Pogorelsky, Barbara; Renz, Carolina Right coideal subalgebras of a bosonization of the Fomin-Kirillov algebra \(\mathcal{FK}_3\). (English) Zbl 1470.16063 Commun. Algebra 49, No. 2, 567-578 (2021). The classification of all right coideal subalgebras has been achieved for some quantized enveloping algebras of semisimple Lie algebras and for some bosonizations of Nichols algebras by Hopf algebras with bijective antipode, and in each case a distinction is made between those right coideal subalgebras containing the coradical and those not containing it. The work at hand develops a similar classification for a Hopf algebra that is not a quantized enveloping algebra of a semisimple Lie algebra: \({\mathcal FK}_3\#\Bbbk\mathbb{S}_3\), the bosonization of the Fomin-Kirillov algebra \({\mathcal FK}_3\) over the symmetric group \({\mathbb S}_3\).As explained in Section 2, given a finite-dimensional Hopf algebra and a Yetter-Drinfeld module \(V\) over it (i.e., an \(H\)-module and \(H\)-comodule satisfying a compatibility relation between the \(H\)-coaction and the coproduct), one can construct the Nichols algebra of \(V\) as a quotient of the tensor algebra \(T(V)\) by its maximal ideal and coideal generated by homogeneous elements of degree greater than 2. If one picks the Yetter-Drinfeld module \(V=\Bbbk [x_{(12)}, x_{(23)},x_{(13)}]\) over \(\Bbbk {\mathbb S}_3\), one gets an algebra that is isomorphic to the Fomin-Kirillov algebra \({\mathcal FK}_3\). Using this fact, the authors explore the properties of the Nichols algebra of this specific \(V\), and present a basis for it and the defining relations among those elements. An appendix describes at length the coproduct on this Hopf algebra.The bosonization of \({\mathcal FK}_3\) (again considering the Nichols algebra of the previous \(V\)) is made explicit towards the end of Section 2, and Sections 3 and 4 follow the lengthy calculations of all right coideal subalgebras of \(\mathcal{H} = {\mathcal FK}_3\#\Bbbk\mathbb{S}_3\). As for the examples alluded to in the Introduction, the process is done in two distinct phases.First, in Section 3, one gets those right coideal subalgebras containing the coradical \(\Bbbk\mathbb{S}_3\). Besides the full \(\mathcal{H}\), one gets \(\mathcal{A}_0 = \Bbbk\mathbb{S}_3\), \(\mathcal{A}_1\) (the subalgebra generated by \(x_{(12)}+x_{(13)}+x_{(23)}\) and \(\Bbbk\mathbb{S}_3\)), and \(\mathcal{A}_2\) (the subalgebra generated by \(x_{(12)}-x_{(13)}\) and \(\Bbbk\mathbb{S}_3\)). The reasoning involves not only proving that these are indeed right coideal subalgebras, but also that \(\mathcal{A}_1\) and \(\mathcal{A}_2\) are maximal.Section 4 deals with right coideal subalgebras not containing the coradical. Here, the situation is more complex, and there exists a total of 16 such subalgebras, mostly distributed among 8 maximal chains of right coideal subalgebras. As for the previous section, the work involves proving that these objects are right coideal subalgebras of \(\mathcal{H}\) and that they are either maximal or below a maximal subalgebra in one of the presented chains. Reviewer: Rui Miguel Saramago (Porto Salvo) MSC: 16T05 Hopf algebras and their applications 57T05 Hopf algebras (aspects of homology and homotopy of topological groups) 16T15 Coalgebras and comodules; corings 17B37 Quantum groups (quantized enveloping algebras) and related deformations Keywords:coideal subalgebra; Hopf algebras; Nichols algebras PDFBibTeX XMLCite \textit{B. Pogorelsky} and \textit{C. Renz}, Commun. Algebra 49, No. 2, 567--578 (2021; Zbl 1470.16063) Full Text: DOI References: [1] Andruskiewitsch, N.; Graña, M., Braided Hopf algebras over non abelian finite groups, Bol. Acad. Ciencias (Córdoba), 63, 45-78 (1999) · Zbl 1007.17010 [2] Andruskiewitsch, N.; Schneider, H. J., New Directions in Hopf Algebras, Pointed Hopf algebras, 1-68 (2002), MSRI Publications Cambridge University Press [3] Ferreira, V. O.; Murakami, L. S.I.; Paques, A., A Hopf-Galois correspondence for free algebras, J. Algebra, 276, 1, 407-416 (2004) · Zbl 1064.16039 [4] Fomin, S.; Kirillov, A. 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