Roots of unity as a Lie algebra
Ross Street – 7 November 2001
Joint with Alexei Davydov and Brian Day
Write V for the category of complex vector spaces, write S for the symmetric groupoid (a skeleton of the category of finite sets and permutations), and write C for the cyclic groupoid (whose objects are natural numbers and arrows n --> n are elements of the cyclic group of order n. There is a faithful functor J : C --> S which is the identity on objects.
We use the term "substitude" for lax procomonoidal V-category. It is possible to define Lie algebra objects in symmetric (or even cyclic) substitudes. Representable substitudes are oplax monoidal V-categories.
Addition of natural numbers gives a monoidal structure on S which "restricts" (see my last talk) to a substitude structure on C. Since S and C are self dual, we also can regard these structures on Sop and Cop.
The main result is that the convolution oplax monoidal structure on the category [C,V] of linear representations of C is identical to the "restriction" of the convolution structure on the category [S,V] of linear representations of S (this is the category of "tensorial species" in Joyal's terminology of SLNM 1234) along the functor LanJ : [C,V] --> [S,V].
Consider the free Lie algebra functor L : V --> V; it is a Lie algebra in [V,V] for the pointwise tensor product. "Taylor series" T : [S,V] --> [V,V] takes the convolution structure to pointwise tensor product. Joyal (loc. cit.) shows that there is a Lie algebra lie in [S,V] such that T(lie) = L.
A classical theorem of A.A. Klyachko ["Lie elements in a tensor algebra" Sibirsk. Mat. Z. 15 (1974) 1296-1304, 1430; MR# 51 8178] provides an object l of [C,V] taken to lie by Lan_J. Indeed, the linear representation ln of the cyclic group Cn is the one-dimensional vector space acted on by all the n-th roots of unity. See also H. Barcelo and S. Sundaram ["On some submodules of the action of the symmetric group on the free Lie algebra" J. Algebra 154 (1993) 12-26].
Our main result, together with the fact that LanJ is faithful (since the inclusion H --> G of any subgroup is a split monic at the H-module level), implies that we obtain a Lie algebra object structure on l in the oplax monoidal category [C,V]. We have (LanJ)l = lie as Lie algebras.