Root Systems In Finite Symplectic Vector Spaces

Abstract

We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽₂. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group.

Key Words:

• Nichols algebra
• Root system
• Symplectic vector space

2010 Mathematics Subject Classification:

• 17B22

ACKNOWLEDGMENTS

We thank Sergei Chmutov for helpful comments.

Notes

We thank the referee for pointing out this significantly easier approach.

Communicated by M. Cohen