article · Tsukuba Journal of Mathematics · 1982
On observable and strongly observable Hopf ideals
Cited by 3Open access
Abstract
Let $\mathfrak{G}=\mathfrak{S}\mathfrak{p}_{k}A$ be an affine group scheme over a ground field $k,$ $\mathfrak{H}=\mathfrak{S}\mathfrak{p}_{k}A/I$ a closed subgroup scheme and $\overline{k}$ denote an algebraic closure of $k$ . When $\mathfrak{G}$ is algebraic and $\mathfrak{G}(k)$ is dense in $\mathfrak{G}(\overline{k})$ , Bialynichi-Birula, Hochschild and Mostow defined in their paper When $\mathfrak{G}$ is algebraic and reduced, and $k=\overline{k}$ , Cline, Parshall and Scott defined in their paper [2] that $\mathfrak{H}(k)$ is strongly observable in $\mathfrak{G}(k)$ if every rational $\mathfrak{H}(k)$ -module $N$ can be embedded in a rational $\mathfrak{G}(k)$ -module $M$ as a $\mathfrak{H}(k)$ -module such that $N^{\mathfrak{H}(k)}=M^{\mathfrak{G}(k)}$ , and showed that one of its necessary and sufficient conditions is $\mathfrak{G}(k)/\mathfrak{H}(k)$ being an affine variety. Since these concepts are the representation-theoretic ones, we can extend them to a Hopf ideal of an arbitrary (not necessarily commutative) Hopf algebra. In this paper, we characterize a strongly observable Hopf ideal of an arbitrary Hopf algebra; a Hopf ideal $I$ of a Hopf algebra $A$ is strongly observable if and only if $A$ is an injective $A/I$ -comodule. This result and the one of M. Takeuchi This is a generalization of a main theorem in [2]. In case of observable Hopf ideals, we do not have such a general characterization except the case when Hopf algebras are commutative. So we can extend some results in One of them is that $\mathfrak{G}/\mathfrak{H}\approx$ is quasi-affine if and only if $I$ is observable in $A$ and $R=A\coprod_{A/I}k$ contains a simple left coideal $M$ such that $M\subset\sqrt{\mathfrak{A}}$ for any left coideal-ideal $\mathfrak{A}$ in $R$ , where $\square $ denotes the cotensor product (see the first section).
Cite this paper
Shigano, H. (1982). On observable and strongly observable Hopf ideals. Tsukuba Journal of Mathematics, 6(1). https://doi.org/10.21099/tkbjm/1496159455
Shigano, Hiroshi. “On observable and strongly observable Hopf ideals.” Tsukuba Journal of Mathematics, vol. 6, no. 1, 1982, https://doi.org/10.21099/tkbjm/1496159455.
Shigano, Hiroshi. 1982. “On observable and strongly observable Hopf ideals.” Tsukuba Journal of Mathematics 6 (1). https://doi.org/10.21099/tkbjm/1496159455.
Shigano, H. (1982) ‘On observable and strongly observable Hopf ideals’, Tsukuba Journal of Mathematics, 6(1). Available at: https://doi.org/10.21099/tkbjm/1496159455.
@article{shigano1982observable,
title = {{On observable and strongly observable Hopf ideals}},
author = {Shigano, Hiroshi},
journal = {Tsukuba Journal of Mathematics},
volume = {6},
number = {1},
year = {1982},
doi = {10.21099/tkbjm/1496159455}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.