Invariants for isomorphism classes in the category $\bcalNT$

TL;DR

This paper introduces invariants for classifying isomorphism classes in a category of graded algebras, providing new criteria and bounds that improve previous results in algebraic classification.

Contribution

It defines new invariants for isomorphism classification in the category bcalNT and uses them to improve bounds on the number of classes.

Findings

Defined invariants for isomorphism classes in bcalNT.

Established new isomorphism criteria using these invariants.

Provided a tighter lower bound for the number of isomorphism classes.

Abstract

The category $\bcalNT$ is a category of certain commutative graded algebras over a field. It was introduced in \cite{Lobos2} as a generalization of algebras generated by Jucys-Murphy elements in the many \textbf{End} algebras of the diagrammatic Soergel category of Elias and Williamson. In the first part of this article we define certain \emph{Invariants} for the isomorphism classes in $\bcalNT,$ following in the same spirit of \cite{Lobos3}, where a series of \emph{Isomorphism criteria} were found. At the end, we use our invariants to provide a new lower bound for the number of isomorphism classes, improving a similar result obtained in \cite{Lobos3}.