# exponential / logarithmic for unipotent algebraic groups

Let $$k$$ to be a field (of possibly positive characteristic), let $$U_n$$ denotes the space of all $$n times n$$ unipotent upper triangular matrices on $$k$$and let $$G$$ to be an algebraic subgroup of $$U_n$$ (from where a unipotent algebraic group itself). Then each $$X in text {Lie} (G)$$ (thought of as a member of $$text {Lie} (U_n)$$, that is to say, a strictly superior triangle $$n times n$$ matrix) is nilpotent, so it makes sense to define

$$text {exp} (X) = 1 + X + X ^ 2/2! + points + X ^ {n-1} / (n-1)!$$

(This definition makes sense even in characteristic cases $$p> 0$$ as long as $$p geq n$$, that is to say that $$p$$ never divides $$1!, 2!, Points, (n-1)!$$). We can also define, for $$g in G$$,
since $$g-1$$ is nilpotent,

$$log (g) = (g-1) – (g-1) ^ 2/2 + (g-1) ^ 3/3 – dots pm (g-1) ^ {n-1} / ( n-1)$$

obviously $$text {exp}$$ and $$log$$ define maps from $$text {Lie} (U_n)$$ at $$U_n$$ and back to $$text {Lie} (U_n)$$and are bijective, being inverses of each other.

My question: Yes $$g in G$$, is $$log (g) in text {Lie} (G)$$? Or, equally, for $$X in text {Lie} (G)$$, is $$text {exp} (X) in G$$?

I think there should be some obvious proof of that, but I do not see it. Yes $$G$$ were a Lie group, the Lie algebra of $$G$$ would often just be defined to
To be everything $$X$$ such as $$e ^ {tX} in G$$ for everyone $$t in mathbb {R}$$and so for the Lie groups $$text {exp}$$ cards of $$text {Lie} (G)$$ at $$G$$ simply by definition. In the context of the algebraic group, this definition no longer makes sense in general, and even when it does, it is not used in the literature (as far as I've seen),
so I tried to use each of the following equivalent definitions of $$text {Lie} (G)$$, without success:

• $$text {Lie} (G) = text {Dist} _1 ^ + (G)$$ (distributions of order no greater than $$1$$ without constant term)

• $$text {Lie} (G) =$$ the subspace of $$text {Lie} (U_n) = text {Dist} _1 ^ + (U_n)$$ who kills $$I = ( text {setting G polynomials})$$

• $$text {Lie} (G) = {M in text {Lie} (U_n): 1 + tau M in G (k[tau]) }$$ or $$tau ^ 2 = 0$$

• $$text {Lie} (G) = {M in text {Lie} (U_n): 1 + tau M text {satisfies the definition polynomials of} G }$$, where again $$tau ^ 2 = 0$$

• $$text {Lie} (G) =$$ left invariant derivations on Hopf algebra $$G$$

This is certainly credible on his face; we have that $$text {Lie} (G) stackrel { text {exp}} { longrightarrow} U_n {stack} { log} { longrightarrow} text {Lie} (G)$$
composes with the identity, the same for $$G stackrel { log} {longrightarrow} text {Lie} (U_n) stackrel { text {exp}} { longrightarrow}$$but I do not see why in the
waiting for $$log (G) subset text {Lie} (G)$$ where $$text {exp} ( text {Lie} (G)) subset G$$.

If that makes a difference, I'm only interested in the case where the polynomials of definition of $$G$$ have an integer (maybe mod $$p$$) coefficients.