The representation theory of Brauer categories I: triangular categories
Steven V Sam, Andrew Snowden
TL;DR
This paper introduces the concept of triangular categories, including the Brauer category, and develops a highest weight theory to study their representations, connecting diagram categories with Lie algebra structures.
Contribution
It defines a general notion of triangular category and applies it to the Brauer and related categories, establishing a new framework for their representation theory.
Findings
Brauer category admits a triangular structure
Develops a highest weight theory for these categories
Connects diagram categories with Lie algebra decomposition
Abstract
This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra