A Characterization of Linearly Semisimple Groups

TL;DR

This paper characterizes linearly semisimple affine group schemes over a field by the non-degeneracy of a trace form on their coordinate rings, linking algebraic properties to trace form behavior.

Contribution

It establishes a precise criterion connecting linear semisimplicity of affine group schemes with the non-degeneracy of a trace form on associated algebraic structures.

Findings

Linearly semisimple groups correspond to non-degenerate trace forms.

Provides an algebraic criterion for linear semisimplicity.

Connects trace form properties with group scheme reductivity.

Abstract

Let $G = Spec A$ be an affine $K$-group scheme and $\tilde{A} = \{w \in A*: dim_K A^* \cdot w \cdot A^* < \infty \}$. Let $< -,-> : A^* \times \tilde{A} \to K, (w,\tilde{w}) := tr(w \tilde{w})$, be the trace form. We prove that $G$ is linearly reductive if and only if the trace form is non-degenerate on $A^*$.