Many possibilities for predual to d1 d0 - d0 d1

Stream: maths

[Dean Young]

An affine algebraic group over a field k with structure ring A results in a lie algebra most naturally inherited from Derₖ(A,A) as those derivations which are left invariant under the group action.

A co-Lie-algebra could be defined as a k-linear map Ω¹(G) → Ω¹(G) ⊗ Ω¹(G) such that dualization by A results in the lie-bracket operation d1 d0 - d0 d1 on Derₖ(A,A).

If Ω¹(G) is projective then this is possible.

My question is to come up with an example of a field k and an algebraic group over with structure ring A with two co-Lie-algebra structures on Ω¹(G) resulting in the same Lie-algebra structure inherited as the G-invariant elements of the Lie-algebra on Derₖ(A,A) and to prove that they are not equal.

This follows my self-correction after learning about how trace and the characteristic polynomial is related to projective property on the Kähler differentials, which as mentioned can ensure the existence of a co-Lie-algebra of an algebraic group.

[Notification Bot]

This topic was moved here from #mathlib4 > Many possibilities for predual to d1 d0 - d0 d1 by Kevin Buzzard.