Graded Group Schemes and Graded Group Varieties

TL;DR

This paper introduces and develops the theory of graded group schemes and varieties, establishing their properties, examples, and classifications, and generalizing key results from classical group scheme theory to the graded setting.

Contribution

It defines graded group schemes and varieties, extends fundamental theorems to graded Hopf algebras, and provides classifications, broadening the understanding of algebraic structures in the graded context.

Findings

Established correspondence between graded group schemes and graded Hopf algebras

Generalized that connected graded bialgebras are graded Hopf algebras

Provided classification results for graded group algebras and varieties

Abstract

We define graded group schemes and graded group varieties and develop their theory. Graded group schemes are the graded analogue of group schemes and are in correspondence with graded Hopf algebra. In this setting, graded group varieties take the place of infinitesimal group schemes; an important and well studied class of group schemes. We give some examples, derive their properties, and define some special type of graded group schemes, like connected and \'{e}tale graded group schemes, among others. We give a generalization of the result that connected graded bialgebras are graded Hopf algebra. Our result is given in for a broader class of graded Hopf algebras; the coordinate rings of graded group varieties, In addition, this result can also be interpreted as the algebraic analogue of a well known geometric result regarding projective group schemes. We also give a classification for graded group algebras and graded group varieties.