On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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The purpose of this paper is to investigate finite-dimensional superbialgebras and Hopf superalgebras. We study connected superbialgebras and provide a classification of non-trivial superbialgebras and Hopf superalgebras in dimension $n$ with $n\leq 4$.
Keywords: superalgebra; superbialgebra; Hopf superalgebra; classification.
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     author = {Said Aissaoui and Abdenacer Makhlouf},
     title = {On {Classification} of {Finite-Dimensional} {Superbialgebras} and {Hopf} {Superalgebras}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a0/}
}
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Said Aissaoui; Abdenacer Makhlouf. On Classification of Finite-Dimensional Superbialgebras and Hopf Superalgebras. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a0/

[1] Abe E., Hopf algebras, Cambridge Tracts in Mathematics, 74, Cambridge University Press, Cambridge, 1980 | MR | Zbl

[2] Andruskiewitsch N., “About finite dimensional {H}opf algebras”, Quantum Symmetries in Theoretical Physics and Mathematics ({B}ariloche, 2000), Contemp. Math., 294, Amer. Math. Soc., Providence, RI, 2002, 1–57 | DOI | MR | Zbl

[3] Andruskiewitsch N., Angiono I., Yamane H., “On pointed {H}opf superalgebras”, New developments in {L}ie theory and its applications, Contemp. Math., 544, Amer. Math. Soc., Providence, RI, 2011, 123–140, arXiv: 1009.5148 | DOI | MR | Zbl

[4] Andruskiewitsch N., Etingof P., Gelaki S., “Triangular {H}opf algebras with the {C}hevalley property”, Michigan Math. J., 49 (2001), 277–298, arXiv: math.QA/0008232 | DOI | MR | Zbl

[5] Andruskiewitsch N., Schneider H. J., “On the classification of finite-dimensional pointed {H}opf algebras”, Ann. of Math., 171 (2010), 375–417, arXiv: math.QA/0502157 | DOI | MR | Zbl

[6] Armour A., The algebraic and geometric classification of four dimensional super-algebras, Master Thesis, Victoria University of Wellington, 2006

[7] Armour A., Chen H. X., Zhang Y., “Classification of 4-dimensional graded algebras”, Comm. Algebra, 37 (2009), 3697–3728 | DOI | MR | Zbl

[8] Beattie M., D{ă}sc{ă}lescu S., “Hopf algebras of dimension 14”, J. London Math. Soc., 69 (2004), 65–78, arXiv: math.QA/0205243 | DOI | MR | Zbl

[9] Beattie M., D{ă}sc{ă}lescu S., Grünenfelder L., “On the number of types of finite-dimensional {H}opf algebras”, Invent. Math., 136 (1999), 1–7 | DOI | MR | Zbl

[10] Beattie M., García G. A., “Classifying {H}opf algebras of a given dimension”, Hopf Algebras and Tensor Categories, Contemp. Math., 585, Amer. Math. Soc., Providence, RI, 2013, 125–152, arXiv: 1206.6529 | DOI | MR

[11] Cheng Y. L., Ng S. H., “On {H}opf algebras of dimension {$4p$}”, J. Algebra, 328 (2011), 399–419 | DOI | MR | Zbl

[12] Connes A., Kreimer D., “Hopf algebras, renormalization and noncommutative geometry”, Comm. Math. Phys., 199 (1998), 203–242, arXiv: hep-th/9808042 | DOI | MR | Zbl

[13] Deguchi T., Fujii A., Ito K., “Quantum superalgebra {$U_q{\rm osp}(2,2)$}”, Phys. Lett. B, 238 (1990), 242–246 | DOI | MR

[14] Dekkar K., Makhlouf A., “Bialgebra structures of 2-associative algebras”, Arab. J. Sci. Eng. Sect. C, 33 (2008), 137–151 | MR | Zbl

[15] Etingof P., Gelaki S., “The classification of finite-dimensional triangular {H}opf algebras over an algebraically closed field of characteristic 0”, Mosc. Math. J., 3 (2003), 37–43 | MR | Zbl

[16] Fukuda N., “Semisimple {H}opf algebras of dimension {$12$}”, Tsukuba J. Math., 21 (1997), 43–54 | MR | Zbl

[17] Gabriel P., “Finite representation type is open”, Proceedings of the {I}nternational {C}onference on {R}epresentations of {A}lgebras ({C}arleton {U}niv., {O}ttawa, {O}nt., 1974), Lecture Notes in Math., 488, Springer, Berlin, 1975, 132–155 | DOI | MR

[18] Gould M. D., Zhang R. B., Bracken A. J., “Quantum double construction for graded {H}opf algebras”, Bull. Austral. Math. Soc., 47 (1993), 353–375 | DOI | MR | Zbl

[19] Guichardet A., Groupes quantiques. Introduction au point de vue formel, Savoirs Actuels, InterEditions, Paris, 1995 | MR | Zbl

[20] Holtkamp R., “Comparison of {H}opf algebras on trees”, Arch. Math. (Basel), 80 (2003), 368–383 | DOI | MR | Zbl

[21] Kassel C., Quantum groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995 | DOI | MR | Zbl

[22] Kreimer D., “On the {H}opf algebra structure of perturbative quantum field theories”, Adv. Theor. Math. Phys., 2 (1998), 303–334, arXiv: q-alg/9707029 | MR | Zbl

[23] Kulish P. P., Reshetikhin N. Y., “Universal {$R$}-matrix of the quantum superalgebra {${\rm osp}(2\vert 1)$}”, Lett. Math. Phys., 18 (1989), 143–149 | DOI | MR | Zbl

[24] Majid S., “Cross products by braided groups and bosonization”, J. Algebra, 163 (1994), 165–190 | DOI | MR | Zbl

[25] Majid S., Foundations of quantum group theory, Cambridge, Cambridge University Press, 1995 | DOI | MR | Zbl

[26] Makhlouf A., “Degeneration, rigidity and irreducible components of {H}opf algebras”, Algebra Colloq., 12 (2005), 241–254, arXiv: math.RA/0211187 | DOI | MR | Zbl

[27] Makhlouf A., “Algèbre de Hopf et renormalisation en théorie quantique des champs”, Théorie Quantique des Champs: Méthodes et Applications, Travaux en Cours, eds. T. Boudjedaa, A. Makhlouf, Hermann, Paris, 2007, 191–242

[28] Masuoka A., “Semisimple {H}opf algebras of dimension 6, 8”, Israel J. Math., 92 (1995), 361–373 | DOI | MR | Zbl

[29] Milnor J. W., Moore J. C., “On the structure of {H}opf algebras”, Ann. of Math., 81 (1965), 211–264 | DOI | MR | Zbl

[30] Montgomery S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82, Conference Board of the Mathematical Sciences, Washington, DC, 1993 | MR | Zbl

[31] Montgomery S., “Classifying finite-dimensional semisimple {H}opf algebras”, Trends in the Representation Theory of Finite-Dimensional Algebras (Seattle, {WA}, 1997), Contemp. Math., 229, Amer. Math. Soc., Providence, RI, 1998, 265–279 | DOI | MR | Zbl

[32] Natale S., “Hopf algebras of dimension 12”, Algebr. Represent. Theory, 5 (2002), 445–455 | DOI | MR | Zbl

[33] Ng S. H., “Non-semisimple {H}opf algebras of dimension {$p^2$}”, J. Algebra, 255 (2002), 182–197, arXiv: math.QA/0110223 | DOI | MR | Zbl

[34] Scheunert M., Zhang R. B., “Integration on {L}ie supergroups: a {H}opf superalgebra approach”, J. Algebra, 292 (2005), 324–342, arXiv: math.RA/0012052 | DOI | MR | Zbl

[35] Shnider S., Sternberg S., Quantum groups. From coalgebras to Drinfel'd algebras. A guided tour, Graduate Texts in Mathematical Physics, 2, International Press, Cambridge, MA, 1993 | MR

[36] {Ş}tefan D., “The set of types of {$n$}-dimensional semisimple and cosemisimple {H}opf algebras is finite”, J. Algebra, 193 (1997), 571–580 | DOI | MR

[37] {Ş}tefan D., “Hopf algebras of low dimension”, J. Algebra, 211 (1999), 343–361 | DOI | MR

[38] Williams R. E., Finite dimensional {H}opf algebras, Ph.D. Thesis, The Florida State University, 1988

[39] Zhu Y., “Hopf algebras of prime dimension”, Int. Math. Res. Not., 1994 (1994), 53–59 | DOI | MR | Zbl