TY - JOUR A1 - Helbig, Michael T1 - Ideals for character Hopf algebras JF - Journal of Pure and Applied Algebra N2 - We give a description in terms of generators and relations of character Hopf algebras. These Hopf algebras play a central role in the classification of pointed Hopf algebras and can be seen as generalized quantum groups. Furthermore, we get as a consequence an analog presentation of Nichols algebras of diagonal type. Y1 - 2025 U6 - https://doi.org/10.1016/j.jpaa.2013.08.003 VL - 218 IS - 4 SP - 634 EP - 641 PB - Elsevier ER - TY - JOUR A1 - Helbig, Michael T1 - On the Lifting of Nichols Algebras JF - Communications in Algebra N2 - Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. Based on recent work of Heckenberger about Nichols algebras of diagonal type we compute explicitly the liftings of all Nichols algebras with generalized Cartan matrix of type A 2, some Nichols algebras with generalized Cartan matrix of type B 2, and some Nichols algebras of two Weyl equivalence classes of non-standard type giving new classes of finite-dimensional pointed Hopf algebras. Y1 - 2012 U6 - https://doi.org/10.1080/00927872.2011.568029 VL - 40 IS - 9 SP - 3317 EP - 3351 PB - Informa UK Limited ER - TY - GEN A1 - Helbig, Michael A1 - Seliger, Norbert T1 - Combinatorics of Inductance States in Reconfigurable Air Coils T2 - Proceedings 2nd International Conference on Recent Advances in Mathematics and Informatics (ICRAMI'25) N2 - We propose a group theory based mathematical model for unambiguous inductance states of a reconfigurable air coil inductor utilizing elastic snapping of wire segments. Vertical (axial) and horizontal winding symmetries under group actions reveal distinguishable tuning regions and individual variation states therein. The model is applied to coils of prismatic (cylinder like) and truncated pyramidal (conical like) shape with edges forming a regular polygon as basis, and verified by numerical magnet field simulation. KW - Inductance KW - Magnetic devices KW - Matrices KW - Group theory KW - Combinatorial mathematics Y1 - 2025 SN - 978-9909-00-770-9 ER - TY - THES A1 - Helbig, Michael T1 - Lifting of Nichols Algebras N2 - Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classi cation program of nite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider [6] is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. The classi cation was carried out in this way in [11] when the group of group-like elements is abelian and the prime divisors of the order of the group are > 7. In this case the appearing Nichols algebras are of Cartan type. Based on recent work of Heckenberger about diagonal Nichols algebras [29, 28, 27] we compute explicitly the liftings of some Nichols algebras not treated in [11]; namely we lift - all Nichols algebras with Cartan matrix of type A2 (Theorem 6.3.3), - some Nichols algebras with Cartan matrix of type B2 (Theorem 6.4.3), and - some Nichols algebras of two Weyl equivalence classes of non-standard type (Theorem 6.5.3), giving new classes of nite-dimensional pointed Hopf algebras. Crucial is the knowledge of a good presentation of the Nichols algebra and its liftings: We want to have an explicit description in terms of generators and (non-redundant) relations, and a basis; this requires new ideas and methods that generalize those in [11]. In this spirit, we describe Hopf algebras generated by skew-primitive elements and an abelian group with action given via characters (including Nichols algebras and their liftings) in Theorem 5.4.1. The relations form a Grobner basis and are given by a combinatorial property involving the theory of Lyndon words. Furthermore, in Theorem 7.3.1 we give a necessary and su cient criterion to check whether a given set of iterated q-commutators establishes a restricted PBW basis for a given realization of the relations. Also with the help of this criterion we determine the redundant relations in the examined Nichols algebras and their liftings. KW - Quantum Algebra KW - Hopf algebra KW - Nichols algebra Y1 - 2025 U6 - https://doi.org/10.5282/edoc.10378 ER - TY - BOOK A1 - Helbig, Michael T1 - Lifting of Nichols Algebras - A Contribution to the Classification of Pointed Hopf Algebras N2 - Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. The classification was carried out in this way in several special cases when the appearing Nichols algebras are of Cartan type. We compute explicitly the liftings of some Nichols algebras not of Cartan type or of Cartan type without restrictions on the group order. These build up new classes of finite-dimensional pointed Hopf algebras. Furthermore, we give a necessary and sufficient PBW basis criterion for a class of pointed Hopf algebras and present them in terms of generators and relations. These Hopf algebras can be seen as generalized Quantum groups. KW - Quantum Algebra KW - Hopf algebra KW - Nichols algebra Y1 - 2025 SN - 978-3-8381-1084-4 PB - Südwestdeutscher Verlag für Hochschulschriften CY - Saarbrücken ER - TY - INPR A1 - Helbig, Michael T1 - A PBW basis criterion for pointed Hopf algebras N2 - We give a necessary and sufficient PBW basis criterion for Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. We apply the criterion to classical examples and liftings of Nichols algebras which were determined in arxiv.org:1003.5882. KW - Quantum Algebra KW - Hopf algebra KW - Nichols algebra Y1 - 2025 U6 - https://doi.org/10.48550/arxiv.1011.0140 SP - 1 EP - 33 ER - TY - INPR A1 - Helbig, Michael T1 - On the presentation of pointed Hopf algebras N2 - We give a presentation in terms of generators and relations of Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. As a consequence we get an analog presentation of Nichols algebras of diagonal type. KW - Quantum Algebra KW - Hopf algebra KW - Nichols algebra Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.1003.5879 SP - 1 EP - 13 ER -