Iyer, Uma N.; Smith, Jonathan D. H.; Taft, Earl J. One-sided Hopf algebras and quantum quasigroups. (English) Zbl 1398.16030 Commun. Algebra 46, No. 11, 4590-4608 (2018). Summary: We study the connections between one-sided Hopf algebras and one-sided quantum quasigroups, tracking the four possible invertibility conditions for the left and right composite morphisms that combine comultiplications and multiplications in these structures. The genuinely one-sided structures exhibit precisely two of the invertibilities, while it emerges that imposing one more condition often entails the validity of all four. A main result shows that under appropriate conditions, just one of the invertibility conditions is sufficient for the existence of a one-sided antipode. In the left Hopf algebra which is a variant of the quantum special linear group of two-dimensional matrices, it is shown explicitly that the right composite is not injective, and the left composite is not surjective. Cited in 3 Documents MSC: 16T05 Hopf algebras and their applications 20N05 Loops, quasigroups Keywords:Hopf algebra; left Hopf algebra; left quasigroup; quantum group; quantum quasigroup Software:QuantumMACMAHON PDFBibTeX XMLCite \textit{U. N. Iyer} et al., Commun. Algebra 46, No. 11, 4590--4608 (2018; Zbl 1398.16030) Full Text: DOI References: [1] Barr, M., *-Autonomous Categories, (1979), Springer, Berlin [2] Benkart, G.; Madaraga, S.; Pérez-Izquierdo, J. M., Hopf algebras with triality, Trans. Am. Math. Soc., 365, 1001-1023, (2012) · Zbl 1278.16032 [3] Dǎscǎlescu, S.; Nǎstǎsescu, C.; Raianu, Ş., Hopf Algebras: An Introduction, (2001), Dekker, New York [4] Davey, B. A.; Davis, G., Tensor products and entropic varieties, Algebra Univer., 21, 68-88, (1985) · Zbl 0604.08004 [5] Etingof, P.; Gelaki, S.; Nikshych, D.; Ostrik, V., Tensor Categories, (2015), American Mathematical Society, Providence, RI · Zbl 1365.18001 [6] Foata, D.; Han, G.-N., A new proof of the garoufalidis–Lê–Zeilberger quantum macmahon master theorem, J. Algebra, 307, 424-431, (2007) · Zbl 1108.05014 [7] Foata, D.; Han, G.-N., A basis for the right quantum algebra and the “1 = q” principle,, J. Algebr. Comb., 27, 163-172, (2008) · Zbl 1145.05004 [8] Garofalidis, S.; Lê, T. T. Q.; Zeilberger, D., The quantum macmahon master theorem, Proc. Nat. Acad. Sci., 103, 13928-13931, (2006) · Zbl 1170.05012 [9] Green, J. A.; Nichols, W. D.; Taft, E. J., Left Hopf algebras, J. Algebra, 65, 399-411, (1980) · Zbl 0439.16008 [10] Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, 23, 37-65, (1982) · Zbl 0474.57003 [11] Kassel, C., Quantum Groups, (1995), Springer, New York · Zbl 0808.17003 [12] Klim, J.; Majid, S., Hopf quasigroups and the algebraic 7-sphere, J. Algebra, 323, 3067-3110, (2010) · Zbl 1200.81087 [13] Klim, J.; Majid, S., Bicrossproduct Hopf quasigroups, Comment. Math. Univ. Carolin., 51, 287-304, (2010) · Zbl 1224.81014 [14] Lauve, A.; Taft, E. J., A class of left quantum groups modeled after SL_{q}(n), J. Pure Appl. Algebra, 208, 797-803, (2007) · Zbl 1125.16028 [15] Nichols, W. D.; Taft, E. J., The left antipodes of a left Hopf algebra, Algebraists’ Homage, (1982), American Mathematical Society, Providence, RI · Zbl 0501.16013 [16] Pérez-Izquierdo, J. M., Algebras, hyperalgebras, nonassociative bialgebras and loops, Adv. Math., 208, 834-876, (2007) · Zbl 1186.17003 [17] Radford, D. E., Hopf Algebras, (2012), World Scientific, Singapore [18] Rodríguez-Romo, S.; Taft, E. J., A left quantum group, J. Algebra, 286, 154-160, (2005) · Zbl 1073.16033 [19] Romanowska, A. B.; Smith, J. D. H., On Hopf algebras in entropic Jónsson-Tarski varieties, Bull. Korean Math. Soc., 52, 1587-1606, (2015) · Zbl 1328.08006 [20] Smith, J. D. H., An Introduction to Quasigroups and Their Representations, (2007), Chapman and Hall/CRC, Boca Raton, FL · Zbl 1122.20035 [21] Smith, J. D. H., One-sided quantum quasigroups and loops, Demonstr. Math., 48, 619-635, (2015) · Zbl 1352.20045 [22] Smith, J. D. H., Quantum quasigroups and loops, J. Algebra, 456, 46-75, (2016) · Zbl 1350.20051 [23] Smith, J. D. H.; Romanowska, A. B., Post-Modern Algebra, (1999), Wiley, New York · Zbl 0946.00001 [24] Van Daele, A., Multiplier Hopf algebras, Trans. Am. Math. Soc., 342, 917-932, (1994) · Zbl 0809.16047 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.