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

ABSTRACT

The Gelfand-Kirillov dimension measures the asymptotic rate of growth of algebras. For every associative dialgebra $\mathcal{D}$, the quotient ${\mathcal{A}}_{\mathcal{D}}:=\mathcal{D}/Id\left(S\right)$, where $Id\left(S\right)$ 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 $\mathcal{D}$. We show that $GKdim\left(\mathcal{D}\right)\le 2GKdim\left({\mathcal{A}}_{\mathcal{D}}\right)$. Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2.

KEYWORDS:

• Associative algebra
• dialgebra
• Gelfand-Kirillov dimension

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 16S15
• 16P90
• 17A30

Acknowledgments

The authors are grateful to the anonymous referee for many valuable comments, especially for Example 3.9. The authors are also grateful to Xiangui Zhao for valuable discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (NNSF of China) under Grant 11571121; the NSF of Guangdong Province under Grant 2017A030313002; the Science and Technology Program of Guangzhou under Grant 201707010137.