David, Marie-Claude; Thiéry, Nicolas M. Exploration of finite-dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions. (English) Zbl 1241.46043 J. Algebra Appl. 10, No. 5, 995-1106 (2011). Summary: We study the four infinite families \(KA(n), KB(n), KD(n)\), and \(KQ(n)\) of finite-dimensional Hopf (in fact Kac) algebras constructed, respectively, by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal sub-algebras. We reduce the study to \(KD(n)\) by proving that the others are isomorphic to \(KD(n)\), its dual, or an index 2 subalgebra of \(KD(2n)\). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we describe. Cited in 3 Documents MSC: 46L65 Quantizations, deformations for selfadjoint operator algebras 46L37 Subfactors and their classification 16T05 Hopf algebras and their applications 16-04 Software, source code, etc. for problems pertaining to associative rings and algebras Keywords:finite-dimensional Kac algebras; Hopf algebras; quantum groupoids; lattices of intermediate subfactors; principal graphs; computer exploration Software:MuPAD; SageMath PDFBibTeX XMLCite \textit{M.-C. David} and \textit{N. M. Thiéry}, J. Algebra Appl. 10, No. 5, 995--1106 (2011; Zbl 1241.46043) Full Text: DOI arXiv References: [1] DOI: 10.1090/S0894-0347-08-00602-4 · Zbl 1215.20022 [2] DOI: 10.5802/aif.1719 · Zbl 0938.46050 [3] Bisch D., J. Reine Angew. Math. 455 pp 21– [4] DOI: 10.2140/pjm.1994.163.201 · Zbl 0814.46053 [5] DOI: 10.1006/jabr.1999.7984 · Zbl 0949.16037 [6] DOI: 10.1016/j.jalgebra.2003.08.006 · Zbl 1042.16022 [7] DOI: 10.2140/pjm.1996.172.331 · Zbl 0852.46054 [8] David M.-C., J. Operat. Theor. 54 pp 27– [9] DOI: 10.1007/BF02352496 · Zbl 0760.57002 [10] DOI: 10.1007/978-3-662-02813-1 [11] DOI: 10.1007/BF02108816 · Zbl 0876.46042 [12] DOI: 10.1007/978-1-4613-9641-3 [13] Hivert F., Sém. Lothar. Combin. 51 pp 1– [14] DOI: 10.1006/jfan.1996.0129 · Zbl 0923.46062 [15] Izumi M., Mem. Amer. Math. Soc. 158 pp 198– [16] DOI: 10.1007/BF01389127 · Zbl 0508.46040 [17] DOI: 10.1017/CBO9780511566219 [18] Kac G. I., Trudy Moskov. Mat. Obšč. 15 pp 224– [19] DOI: 10.1007/978-1-4612-0783-2 [20] DOI: 10.1090/conm/267/04271 [21] DOI: 10.1080/00927879808826132 · Zbl 0912.16018 [22] D. Nikshych and L. Vainerman, Hopf Algebras and Quantum Groups, Lecture Notes in Pure and Appl. Math 209 (Dekker, New York, 2000) pp. 189–220. · Zbl 1032.46537 [23] DOI: 10.1006/jfan.1999.3522 · Zbl 1010.46063 [24] DOI: 10.1006/jfan.2000.3650 · Zbl 0995.46041 [25] D. Nikshych and L. Vainerman, New Directions in Hopf Algebras, Mathematical Sciences Research Institute Publications 43 (Cambridge University Press, Cambridge, 2002) pp. 211–262. [26] Pimsner M., Ann. Sci. École Norm. Sup. (4) 19 pp 57– [27] Popa S., C. R. Acad. Sci. Paris Sér. I Math. 311 pp 95– [28] DOI: 10.1007/978-3-322-96649-0 [29] DOI: 10.1007/s002200050284 · Zbl 0946.46058 [30] Watatani Y., Mem. Amer. Math. Soc. 83 pp vi+117– [31] DOI: 10.1006/jfan.1996.0110 · Zbl 0899.46050 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.