Résume | The study of finite-dimensional Nichols algebras of diagonal type is a fundamental step towards the classification of finite-dimensional pointed Hopf algebras. But these algebras are interesting by themselves. This family of Nichols algebrasinclude the positive parts of the quantized enveloping superalgebras of finite-dimensional simple Lie superalgebras over a field of characteristic 0.
The classification of Nichols algebras of diagonal type is possible thanks to the introduction of a powerful tool, the Weyl groupoid. This groupoid relates the different Nichols algebras with the same Drinfeld double.
Contragredient Lie superalgebras have an analogous phenomenon; there exist different matrices (and parities of the generators) such that the attached Lie superalgebras are isomorphic.
In this talk we recall the definition of these three objects: Nichols algebras, Weyl groupoid and contragredient Lie superalgebras (over a field of arbitrary characteristic). We will explain a close relation between some exceptional examples in positive characteristic, i.e. with no analogues in characteristic zero, and Nichols algebras of unidentified type, that is, neither coming from quantized enveloping superalgebras over a field of characteristic zero nor of standard type. |