TITLE: Is there a definition of analogue Weyl group for Lie super algebra?
QUESTION [12 upvotes]: I heard from some people working in Lie super algebra that there was no proper definition for Weyl group of Lie super algebra. I do not know Lie super algebra at all. But When I searched on Google, I found that it seems there still exists some definitions of Weyl group. 
I wonder whether there is a well-accepted definition for it. 
The reason I want to ask this question is that I need Weyl group for Lie super algebra to play with the geometry related to super Lie algebra. 
Another question is that I heard from some experts in Lie super algebra that there was no well-accepted super geometry related to Lie super algebra. 
However, It seems that one of the students of Manin, who is Dimitry Leites ever developed supergeometry.

REPLY [2 votes]: As it is clear from the other answers there are several viewpoints on Weyl group in super case.
Let me mention papers by Sergeev and Veselov, who also stands on the point of the Weyl groupoid in the super case. 
As far as I understand such viewpoint agrees with their works on 
Calogero-Moser integrable system.
https://arxiv.org/abs/1504.08310
"Orbits and invariants of super Weyl groupoid"
https://arxiv.org/abs/0704.2250
"Grothendieck rings of basic classical Lie superalgebras"
Abstract:
"The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalised root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called Weyl groupoids."

PS
By the way, GL(n,F_q) for q=1 "is" Weyl group (see e.g. field with one element). I wonder is there some kind of super analog of that heuristics ?
Is there some kind of "bijection" between "irreps" and "conjugacy classes"
for Weyl grouppoid (see  MO270916) ?