nLab Nichols algebra

Idea

Nichols algebra is a particular graded braided Hopf algebra attached to a Yetter–Drinfeld module over a Hopf algebra or, more generally, a braided vector space.

They play a major role in a program of classification of pointed Hopf algebras. The quantum analogue of the unipotent part of the Borel subalgebra of the universal enveloping algebra as defined by Lusztig and Rosso can be constructed as an example.

References

It is introduced in

• W. D. Nichols, Bialgebras of type one, Comm. Algebra 6 (1978) 1521–1551

Braided symmetrizers for a given braiding used in the construction are due

• S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122:1 (1989) 125–170

Surveys

• Mitsuhiro Takeuchi, A survey on Nichols algebras, in: Contemporary Math. 376 (2005)
• Nicolás Andruskiewitsch, An introduction to Nichols algebras, pdf In: Cardona, A., Morales, P., Ocampo, H., Paycha, S., Reyes Lega, A. (eds) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Mathematical Physics Studies. Springer, Cham. doi

Role in the lifting program in classification

• N. Andruskiewitsch, H.-J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order $p^3$, doi

• N. Andruskiewitsch, H.-J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), 1–45

• N. Andruskiewitsch, H.-J. Schneider, Pointed Hopf algebras, MSRI Publ. 43 (2002), 1–68.

• Nicolás Andruskiewitsch, István Heckenberger, Hans-Jürgen Schneider, The Nichols algebra of a semisimple Yetter–Drinfeld module, American J. of Math. 132:6, (2010) 1493–1547 doi

• I. Heckenberger, H.-J. Schneider, Root systems and Weyl groupoids for Nichols algebras, Proc. Lond. Math. Soc., 101 (2010) 623–654

• I. Heckenberger, L. Vendramin, A classification of Nichols algebras of semi-simple Yetter-Drinfeld modules over non-abelian groups, J. Eur. Math. Soc. 19 (2) (2014) doi

Appearance in conformal field theory,

• A. M. Semikhatov, I. Y. Tipunin, The Nichols algebra of screenings, Commun. Contemp. Math. 14, 1250029, 66 (2012) arXiv:1101.5810
• Simon D. Lentner, Quantum groups and Nichols algebras acting on conformal field theories, Adv. Math. 378 (2021) 107517 doi arXiv:1702.06431

Other articles

• D. Bagio, G. A. Garcia, J. M. J. Giraldi, P. Marquez, Finite-dimensional Nichols algebras over dual Radford algebras, J. Alg. Appl. 2020 doi

category: algebra