**Existence of a [root system](http://en.wikipedia.org/wiki/Root_system)** has been established for [Nichols algebras](http://en.wikipedia.org/wiki/Nichols_algebra) $B(V)$ of a [Yetter-Drinfel'd-module](http://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category) $V$ (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](http://arxiv.org/abs/math/0005101). In 2008 [Schneider and Heckenberger](http://arxiv.org/abs/0807.0691) 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?**


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



Dimension formula for Cartan-type abelian.group Nichols algebra?
Yes, careful application of the theory of rootsystems verifies indeed these formulae.....farily streight-forward!


