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arXiv · 1904.12677

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

Zerui Zhang·Yuqun Chen·Bing Yu
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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.

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Zerui Zhang, Yuqun Chen, Bing Yu. 2019-04-26. No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2. https://arxiv.org/abs/1904.12677

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