A characterization of linearly semisimple groups (Q546284)

Let \(K\) be any field and \(G\) be any finite group of order \(n\). Maschke's theorem implies that if \(\text{g.c.d}(\text{char}(K), n) = 1\) then finite dimensional representations of \(G\) over \(K\) vector spaces are completely reducible (or semi-simple). In this case, the bilinear pairing \(\mathrm{tr}: K[G] \otimes_{K} K[G] \rightarrow K\) defined by \(\mathrm{tr}(a, b) := \mathrm{trace}_{K[G]}(a.b)\) (\(K[G]\) is a finite dimensional vector space/ \(K\)) is non-degenerate. Consider case of an finite-dimensional affine group scheme \(G = \mathrm{Spec}(A)\) over \(\mathrm{Spec}(K)\). Let \(A^{\star}\) denote the dual vector space endowed with an algebra structure coming from point-wise multiplication. Consider the two-sided ideal \(\tilde{A} := \{ f\in A^{\star} : \dim_{K}(f.A^{\star}) < \infty\text{ and }\dim_{K}(A^{\star}. f) < \infty \}\). For any \(f \in \tilde{A}\) the left-multiplication map \(m(a): \tilde{A} \rightarrow \tilde{A}\) has finite rank and hence \(\mathrm{tr}(a):= \mathrm{trace}(m(a))\) is well-defined. This map extends to a bilinear pairing \(\mathrm{Tr}: A^{\star} \otimes_{K} \tilde{A} \rightarrow K\) defined by \(\mathrm{Tr}(f,g) =\mathrm{tr}(m(fg))\). The main-result of this paper is that the category of \(K\)-rational representations of the affine group scheme \(G = \mathrm{Spec}(A)\) is semi-simple if and only if the trace form \(\mathrm{Tr}(,)\) is non-degenerate on \(A^{\star}\) (i.e. \(\mathrm{Tr} : A^{\star} \rightarrow (\tilde{A})^{\star} \) is injective). When \(\tilde{A}\) is dense in \(A^{\star}\) (the topology under consideration is not clearly enunciated) and \(K\)-rational representations of \(G\) are semi-simple then the authors construct a geometric parametrization of irreducible (\(K\)-rational) representations of \(G\). References: [\textit{W. Murray}, ``Bilinear forms on Frobenius algebras'', J. Algebra 293, No. 1, 89--101 (2005; Zbl 1121.16019), \textit{M. E. Sweedler}, Hopf algebras. New York: W.A. Benjamin, Inc. (1969; Zbl 0194.32901), \textit{A. Alvarez, C. Sancho} and \textit{P. Sancho}, ``Algebra schemes and their representations'', J. Algebra 296, No. 1, 110--144 (2006; Zbl 1095.14043)].