**Existence of a root system** has been established for Nichols algebras $B(V)$ of a Yetter-Drinfel'd-module $V$ (resp. braided vectorspaces $V$) over abelian groups (resp. with diagonal braiding $x_i\otimes x_j\mapsto q_{ij}x_j\otimes x_i$) already in 2000 by Kharchenko. In 2008 Schneider and Heckenberger established root systems over nonabelian groups, under the condition that the root system / Nichols algebra be finite (-dimensional)!

**Having a root system means** among others that one has a set of irreducible sub-Yetter-modules $X_1\ldots X_n\subset B(V)$, with **multiplying as below a bijection** (no algebra morphism!):

$$B(V)\cong\bigotimes_{i=1}^nB(X_i)$$

**Dynkin-diagrams** can be drawn with Cartan Matrix $$q_{ij}q_{ji}=q_{ii}^{-A_{ij}}$$ but there are sporadic cases not corresponding to finite semisimple Lie algebras (triangle, exotic edge...) for low prime exponents. In these cases the classification of pointed Hopf algebras (Schneider /
Andruskiewitsch) fails; the Weyl group is replaced by a -groupoid between different Dynkin diagrams.

**But in Cartan-type** it's the same as in the semisimple case - a root system $\Phi$!

**Is is generally true that the "roots"**in both formulas coicide? $$|\{X_1,X_2,\ldots X_n\}|=n=|\Phi^+|\qquad or=\ldots?$$**So can I calculate the dimension of $B(X)$ like that?**

$$dim(B(X))=\prod_{\alpha\in\Phi^+}dim(X_i)$$

...clearly $B(X_i)\cong k[x]/(x^{ord(q)})$ with dimension $ord(q_{ii})$) or infinite for $q=1$ (the *bosonic* $k[x]$).

**So I just multiply the orders of respective the $q_{ii}$ for all roots?**

$$dim(B(X))=\prod_{\alpha\in\Phi^+}ord(q_{\alpha\alpha})$$

**So especially for $G=\mathbb{Z}_2^k$ the dimension is the following?**

$$dim(B(X))=2^{|\Phi^+|}$$