Abstract
In this chapter, we prove R. Steinberg’s Theorem [Ste62] that the direct sum of the tensor powers of a faithful representation of a monoid yields a faithful representation of the monoid algebra. We also commence the study of a special family of ideals, called bi-ideals, which will be at the heart of the next chapter.The results of this chapter should more properly be viewed as about bialgebras, but we have chosen not to work at that level of generality in order to keep things more concrete. The approach we follow here is influenced by Passman [Pas14]. The bialgebraic approach was pioneered by Rieffel [Rie67].
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These are precisely the ideals for which A∕I is a quotient bialgebra [Pas14].
References
D.S. Passman, Elementary bialgebra properties of group rings and enveloping rings: an introduction to Hopf algebras. Commun. Algebra 42 (5), 2222–2253 (2014)
M.A. Rieffel, Burnside’s theorem for representations of Hopf algebras. J. Algebra 6, 123–130 (1967)
R. Steinberg, Complete sets of representations of algebras. Proc. Am. Math. Soc. 13, 746–747 (1962)
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Steinberg, B. (2016). 10 Bi-ideals and R. Steinberg’s Theorem. In: Representation Theory of Finite Monoids. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-43932-7_10
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DOI: https://doi.org/10.1007/978-3-319-43932-7_10
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