K-projectors

TL;DR

This paper develops an algebraic framework for studying specific representations of a free associative algebra, focusing on their structure and providing numerous examples.

Contribution

It introduces a new algebraic theory for representations with particular tensor product properties and constructs many explicit examples.

Findings

Established a classification of such representations

Developed methods to construct explicit examples

Provided insights into the structure of these algebraic objects

Abstract

We study representations of a free associative algebra $T^*(W\otimes W^*)$ in a vector space $V$ with the property $V\otimes V\cong V\oplus V_0$ where $T^*(W\otimes W^*)$ acts by zero on $V_0$ and the tensor product $V\otimes V$ of representations corresponds to the natural homomorphism $W\otimes W^*\to W\otimes W^* \otimes W\otimes W^*$. We develop an algebraic theory of such objects and construct a lot of examples.