On the support of (R,+)-gradings on the Grassmann algebra

Abstract

Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic different from 2. Let $E_{(r_1,\dots ,r_n)}^{(v_1,\dots ,v_n)}$ be a n-induced $(\mathbb{R},+)$-grading on E and let $S_{(r_1,\dots ,r_n)}^{(v_1,\dots ,v_n)}$ be its support. In this paper, we study the properties of the set $S_{(r_1,\dots ,r_n)}^{(v_1,\dots ,v_n)}$ as a subset of $\mathbb{R}.$ Assuming that this set is bounded, we describe the graded identities of the respective graded algebra. We also show that $S_{(r_1,r_2)}^{(v_1,v_2)}$ is a subgroup of $(\mathbb{R},+)$ if and only if it is cyclic, and we describe the graded identities in this case.

Keywords:

• Graded algebra
• graded identity
• Grassmann algebra
• subgroups of the real line

MSC Classification 2010:

• 15A75
• 16R10
• 16W50
• 20K27
• 20K99
• 26A03
• 26A99

Acknowledgments

We thank Claudemir Fidelis for the useful discussions and for his valuable comments. Additionally, we are grateful to the anonymous referee for his/her useful comments and suggestions which helped us to substantially clarify and improve the manuscript.