# zbMATH — the first resource for mathematics

Simple modules of the quantum double of the Nichols algebra of unidentified diagonal type $$\mathfrak{ufo}(7)$$. (English) Zbl 1440.16037
Summary: The finite-dimensional simple modules over the Drinfeld double of the bosonization of the Nichols algebra $$\mathfrak{ufo}(7)$$ are classified.

##### MSC:
 16T20 Ring-theoretic aspects of quantum groups 17B37 Quantum groups (quantized enveloping algebras) and related deformations
Full Text:
##### References:
 [1] Andersen, H. H.; Jantzen, J. C.; Soergel, W., Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p. Astérisque 220:321 (Paris: Soc. Math. France), (1994) · Zbl 0802.17009 [2] Andruskiewitsch, N.; Cardona, A.; Morales, P.; Ocampo, H.; Paycha, S.; Reyes, A., An introduction to nichols algebras, Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Mathematical Physics Studies. Springer (to appear.) [3] Andruskiewitsch, N.; Angiono, I. [4] Andruskiewitsch, N.; Angiono, I.; Rossi Bertone, F., The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type. Math. Res. Lett. arXiv:1501.04518 (in press) · Zbl 1406.16035 [5] Andruskiewitsch, N.; Angiono, I.; Yamane, H., On pointed Hopf superalgebras, Contemp. Math., 544, 123-140, (2011) · Zbl 1245.16022 [6] Andruskiewitsch, N.; Radford, D.; Schneider, H.-J., Complete reducibility theorems for modules over pointed Hopf algebras, J. Algebra, 324, 2932-2970, (2010) · Zbl 1223.16010 [7] Andruskiewitsch, N.; Schneider, H.-J., Finite quantum groups and Cartan matrices, Adv. Math., 154, 1-45, (2000) · Zbl 1007.16027 [8] Angiono, I., On nichols algebras with standard braiding, Algebra Number Theory, 3, 35-106, (2009) · Zbl 1183.17005 [9] Angiono, I., Nichols algebras of unidentified diagonal type, Comm. Algebra, 41, 4667-4693, (2013) · Zbl 1297.16028 [10] Angiono, I., On nichols algebras of diagonal type, J. Reine Angew. Math., 683, 189-251, (2013) · Zbl 1331.16023 [11] Angiono, I., A presentation by generators and relations of nichols algebras of diagonal type and convex orders on root systems, J. Eur. Math. Soc., 17, 2643-2671, (2015) · Zbl 1343.16022 [12] Angiono, I., Distinguished pre-nichols algebras, Transf. Groups, 21, 1-33, (2016) · Zbl 1355.16028 [13] Heckenberger, I., Classification of arithmetic root systems, Adv. Math., 220, 59-124, (2009) · Zbl 1176.17011 [14] Heckenberger, I., Lusztig isomorphism for drinfel’d doubles of bosonizations of nichols algebras of diagonal type, J. Algebra, 323, 2130-2182, (2010) · Zbl 1238.17010 [15] Heckenberger, I.; Yamane, H., Drinfel’d doubles and Shapovalov determinants, Rev. UMA, 51, 107-146, (2010) · Zbl 1239.17010 [16] Kauffman, L.; Radford, D., A necessary and suﬃcient condition for a finite dimensional drinfel’d double to be a ribbon Hopf algebra, J. Algebra, 159, 98-114, (1993) · Zbl 0802.16035 [17] Lusztig, G., Introduction to quantum groups. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser/Springer, New York, xiv+346 pp. ISBN: 978-0-8176-4716-2, (2010) [18] Pogorelsky, B.; Vay, C., Verma and simple modules for quantum groups at non-abelian groups, Adv. Math., 301, 423-457, (2016) · Zbl 1393.17027 [19] Radford, D.; Schneider, H.-J., On the simple representations of generalized quantum groups and quantum doubles, J. Algebra, 319, 3689-3731, (2008) · Zbl 1236.17026
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.