No 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 $S:=\{x⊢y-x⊣y\mid x,y\in \mathcal{D}\}$, 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.

12 pages