**
A. Davydov, R. Street**

##
Roots of Unity as a Lie Algebra

**abstract:**

This note gives a categorical development arising from a theorem of A. A.
Klyachko relating the Lie operad to roots of unity. We examine the "substitude"
structure on the groupoid C whose homsets are the cyclic groups. The roots of
unity representations of the cyclic groups form a Lie algebra for a certain
oplax monoidal structure on the category of linear representations of C.