Infinitely dimensional, affine nil algebras $A\otimes A^{op}$ and $A\otimes A$ exist

TL;DR

This paper constructs finitely generated, infinite-dimensional algebras over algebraically closed fields where their tensor products with themselves and their opposites are nil, answering previously open questions.

Contribution

It demonstrates the existence of such algebras, providing new examples and insights into the structure of nil algebras and tensor products.

Findings

Existence of finitely generated, infinite-dimensional nil algebras with nil tensor products

Answers two open questions from prior research

Provides explicit constructions over algebraically closed fields

Abstract

In this paper we answer two questions from [16], by showing that, over any algebraically closed field, $K$, there is a finitely generated, infinitely dimensional algebra $A$ such that algebras $A\otimes_{K}A$ and $A\otimes_{K} A^{op}$ are nil.