra and Algebraic Geometry
July 25, 17:30 ~ 18:00
Algebras with a negation map
Louis Rowen
Bar-Ilan University, Israel - rowen@macs.biu.ac.il
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, called systems in the context of universal algebra, leading to more viable (super)tropical versions of these algebraic structures. Some basic results are obtained in linear algebra, linking determinants to linear independence. This approach also is applied to other theories, such as hyper fields.
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. Next, we use this functor to analyze some tropical structures and propose tropical analogs of classical algebraic notions.