# No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2

## TL;DR

This paper investigates the Gelfand-Kirillov dimension of associative dialgebras, establishing an upper bound related to associated algebras and proving that no dialgebra has a dimension strictly between 1 and 2.

## Contribution

It introduces bounds on the Gelfand-Kirillov dimension of dialgebras and proves the non-existence of dialgebras with dimensions between 1 and 2.

## Key findings

- Gelfand-Kirillov dimension of dialgebras is at most twice that of associated algebras.
- No associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2.

## Abstract

The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of $\mathcal{D}$ generated by the set $S:=\{x \vdash y-x\dashv y \mid x,y\in \mathcal{D}\}$, is called the associative algebra associated to $\mathcal{D}$. Here we show that the Gelfand--Kirillov dimension of $\mathcal{D}$ is bounded above by twice the Gelfand--Kirillov dimension of $\mathcal{A}_\mathcal{D}$. Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2.